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Hindawi Publishing CorporationISRN Chemical EngineeringVolume 2013 Article ID 846826 8 pageshttpdxdoiorg1011552013846826
Research ArticleRadiation and Chemical Reaction Effects on UnsteadyDouble Diffusive Convective Flow past an Oscillating Surfacewith Constant Heat Flux
Arpita Jain
Department of Mathematics JECRC UDML College of Engineering Jaipur 302028 India
Correspondence should be addressed to Arpita Jain arpita 252rediffmailcom
Received 16 June 2013 Accepted 13 July 2013
Academic Editors J Ancheyta and M Kostoglou
Copyright copy 2013 Arpita Jain This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper presents an analysis of combinedheat andmass transfer flowpast an oscillating vertical plate under the action of radiationeffects and chemical reaction when heat is supplied to the plate at constant rate The governing equations are solved in closed formby Laplace-transform technique The results are obtained for temperature concentration velocity skin friction Nusselt numberand Sherwood number The effects of various parameters on flow variables are illustrated graphically and the physical aspects ofthe problem are discussed
1 Introduction
Free convection arises in the fluid when temperature changescause density variation leading to buoyancy forces acting onthe fluid elementsThemost common example of free convec-tion is the atmospheric flow which is driven by temperaturedifferences Natural convection has been analyzed extensivelybymany investigators Some of them are Revankar [1] Li et al[2]
Sometimes along with the free convection currentscaused by difference in temperature the flow is also affectedby the differences in concentration or material constitutionThere are many situations where convection heat transferphenomena are accompanied by mass transfer also Whenmass transfer takes place in a fluid at rest the mass istransferred purely by molecular diffusion resulting fromconcentration gradients For low concentration of the massin the fluid and low mass transfer rates the convectiveheat and mass transfer processes are similar in nature Anumber of investigations have already been carried out withcombined heat and mass transfer under the assumption ofdifferent physical situationsThe illustrative examples ofmasstransfer can be found in the book of Cussler [3] Combinedheat and mass transfer flow past a surface are analyzed byChaudhary and Arpita [4] Muthucumaraswamy et al [5 6]
and Rajput and Kumar [7] with different physical conditionsJuncu [8] pioneered unsteady heat andmass transfer flowpasta surface by numerical method
Combined heat and mass transfer problems with chemi-cal reaction are of importance in many processes and havetherefore received a considerable amount of attention inrecent years Chemical reaction can be codified as eitherhomogeneous or heterogeneous processes A homogeneousreaction is one that occurs uniformly through a givenphase In contrast a heterogeneous reaction takes place ina restricted region or within the boundary of a phase Areaction is said to be first order if the rate of reaction is directlyproportional to the concentration itself which has manyapplications in different chemical engineering processes andother industrial applications such as polymer productionmanufacturing of ceramics or glassware and food processingDas et al [9] considered the effects of first order chemicalreaction on the flow past an impulsively started infinitevertical plate with constant heat flux and mass transferMuthucumaraswamy and Meenakshisundaram [10] studiedchemical reaction effects on vertical oscillating plate withvariable temperature
In the above mentioned studies the effect of radia-tion on flow has not been considered Actually many pro-cesses in new engineering areas occur at high temperature
2 ISRN Chemical Engineering
and knowledge of radiation heat transfer becomes imperativefor the design of the pertinent equipment Nuclear powerplants gas turbines and the various propulsion devices foraircraft missiles satellites and space vehicles are examplesof such engineering areas Unsteady free convection flowpast a vertical plate with chemical reaction under differenttemperature condition on the plate is elucidated by Neog andDas Rudra [11] and Rajesh et al [12]Thermal radiation effecton flow past a vertical plate withmass transfer is examined byMuralidharan and Muthucumaraswamy [13] and Rajput andKumar [14] Natural convective flow past an oscillating platewith constantmass flux in the presence of radiation is studiedby Chaudhary and Jain [15]
The effects of radiation on free convection on the accel-erated flow of a viscous incompressible fluid past an infinitevertical plate havemany important technological applicationsin the astrophysical geophysical and engineering problem
However it seems less attention was paid on hydromag-netic free convection flows near a vertical plate subjectedto a constant heat flux boundary condition even thoughthis situation is involved in many engineering applicationsOgulu and Makinde [16] and Narahari and Debnath [17]examined the flow past a surface with constant heat flux Freeconvection effects on flow past an infinite vertical acceleratedwith constant heat flux are pioneered by Chaudhary andArpita [18]
Hence the objective of this paper is to study radiationand chemical reaction effects on unsteady double diffusiveconvective flow past an oscillating vertical plate when heat issupplied to it at constant rate
2 Mathematical Analysis
We consider a two-dimensional flow of an incompressibleand electrically conducting viscous fluid along an infinitevertical plate The 119909
1015840-axis is taken on the infinite plate andparallel to the free stream velocity and 119910
1015840-axis normal to itInitially the plate and the fluid are at same temperature 1198791015840
infin
with concentration level 1198621015840infin
at all points At time 1199051015840gt 0
the plate concentration is changed to 1198621015840
119908with heat supplied
at a Constant rate to the plate and it starts oscillating witha velocity 119880
119877cos12059610158401199051015840 in its own plane It is assumed that
there exists a homogeneous chemical reaction of first orderwith constant rate 119870
119897between the diffusing species and
the fluid Since the plate is infinite in extent therefore theflow variables are the functions of 1199101015840 and 119905
1015840 only The fluidis considered to be gray absorbing-emitting radiation butnon scattering medium The radiative heat flux in the 119909
1015840-direction is considered negligible in comparison with of 1199101015840-directionThen neglecting viscous dissipation heat producedby chemical reaction and assuming variation of density in thebody force term (Boussinesqrsquos approximation) the problemcan be governed by the following set of equations
1205971198791015840
1205971199051015840=
120581
120588119862119901
12059721198791015840
12059711991010158402
minus
1
120588119862119901
120597119902119903
1205971199101015840 (1)
1205971198621015840
1205971199051015840= 119863
12059721198621015840
12059711991010158402
minus 1198961198971198621015840 (2)
1205971199061015840
1205971199051015840= ]
12059721199061015840
12059711991010158402
+ 119892120573 (1198791015840minus 1198791015840
infin) + 119892120573
119888(1198621015840minus 1198621015840
infin
1015840
) (3)
with following initial and boundary conditions
1199061015840= 0 119879
1015840= 1198791015840
infin 1198621015840= 1198621015840
infinforall1199101015840 1199051015840le 0
1199061015840= 119880119877cos12059610158401199051015840 120597119879
1015840
1205971199101015840= minus
119902
120581
1198621015840= 1198621015840
119908at 1199101015840 = 0 119905
1015840gt 0
1199061015840997888rarr 0 119879
1015840997888rarr 119879
1015840
infin 1198621015840997888rarr 119862
1015840
infin
as 1199101015840 997888rarr infin 1199051015840gt 0
(4)
The radiative heat flux term by using Rosselandrsquos approx-imation is given by
119902119903= minus
41205901015840
3120581lowast
12059711987910158404
1205971199101015840 (5)
We assume that the temperature differences within theflow are such that11987910158404may be expressed as a linear function ofthe temperature1198791015840This is accomplished by expanding 11987910158404 ina Taylor series about 1198791015840
infinand neglecting higher-order terms
One has
11987910158404≃ 411987910158403
infin1198791015840minus 311987910158404
infin (6)
By using (5) and (6) (1) gives
120588119862119901
1205971198791015840
1205971199051015840= 120581
12059721198791015840
12059711991010158402
+
16120590101584011987910158403
infin
3120581lowast
12059721198791015840
12059711991010158402 (7)
Introducing the following dimensionless quantities
119905 =
1199051015840
119905119877
119910 =
1199101015840
119871119877
119906 =
1199061015840
119880119877
120596 = 1205961015840119905119877 119896 =
1198802
119877119896119897
]2
Pr =120583C119901
120581
Sc = ]
119863
120579 =
1198791015840minus 1198791015840
infin
119902]120581119880119877
119866 =
119892120573119902]2
1205811198804
119877
119862 =
1198621015840minus 1198621015840
infin
1198621015840
119908minus 1198621015840
infin
Gm =
]119892120573119862(1198621015840
119908minus 1198621015840
infin)
1198803
119877
119896 =
]119896119897
1198802
119877
119877 =
120581lowast120581
4120590101584011987910158403
infin
Δ119879 = 1198791015840
119908minus 1198791015840
infin 119880
119877= (]119892120573Δ119905)
13
119871119877= (
119892120573Δ119879
]2)
minus13
119905119877= (119892120573Δ119879)
minus23
]13
(8)
ISRN Chemical Engineering 3
the governing equations (1) to (3) reduce to the followingnondimensional form
Pr120597120579120597119905
= (1 +
4
3119877
)
1205972120579
1205971199102
120597119862
120597119905
=
1
Sc1205972119862
1205971199102minus 119896119862
120597119906
120597119905
=
1205972119906
1205971199102+ 119866120579 + Gm119862
(9)
with the following initial and boundary conditions
119906 = 0 120579 = 0 119862 = 0 forall119910 119905 le 0
119906 = cos120596t 120597120579
120597119910
= minus1 119862 = 1 at 119910 = 0 119905 gt 0
119906 997888rarr 0 120579 997888rarr 0 119862 997888rarr 0 as 119910 997888rarr infin 119905 gt 0
(10)
On taking Laplace-transformof (9) and boundary conditions(10) we get
(1 +
4
3119877
)
1198892120579
1198891199102minus 119901Pr 120579 = 0
1198892120601
1198891199102(119896 + 119901 Sc) = 0
1198892119906
1198891199102minus 119901119906 = minus119866120579 (119910 119901) minus Gm997888
119862
(11)
119906 =
119901
1199012+ 1205962
119889120579
119889119910
= minus
1
119901
119862 =
1
119901
at 119910 = 0 119905 gt 0
119906 997888rarr 0 120579 997888rarr 0 119862 997888rarr 0 as 119910 997888rarr infin 119905 gt 0
(12)
where 119901 is the Laplace-transform parameterSolving (11) with the help of boundary condition (12) we
get
120579 (119910 119901) = radic119867
Prexp (minus119910radic119901Pr119867)
11990132
119862 (119910 119901) =
exp(minus119910radic(119896 Sc + 119901 Sc))
119901
119906 (119910 119901) =
119901 exp (minus119910radic119901)
1199012+ 1205962
+ radic119867
Pr119866 exp (minus119910radic119901)
11990152
((Pr119867) minus 1)
+
Gm exp (minus119910radic119901)
119901 (Sc minus 1) (119901 + (119896 Sc (Sc minus 1)))
minus
Gm119901 (Sc minus 1) (119901 + (119896 Sc (Sc minus 1)))
times exp(minus119910radic119901 Sc + 119896 Sc)
minus radic119867
Pr119866 exp (minus119910radic119901Pr119867)
11990152
(
Pr119867
minus 1)
(13)
On taking inverse Laplace-transform of (13) we get
120579 = radic119905 119867
120587Prerfc(
minus1205782Pr119867
) minus 120578radic119905 erfc(120578radicPr119867
)
119862 =
1
2
exp (2120578radic119896 Sc 119905) erfc (120578radicSc + radic119896119905)
+ exp (minus2120578radic119896 Sc 119905) erfc (120578radicSc minus radic119896119905)
(14)
For Pr = Sc = 1 one has
119906 =
exp (119894120596119905)4
times exp (2120578radic119894120596119905) erfc (120578 + radic119894120596119905)
+ exp (minus2120578radic119894120596119905) erfc (120578 minus radic119894120596119905) +
exp (minus119894120596119905)4
times exp (2120578radicminus119894120596119905) erfc (120578 + radicminus119894120596119905)
+ exp (minus2120578radicminus119894120596119905) erfc (120578 minus radicminus119894120596119905)
+ radic119867
Pr11986611990532
3 ((Pr119867) minus 1)
4
radic120587
(1 + 1205782) exp (minus1205782)
minus 120578 (6 + 41205782) erfc (120578)
+ radic119867
Pr11986611990532
3 (1 minus (Pr119867))
times
4
radic120587
(1 + 1205782 Pr119867
) exp (minus1205782 Pr119867
)
minus120578radicPr119867
(6 + 41205782 Pr119867
) erfc(120578radicPr119867
)
4 ISRN Chemical Engineering
+
Gm119896 Sc
erfc (120578) minus Gm2119896 Sc
exp( 119896 Sc 119905
1 minus Sc)
times
(exp(2120578radic119896 Sc 119905
1 minus Sc) erfc(120578 + radic
119896 Sc 119905
1 minus Sc)
+ exp(minus2120578radic119896 Sc 119905
1 minus Sc) erfc(120578 minus radic
119896 Sc 119905
1 minus Sc)
minus
Gm2119896 Sc
exp( 119896 Sc 119905
1 minus Sc)
times
(exp(2120578radic119896 Sc 119905
1 minus Sc) erfc(120578radicSc + radic
119896119905
1 minus Sc)
+ exp(minus2120578radic119896 Sc 119905
1 minus Sc) erfc(120578radicSc minus radic
119896119905
1 minus Sc)
minus
Gm2119896 Sc
(exp (2120578radic119896 Sc 119905) erfc (120578radicSc + radic119896119905)
+ (exp (minus2120578radic119896 Sc 119905) erfc (120578radicSc minus radic119896119905)
(15)
where119867 = 1 + (43119877) and for Pr = Sc = 1 when 119877 rarr infin
119906 =
exp (119894120596119905)4
exp (2120578radic119894120596119905) erfc (120578 + radic119894120596119905)
+ exp (minus2120578radic119894120596119905) erfc (120578 minus radic119894120596119905)
+
exp (minus119894120596119905)4
times exp (2120578radicminus119894120596119905) erfc (120578 + radicminus119894120596119905)
+ exp (minus2120578radicminus119894120596119905) erfc (120578 minus radicminus119894120596119905)
(16)
In expressions erfc (1199091+ 1198941199101) is complementary error
function of complex argument which can be calculated interms of tabulated functions in Abramowitz and Stegun [19]The tables given in Abramowitz and Stegun [19] do not giveerfc (119909
1+ 1198941199101) directly but an auxiliary function119882
1(1199091+ 1198941199101)
which is defined as
erfc (1199091+ 1198941199101) = 119882
1(minus1199101+ 1198941199091) exp minus(119909
1+ 1198941199101)2
(17)
Some properties of1198821(1199091+ 1198941199101) are
1198821(minus1199091+ 1198941199101) = 119882
2(1199091+ 1198941199101)
1198821(1199091minus 1198941199101) = 2 exp minus(119909
1minus 1198941199101)2
minus1198822(1199091+ 1198941199101)
(18)
where1198822(1199091+ 1198941199101) is complex conjugate of119882
1(1199091+ 1198941199101)
3 Skin Friction
From velocity field skin friction at the plate in nondimen-sional form is expressed as
120591 = minus(
120597119906
120597119910
)
119910=0
120591 =
exp (119894120596119905)2radic119905
radic119894120596119905 erf (radic119894120596119905) +
exp (minus119894120596119905)2radic119905
times radicminus119894120596119905 erf (radicminus119894120596119905) +
119866119905119867
(Pr minus 119867)
radic119867
Pr(1 minus radic
Pr119867
)
+
1
radic120587119905
minus
Gm119896 Sc
exp( 119896 Sc1199051 minus Sc
) erf (radic119896 Sc1199051 minus Sc
)radic119896 Sc1 minus Sc
+
Gm119896 Sc
exp( 119896 Sc1199051 minus Sc
) erf (radic119896119905
1 minus Sc)
times radic119896 Sc1 minus Sc
minus
Gm119896 Sc
radic119896 Sc erf (radic119896119905)
(19)
4 Nusselt Number
From temperature field the rate of heat transfer in nondimen-sional form is expressed as
Nu = minus
1
120579 (0)
120597120579
120597119910
10038161003816100381610038161003816100381610038161003816119910=0
= +
1
120579 (0)
= radic120587Pr119905119867
(20)
5 Sherwood Number
From the concentration field the rate of concentrationtransfer expressed in nondimensional form is given by
Sh = minus
120597120593
120597119910
10038161003816100381610038161003816100381610038161003816119910=0
= radic119896 Sc2
(2 +
1
2
radicSc119905
exp (minus119896119905)) (21)
6 Discussion
In order to get physical insight into the problem the valuesof Schmidt number are chosen to represent the presence ofspecies by hydrogen (022) water vapour (060) ammonia(078) and Carbon dioxide (096) at 25∘C temperature and1 atmospheric pressure The values of Pr are chosen 0717 which represent air and water respectively at 20∘C tem-perature and 1 atmospheric pressure The values of radia-tion parameter and chemical reaction parameter are chosenarbitrarily Only real values of solutions are considered forplotting graphs
Figure 1 reveals temperature profiles against 120578 (distancefrom the plate) It is obvious from the figure that themagnitude of temperature is maximum at the plate andthen decays to zero asymptotically Further the magnitudeof the temperature for air is greater than that of water
ISRN Chemical Engineering 5
0
01
02
03
04
05
06
07
0 05 1 15 2 25
minus01
4 02 7
15 02 7
4 06 7
4 02 071
15 02 071
4 06 071
R t Pr R t Pr
120578
120579
Figure 1 Temperature profile
This is due to the fact that thermal conductivity of fluiddecreases with increasing Pr resulting in a decrease in ther-mal boundary layer thickness It is also seen that it decreasessteeply for Pr = 7 than that of Pr = 071 Furthermore itis noticed that an increase in radiation parameter from 4 to15 the temperature marginally decreases It is obvious sincethe radiation parameter defines the relative contribution ofconduction heat transfer to the thermal radiation transferFurther it increases with an increase in time
The species concentration profile against 120578 is plotted inFigure 2 It is observed that the concentration at all points inthe flow field decreases with 120578 and tends to zero as 120578 rarr infinFurthermore an increase in the value of Sc leads to a fall in theconcentration Physically it is true since increase of Sc meansdecrease of molecular diffusivity which results in decrease ofconcentration boundary layer Hence the concentration ofspecies is higher for small values of Sc It is also observed thatconcentration decreases with an increase in time
Figure 3 elucidates the effect of 120596119905 and Pr on the velocityIt is evident from Figure 3 that the velocity increases andattains itsmaximumvalue in the vicinity of the plate (120578 le 05)and then tends to zero asymptotically It is also noted thatboth the velocity and the penetration for Pr = 071 is higherthan those of Pr = 7 Physically it is possible because fluidswith high Prandtl number have high viscosity and hencemove slowly It is also seen that the velocity decreases withan increase in 120596119905 Figure 4 illustrates the influences of Sc119905 and Pr on the velocity It is noticed that the maximumvelocity attains near the plate then decreases and vanishesfar away from the plate When Pr = 7 and 071 it decreasescontinuously to asymptotic value Further they also increaseswith an increase in time at each point in the flow field forhydrogen gas Moreover the effect of time on the velocity is
0
02
04
06
08
1
12
0 1 2 3 4 5 6
Sc
Sc
minus02
022 02
060 02
022 10
t
120578
Figure 2 Concentration profile for 119896 = 02
more dominant than other parameters Finally It is observedthat the velocity for hydrogen (Sc = 022) is higher thanthat of water vapour (Sc = 060) and carbon dioxide (Sc =
096) for both air and water Physically it is possible sincean increase in Sc means increase in kinematic viscosity orviscosity of fluid due to which the velocity of fluid decreasesFigure 5 illustrates the influences of Gm 119866 and Pr on thevelocityThe figure reveals that the maximum velocity attainsnear the plate then decreases and vanish far away from theplate Moreover It is observed that an increase in119866 Gm leadsto an increase in velocity for both Pr = 7 and Pr = 071
when Hydrogen gas is present in the flow The reason is thatthe values of Grashof number and modified Grashof numberhave the tendency to increase the mass buoyancy effect Theincrease in velocity due to increase in 119866 and Gm is nearerthe plate than away from the plate Figure 6 elucidates theeffect of 119896 Pr on velocity profile It is found that the velocitydecreases with an increase in chemical reaction parameter 119896for both Pr = 7 and 071 It is due to the fact that increasein chemical reaction parameter 119896 gives rise to an increasein viscosity of fluid which means velocity boundary layerthickness decreases
Figure 7 elucidates the effects of parameters Sc 119896 119866and 120596119905 on skin friction at the plate at different values of119905 It is noticed that skin friction decreases sharply with anincrease in time and for 120596119905 = 0 it becomes negative whichmeans separation of boundary later occurswhich is for highervalues of 119905 (119905 gt 03) The figure depicts that skin frictionis higher for Sc = 096 in comparison to Sc = 022Physically it is correct since an increase in Sc serves toincrease momentum boundary layer thickness Further itdecreases with an increase in value of Gm and marginallyincreases with an increase in 119896 The increase of 120596119905 gives riseto an increase in the value of skin friction
6 ISRN Chemical Engineering
0 071
0 7
120596t Pr
6 071pi
76pi72pi
2 071pi
u
Figure 3 Velocity profile for Sc = 022 119905 = 02 119877 = 4 119896 = 02119866 = 5 and Gm = 2
0
02
04
06
08
1
12
14
16
0 1 2 3 4
7
7
7
7
Pr
u
120578
022
Sc022 02
060 02
096 02
t
minus02
04
071
071
071
071
Pr
022
Sc022 02
060 02
096 02
t
04
Figure 4 Velocity profile 120596119905 = 1205874 119877 = 4 119896 = 02 119866 = 5 andGm = 2
0
02
04
06
08
1
12
14
0 1 2 3 4
u
120578minus02
2 5 7
2 10 7
4 5 7
2 5 071
2 10 071
4 5 071
Gm G Pr Gm G Pr
Figure 5 Velocity profile for 120596119905 = 1205874 Sc = 022 119905 = 02 119877 = 4and 119896 = 02
0
02
04
06
08
1
12
0 05 1 15 2 25 3 35
u
120578minus02
02 7
15 7
02 071
15 071
k Pr k Pr
Figure 6 Velocity profile for 120596119905 = 1205874 Sc = 022 119905 = 02 119891 = 4119866 = 5 and Gm = 2
Figure 8 exhibits the Nusselt number against time It isconcluded from the figure that there is an increase in it withan increase in the value of radiation parameter Furthermorethe value of Nusselt number for water is greater than airIt is consistent with the fact that smaller values of Pr areequivalent to increasing thermal conductivities and therefore
ISRN Chemical Engineering 7
0
05
1
15
2
25
3
35
4
45
5
0 01 02 03 04
Sc Gm
120591
t
1205874
1205874
1205874
1205874
120596tk Sc Gm 120596tk022
022
022
022096
02
02
02
02
15
2
2
2
2
3
0
minus05
Figure 7 Skin friction for 119877 = 4 119866 = 5 Pr = 7
heat is able to diffuse away from the plate more rapidly thanhigher values of Pr hence the rate of heat transfer is reduced
Figure 9 depicts the effect of chemical reaction parameter119896 and Schmidt number Sc on Sherwood number It isobserved that Sherwood number increases with an increasein 119896 and Sc Since increase in Sc means decrease in moleculardiffusivity this in turn gives rise to increase in Sherwoodnumber as Sherwood number is the ratio of convective anddiffusive mass transfer coefficient Chemical reaction param-eter increases the interfacial mass transfer so Sherwoodnumber increases with an increase in 119896
7 Conclusion
A semi-analytical study has been carried out for flow pasta vertical surface in the presence of radiation and chemicalreaction when heat is supplied to the plate at constantrate The dimensionless governing equations are solved byusing Laplace transform technique Numerical evaluations ofclosed form solutions were performed and some graphicalresults were obtained to illustrate the details of flow heat andmass transfer characteristics and their dependence on somephysical parameters The physical aspects of the problem arealso discussed The results of the flow problem indicates thefollowing
(1) Increasing radiation parameter and Prandtl Numberthe temperature decreases whereas it increases withincreasing time
(2) Concentration profile decreases with an increase intime and Schmidt number
0
1
2
3
4
5
6
0 02 04 06 08 1 12
Nu
02 7
04 7
02 071
04 071
R
t
Pr R Pr
Figure 8 Nusselt number
0
005
01
015
02
025
03
035
04
045
0 2 4 6 8 10
Sh
Sck
022
060
022
02
02
04
t
Figure 9 Sherwood number
(3) Fluid velocity decreases with an increase in Schmidtnumber phase angle Prandtl number and chemicalreaction parameter but increases with an increasein Grashof number modified Grashof number andtime
8 ISRN Chemical Engineering
(4) There is a rise in skin frictionwith increasing Schmidtnumber chemical reaction parameter and phaseangle and fall with an increase in value of modifiedGrashof number The rate of heat transfer increaseswith an increase in the value of radiation param-eter and Prandtl number Sherwood number alsoincreases with an increase in chemical reactionparameter and Schmidt number
Nomenclature
119880119877 Reference velocity
119892 Gravitational acceleration119862119901 Specific heat at constant pressure
119863 Mass diffusivity120573 Thermal expansion coefficient120573119862 Concentration expansion coefficient
120588 Density120581 Thermal conductivity of fluid120581lowast Mean absorption coefficient
120590 Electrical conductivity of fluid] Kinematic viscosity119902119903 Radiative heat flux
1205901015840 Stefan-Boltzmann constant
119871119877 Reference length
119905119877 Reference time
Gm Modified Grashof numberPr Prandtl numberSc Schmidt number120596 Frequency of oscillation119906 Dimensionless velocity component120579 Dimensionless temperature119862 Dimensionless concentration120583 Viscosity of fluid119905 Time in dimensionless coordinate119877 Radiation parameter119896 Chemical reaction parameter
References
[1] S T Revankar ldquoFree convection effect on a flow past an impul-sively started or oscillating infinite vertical platerdquo MechanicsResearch Communications vol 27 no 2 pp 241ndash246 2000
[2] J Li D B Ingham and I Pop ldquoNatural convection froma vertical flat plate with a surface temperature oscillationrdquoInternational Journal of Heat and Mass Transfer vol 44 no 12pp 2311ndash2322 2001
[3] E L Cussler Diffusion Mass Transfer in Fluid Systems Cam-bridge University Press Cambridge UK 1998
[4] R C Chaudhary and J Arpita ldquoCombined heat and masstransfer effect on MHD free convection flow past an oscillatingplate embedded in porous mediumrdquo Romanian Journal ofPhysics vol 52 pp 505ndash524 2007
[5] R Muthucumaraswamy and A Vijayalakshmi ldquoEffects of heatand mass transfer on flow past an oscillating vertical platewith variable temperaturerdquo International Journal of AppliedMathematics and Mechanics vol 4 pp 59ndash65 2008
[6] R Muthucumaraswamy M Raj Sundar and V S A Subra-manian ldquoUnsteady flow past an accelerated infinite vertical
plate with variable temperature and uniform mass diffusionrdquoInternational Journal of Applied Mathematics and Mechanicsvol 5 pp 51ndash56 2009
[7] U S Rajput and S Kumar ldquoMHD flow past an impulsivelystarted vertical plate with variable temperature and mass diffu-sionrdquoAppliedMathematical Sciences vol 5 no 1ndash4 pp 149ndash1572011
[8] G Juncu ldquoUnsteady forced convection heatmass transfer froma flat platerdquoHeat andMass TransferWaerme- und Stoffuebertra-gung vol 41 no 12 pp 1095ndash1102 2005
[9] U N Das R Deka and V M Soundalgekar ldquoEffects of masstransfer on flowpast an impulsively started infinite vertical platewith constant heat flux and chemical reactionrdquo Forschung imIngenieurwesenEngineering Research vol 60 no 10 pp 284ndash287 1994
[10] R Muthucumaraswamy and S Meenakshisundaram ldquoTheoret-ical study of chemical reaction effects on vertical oscillatingplate with variable temperature Theoretrdquo Journal of AppliedMechanics vol 33 pp 245ndash257 2006
[11] B C Neog and K R Das Rudra ldquoUnsteady Free ConvectionMHD Flow past a vertical plate with variable temperatureand chemical reactionrdquo International Journal of EngineeringResearch amp Technology vol 1 pp 1ndash5 2012
[12] V Rajesh ldquoRadiation effects onMHD free convection flow neara vertical plate with ramped wall temperaturerdquo InternationalJournal of Applied Mathematics and Mechanics vol 6 pp 60ndash77 2010
[13] M Muralidharan and R Muthucumaraswamy ldquoThermal radi-ation on linearly accelerated vertical plate with variable tem-perature and uniform mass fluxrdquo Indian Journal of Science andTechnology vol 3 pp 398ndash401 2010
[14] U S Rajput and S Kumar ldquoRadiation effects on MHD flowpast an impulsively started vertical plate with variable heat andmass transferrdquo International Journal of AppliedMathematics andMechanics vol 8 pp 66ndash85 2012
[15] R C Chaudhary and A Jain ldquoUnsteady free convection flowpast an oscillating plate with constant mass flux in the presenceof radiationrdquoActaTechnicaCSAV vol 52 no 1 pp 93ndash108 2007
[16] A Ogulu and O D Makinde ldquoUnsteady hydromagnetic freeconvection flow of a dissipative and radiating fluid past avertical plate with constant heat fluxrdquo Chemical EngineeringCommunications vol 196 no 4 pp 454ndash462 2009
[17] M Narahari and L Debnath ldquoUnsteady magnetohydrody-namic free convection flow past an accelerated vertical platewith constant heat flux and heat generation or absorptionrdquoZeitschrift fur Angewandte Mathematik und Mechanik AppliedMathematics and Mechanics vol 93 pp 38ndash49 2013
[18] R C Chaudhary and J Arpita ldquoFree convection effects onMHD flow past an infinite vertical accelerated plate embeddedin porous media with constant heat fluxrdquoMatematicas pp 73ndash82 2009
[19] B M Abramowitz and I A StegunHandbook of MathematicalFunctions Dover New York NY USA 1970
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2 ISRN Chemical Engineering
and knowledge of radiation heat transfer becomes imperativefor the design of the pertinent equipment Nuclear powerplants gas turbines and the various propulsion devices foraircraft missiles satellites and space vehicles are examplesof such engineering areas Unsteady free convection flowpast a vertical plate with chemical reaction under differenttemperature condition on the plate is elucidated by Neog andDas Rudra [11] and Rajesh et al [12]Thermal radiation effecton flow past a vertical plate withmass transfer is examined byMuralidharan and Muthucumaraswamy [13] and Rajput andKumar [14] Natural convective flow past an oscillating platewith constantmass flux in the presence of radiation is studiedby Chaudhary and Jain [15]
The effects of radiation on free convection on the accel-erated flow of a viscous incompressible fluid past an infinitevertical plate havemany important technological applicationsin the astrophysical geophysical and engineering problem
However it seems less attention was paid on hydromag-netic free convection flows near a vertical plate subjectedto a constant heat flux boundary condition even thoughthis situation is involved in many engineering applicationsOgulu and Makinde [16] and Narahari and Debnath [17]examined the flow past a surface with constant heat flux Freeconvection effects on flow past an infinite vertical acceleratedwith constant heat flux are pioneered by Chaudhary andArpita [18]
Hence the objective of this paper is to study radiationand chemical reaction effects on unsteady double diffusiveconvective flow past an oscillating vertical plate when heat issupplied to it at constant rate
2 Mathematical Analysis
We consider a two-dimensional flow of an incompressibleand electrically conducting viscous fluid along an infinitevertical plate The 119909
1015840-axis is taken on the infinite plate andparallel to the free stream velocity and 119910
1015840-axis normal to itInitially the plate and the fluid are at same temperature 1198791015840
infin
with concentration level 1198621015840infin
at all points At time 1199051015840gt 0
the plate concentration is changed to 1198621015840
119908with heat supplied
at a Constant rate to the plate and it starts oscillating witha velocity 119880
119877cos12059610158401199051015840 in its own plane It is assumed that
there exists a homogeneous chemical reaction of first orderwith constant rate 119870
119897between the diffusing species and
the fluid Since the plate is infinite in extent therefore theflow variables are the functions of 1199101015840 and 119905
1015840 only The fluidis considered to be gray absorbing-emitting radiation butnon scattering medium The radiative heat flux in the 119909
1015840-direction is considered negligible in comparison with of 1199101015840-directionThen neglecting viscous dissipation heat producedby chemical reaction and assuming variation of density in thebody force term (Boussinesqrsquos approximation) the problemcan be governed by the following set of equations
1205971198791015840
1205971199051015840=
120581
120588119862119901
12059721198791015840
12059711991010158402
minus
1
120588119862119901
120597119902119903
1205971199101015840 (1)
1205971198621015840
1205971199051015840= 119863
12059721198621015840
12059711991010158402
minus 1198961198971198621015840 (2)
1205971199061015840
1205971199051015840= ]
12059721199061015840
12059711991010158402
+ 119892120573 (1198791015840minus 1198791015840
infin) + 119892120573
119888(1198621015840minus 1198621015840
infin
1015840
) (3)
with following initial and boundary conditions
1199061015840= 0 119879
1015840= 1198791015840
infin 1198621015840= 1198621015840
infinforall1199101015840 1199051015840le 0
1199061015840= 119880119877cos12059610158401199051015840 120597119879
1015840
1205971199101015840= minus
119902
120581
1198621015840= 1198621015840
119908at 1199101015840 = 0 119905
1015840gt 0
1199061015840997888rarr 0 119879
1015840997888rarr 119879
1015840
infin 1198621015840997888rarr 119862
1015840
infin
as 1199101015840 997888rarr infin 1199051015840gt 0
(4)
The radiative heat flux term by using Rosselandrsquos approx-imation is given by
119902119903= minus
41205901015840
3120581lowast
12059711987910158404
1205971199101015840 (5)
We assume that the temperature differences within theflow are such that11987910158404may be expressed as a linear function ofthe temperature1198791015840This is accomplished by expanding 11987910158404 ina Taylor series about 1198791015840
infinand neglecting higher-order terms
One has
11987910158404≃ 411987910158403
infin1198791015840minus 311987910158404
infin (6)
By using (5) and (6) (1) gives
120588119862119901
1205971198791015840
1205971199051015840= 120581
12059721198791015840
12059711991010158402
+
16120590101584011987910158403
infin
3120581lowast
12059721198791015840
12059711991010158402 (7)
Introducing the following dimensionless quantities
119905 =
1199051015840
119905119877
119910 =
1199101015840
119871119877
119906 =
1199061015840
119880119877
120596 = 1205961015840119905119877 119896 =
1198802
119877119896119897
]2
Pr =120583C119901
120581
Sc = ]
119863
120579 =
1198791015840minus 1198791015840
infin
119902]120581119880119877
119866 =
119892120573119902]2
1205811198804
119877
119862 =
1198621015840minus 1198621015840
infin
1198621015840
119908minus 1198621015840
infin
Gm =
]119892120573119862(1198621015840
119908minus 1198621015840
infin)
1198803
119877
119896 =
]119896119897
1198802
119877
119877 =
120581lowast120581
4120590101584011987910158403
infin
Δ119879 = 1198791015840
119908minus 1198791015840
infin 119880
119877= (]119892120573Δ119905)
13
119871119877= (
119892120573Δ119879
]2)
minus13
119905119877= (119892120573Δ119879)
minus23
]13
(8)
ISRN Chemical Engineering 3
the governing equations (1) to (3) reduce to the followingnondimensional form
Pr120597120579120597119905
= (1 +
4
3119877
)
1205972120579
1205971199102
120597119862
120597119905
=
1
Sc1205972119862
1205971199102minus 119896119862
120597119906
120597119905
=
1205972119906
1205971199102+ 119866120579 + Gm119862
(9)
with the following initial and boundary conditions
119906 = 0 120579 = 0 119862 = 0 forall119910 119905 le 0
119906 = cos120596t 120597120579
120597119910
= minus1 119862 = 1 at 119910 = 0 119905 gt 0
119906 997888rarr 0 120579 997888rarr 0 119862 997888rarr 0 as 119910 997888rarr infin 119905 gt 0
(10)
On taking Laplace-transformof (9) and boundary conditions(10) we get
(1 +
4
3119877
)
1198892120579
1198891199102minus 119901Pr 120579 = 0
1198892120601
1198891199102(119896 + 119901 Sc) = 0
1198892119906
1198891199102minus 119901119906 = minus119866120579 (119910 119901) minus Gm997888
119862
(11)
119906 =
119901
1199012+ 1205962
119889120579
119889119910
= minus
1
119901
119862 =
1
119901
at 119910 = 0 119905 gt 0
119906 997888rarr 0 120579 997888rarr 0 119862 997888rarr 0 as 119910 997888rarr infin 119905 gt 0
(12)
where 119901 is the Laplace-transform parameterSolving (11) with the help of boundary condition (12) we
get
120579 (119910 119901) = radic119867
Prexp (minus119910radic119901Pr119867)
11990132
119862 (119910 119901) =
exp(minus119910radic(119896 Sc + 119901 Sc))
119901
119906 (119910 119901) =
119901 exp (minus119910radic119901)
1199012+ 1205962
+ radic119867
Pr119866 exp (minus119910radic119901)
11990152
((Pr119867) minus 1)
+
Gm exp (minus119910radic119901)
119901 (Sc minus 1) (119901 + (119896 Sc (Sc minus 1)))
minus
Gm119901 (Sc minus 1) (119901 + (119896 Sc (Sc minus 1)))
times exp(minus119910radic119901 Sc + 119896 Sc)
minus radic119867
Pr119866 exp (minus119910radic119901Pr119867)
11990152
(
Pr119867
minus 1)
(13)
On taking inverse Laplace-transform of (13) we get
120579 = radic119905 119867
120587Prerfc(
minus1205782Pr119867
) minus 120578radic119905 erfc(120578radicPr119867
)
119862 =
1
2
exp (2120578radic119896 Sc 119905) erfc (120578radicSc + radic119896119905)
+ exp (minus2120578radic119896 Sc 119905) erfc (120578radicSc minus radic119896119905)
(14)
For Pr = Sc = 1 one has
119906 =
exp (119894120596119905)4
times exp (2120578radic119894120596119905) erfc (120578 + radic119894120596119905)
+ exp (minus2120578radic119894120596119905) erfc (120578 minus radic119894120596119905) +
exp (minus119894120596119905)4
times exp (2120578radicminus119894120596119905) erfc (120578 + radicminus119894120596119905)
+ exp (minus2120578radicminus119894120596119905) erfc (120578 minus radicminus119894120596119905)
+ radic119867
Pr11986611990532
3 ((Pr119867) minus 1)
4
radic120587
(1 + 1205782) exp (minus1205782)
minus 120578 (6 + 41205782) erfc (120578)
+ radic119867
Pr11986611990532
3 (1 minus (Pr119867))
times
4
radic120587
(1 + 1205782 Pr119867
) exp (minus1205782 Pr119867
)
minus120578radicPr119867
(6 + 41205782 Pr119867
) erfc(120578radicPr119867
)
4 ISRN Chemical Engineering
+
Gm119896 Sc
erfc (120578) minus Gm2119896 Sc
exp( 119896 Sc 119905
1 minus Sc)
times
(exp(2120578radic119896 Sc 119905
1 minus Sc) erfc(120578 + radic
119896 Sc 119905
1 minus Sc)
+ exp(minus2120578radic119896 Sc 119905
1 minus Sc) erfc(120578 minus radic
119896 Sc 119905
1 minus Sc)
minus
Gm2119896 Sc
exp( 119896 Sc 119905
1 minus Sc)
times
(exp(2120578radic119896 Sc 119905
1 minus Sc) erfc(120578radicSc + radic
119896119905
1 minus Sc)
+ exp(minus2120578radic119896 Sc 119905
1 minus Sc) erfc(120578radicSc minus radic
119896119905
1 minus Sc)
minus
Gm2119896 Sc
(exp (2120578radic119896 Sc 119905) erfc (120578radicSc + radic119896119905)
+ (exp (minus2120578radic119896 Sc 119905) erfc (120578radicSc minus radic119896119905)
(15)
where119867 = 1 + (43119877) and for Pr = Sc = 1 when 119877 rarr infin
119906 =
exp (119894120596119905)4
exp (2120578radic119894120596119905) erfc (120578 + radic119894120596119905)
+ exp (minus2120578radic119894120596119905) erfc (120578 minus radic119894120596119905)
+
exp (minus119894120596119905)4
times exp (2120578radicminus119894120596119905) erfc (120578 + radicminus119894120596119905)
+ exp (minus2120578radicminus119894120596119905) erfc (120578 minus radicminus119894120596119905)
(16)
In expressions erfc (1199091+ 1198941199101) is complementary error
function of complex argument which can be calculated interms of tabulated functions in Abramowitz and Stegun [19]The tables given in Abramowitz and Stegun [19] do not giveerfc (119909
1+ 1198941199101) directly but an auxiliary function119882
1(1199091+ 1198941199101)
which is defined as
erfc (1199091+ 1198941199101) = 119882
1(minus1199101+ 1198941199091) exp minus(119909
1+ 1198941199101)2
(17)
Some properties of1198821(1199091+ 1198941199101) are
1198821(minus1199091+ 1198941199101) = 119882
2(1199091+ 1198941199101)
1198821(1199091minus 1198941199101) = 2 exp minus(119909
1minus 1198941199101)2
minus1198822(1199091+ 1198941199101)
(18)
where1198822(1199091+ 1198941199101) is complex conjugate of119882
1(1199091+ 1198941199101)
3 Skin Friction
From velocity field skin friction at the plate in nondimen-sional form is expressed as
120591 = minus(
120597119906
120597119910
)
119910=0
120591 =
exp (119894120596119905)2radic119905
radic119894120596119905 erf (radic119894120596119905) +
exp (minus119894120596119905)2radic119905
times radicminus119894120596119905 erf (radicminus119894120596119905) +
119866119905119867
(Pr minus 119867)
radic119867
Pr(1 minus radic
Pr119867
)
+
1
radic120587119905
minus
Gm119896 Sc
exp( 119896 Sc1199051 minus Sc
) erf (radic119896 Sc1199051 minus Sc
)radic119896 Sc1 minus Sc
+
Gm119896 Sc
exp( 119896 Sc1199051 minus Sc
) erf (radic119896119905
1 minus Sc)
times radic119896 Sc1 minus Sc
minus
Gm119896 Sc
radic119896 Sc erf (radic119896119905)
(19)
4 Nusselt Number
From temperature field the rate of heat transfer in nondimen-sional form is expressed as
Nu = minus
1
120579 (0)
120597120579
120597119910
10038161003816100381610038161003816100381610038161003816119910=0
= +
1
120579 (0)
= radic120587Pr119905119867
(20)
5 Sherwood Number
From the concentration field the rate of concentrationtransfer expressed in nondimensional form is given by
Sh = minus
120597120593
120597119910
10038161003816100381610038161003816100381610038161003816119910=0
= radic119896 Sc2
(2 +
1
2
radicSc119905
exp (minus119896119905)) (21)
6 Discussion
In order to get physical insight into the problem the valuesof Schmidt number are chosen to represent the presence ofspecies by hydrogen (022) water vapour (060) ammonia(078) and Carbon dioxide (096) at 25∘C temperature and1 atmospheric pressure The values of Pr are chosen 0717 which represent air and water respectively at 20∘C tem-perature and 1 atmospheric pressure The values of radia-tion parameter and chemical reaction parameter are chosenarbitrarily Only real values of solutions are considered forplotting graphs
Figure 1 reveals temperature profiles against 120578 (distancefrom the plate) It is obvious from the figure that themagnitude of temperature is maximum at the plate andthen decays to zero asymptotically Further the magnitudeof the temperature for air is greater than that of water
ISRN Chemical Engineering 5
0
01
02
03
04
05
06
07
0 05 1 15 2 25
minus01
4 02 7
15 02 7
4 06 7
4 02 071
15 02 071
4 06 071
R t Pr R t Pr
120578
120579
Figure 1 Temperature profile
This is due to the fact that thermal conductivity of fluiddecreases with increasing Pr resulting in a decrease in ther-mal boundary layer thickness It is also seen that it decreasessteeply for Pr = 7 than that of Pr = 071 Furthermore itis noticed that an increase in radiation parameter from 4 to15 the temperature marginally decreases It is obvious sincethe radiation parameter defines the relative contribution ofconduction heat transfer to the thermal radiation transferFurther it increases with an increase in time
The species concentration profile against 120578 is plotted inFigure 2 It is observed that the concentration at all points inthe flow field decreases with 120578 and tends to zero as 120578 rarr infinFurthermore an increase in the value of Sc leads to a fall in theconcentration Physically it is true since increase of Sc meansdecrease of molecular diffusivity which results in decrease ofconcentration boundary layer Hence the concentration ofspecies is higher for small values of Sc It is also observed thatconcentration decreases with an increase in time
Figure 3 elucidates the effect of 120596119905 and Pr on the velocityIt is evident from Figure 3 that the velocity increases andattains itsmaximumvalue in the vicinity of the plate (120578 le 05)and then tends to zero asymptotically It is also noted thatboth the velocity and the penetration for Pr = 071 is higherthan those of Pr = 7 Physically it is possible because fluidswith high Prandtl number have high viscosity and hencemove slowly It is also seen that the velocity decreases withan increase in 120596119905 Figure 4 illustrates the influences of Sc119905 and Pr on the velocity It is noticed that the maximumvelocity attains near the plate then decreases and vanishesfar away from the plate When Pr = 7 and 071 it decreasescontinuously to asymptotic value Further they also increaseswith an increase in time at each point in the flow field forhydrogen gas Moreover the effect of time on the velocity is
0
02
04
06
08
1
12
0 1 2 3 4 5 6
Sc
Sc
minus02
022 02
060 02
022 10
t
120578
Figure 2 Concentration profile for 119896 = 02
more dominant than other parameters Finally It is observedthat the velocity for hydrogen (Sc = 022) is higher thanthat of water vapour (Sc = 060) and carbon dioxide (Sc =
096) for both air and water Physically it is possible sincean increase in Sc means increase in kinematic viscosity orviscosity of fluid due to which the velocity of fluid decreasesFigure 5 illustrates the influences of Gm 119866 and Pr on thevelocityThe figure reveals that the maximum velocity attainsnear the plate then decreases and vanish far away from theplate Moreover It is observed that an increase in119866 Gm leadsto an increase in velocity for both Pr = 7 and Pr = 071
when Hydrogen gas is present in the flow The reason is thatthe values of Grashof number and modified Grashof numberhave the tendency to increase the mass buoyancy effect Theincrease in velocity due to increase in 119866 and Gm is nearerthe plate than away from the plate Figure 6 elucidates theeffect of 119896 Pr on velocity profile It is found that the velocitydecreases with an increase in chemical reaction parameter 119896for both Pr = 7 and 071 It is due to the fact that increasein chemical reaction parameter 119896 gives rise to an increasein viscosity of fluid which means velocity boundary layerthickness decreases
Figure 7 elucidates the effects of parameters Sc 119896 119866and 120596119905 on skin friction at the plate at different values of119905 It is noticed that skin friction decreases sharply with anincrease in time and for 120596119905 = 0 it becomes negative whichmeans separation of boundary later occurswhich is for highervalues of 119905 (119905 gt 03) The figure depicts that skin frictionis higher for Sc = 096 in comparison to Sc = 022Physically it is correct since an increase in Sc serves toincrease momentum boundary layer thickness Further itdecreases with an increase in value of Gm and marginallyincreases with an increase in 119896 The increase of 120596119905 gives riseto an increase in the value of skin friction
6 ISRN Chemical Engineering
0 071
0 7
120596t Pr
6 071pi
76pi72pi
2 071pi
u
Figure 3 Velocity profile for Sc = 022 119905 = 02 119877 = 4 119896 = 02119866 = 5 and Gm = 2
0
02
04
06
08
1
12
14
16
0 1 2 3 4
7
7
7
7
Pr
u
120578
022
Sc022 02
060 02
096 02
t
minus02
04
071
071
071
071
Pr
022
Sc022 02
060 02
096 02
t
04
Figure 4 Velocity profile 120596119905 = 1205874 119877 = 4 119896 = 02 119866 = 5 andGm = 2
0
02
04
06
08
1
12
14
0 1 2 3 4
u
120578minus02
2 5 7
2 10 7
4 5 7
2 5 071
2 10 071
4 5 071
Gm G Pr Gm G Pr
Figure 5 Velocity profile for 120596119905 = 1205874 Sc = 022 119905 = 02 119877 = 4and 119896 = 02
0
02
04
06
08
1
12
0 05 1 15 2 25 3 35
u
120578minus02
02 7
15 7
02 071
15 071
k Pr k Pr
Figure 6 Velocity profile for 120596119905 = 1205874 Sc = 022 119905 = 02 119891 = 4119866 = 5 and Gm = 2
Figure 8 exhibits the Nusselt number against time It isconcluded from the figure that there is an increase in it withan increase in the value of radiation parameter Furthermorethe value of Nusselt number for water is greater than airIt is consistent with the fact that smaller values of Pr areequivalent to increasing thermal conductivities and therefore
ISRN Chemical Engineering 7
0
05
1
15
2
25
3
35
4
45
5
0 01 02 03 04
Sc Gm
120591
t
1205874
1205874
1205874
1205874
120596tk Sc Gm 120596tk022
022
022
022096
02
02
02
02
15
2
2
2
2
3
0
minus05
Figure 7 Skin friction for 119877 = 4 119866 = 5 Pr = 7
heat is able to diffuse away from the plate more rapidly thanhigher values of Pr hence the rate of heat transfer is reduced
Figure 9 depicts the effect of chemical reaction parameter119896 and Schmidt number Sc on Sherwood number It isobserved that Sherwood number increases with an increasein 119896 and Sc Since increase in Sc means decrease in moleculardiffusivity this in turn gives rise to increase in Sherwoodnumber as Sherwood number is the ratio of convective anddiffusive mass transfer coefficient Chemical reaction param-eter increases the interfacial mass transfer so Sherwoodnumber increases with an increase in 119896
7 Conclusion
A semi-analytical study has been carried out for flow pasta vertical surface in the presence of radiation and chemicalreaction when heat is supplied to the plate at constantrate The dimensionless governing equations are solved byusing Laplace transform technique Numerical evaluations ofclosed form solutions were performed and some graphicalresults were obtained to illustrate the details of flow heat andmass transfer characteristics and their dependence on somephysical parameters The physical aspects of the problem arealso discussed The results of the flow problem indicates thefollowing
(1) Increasing radiation parameter and Prandtl Numberthe temperature decreases whereas it increases withincreasing time
(2) Concentration profile decreases with an increase intime and Schmidt number
0
1
2
3
4
5
6
0 02 04 06 08 1 12
Nu
02 7
04 7
02 071
04 071
R
t
Pr R Pr
Figure 8 Nusselt number
0
005
01
015
02
025
03
035
04
045
0 2 4 6 8 10
Sh
Sck
022
060
022
02
02
04
t
Figure 9 Sherwood number
(3) Fluid velocity decreases with an increase in Schmidtnumber phase angle Prandtl number and chemicalreaction parameter but increases with an increasein Grashof number modified Grashof number andtime
8 ISRN Chemical Engineering
(4) There is a rise in skin frictionwith increasing Schmidtnumber chemical reaction parameter and phaseangle and fall with an increase in value of modifiedGrashof number The rate of heat transfer increaseswith an increase in the value of radiation param-eter and Prandtl number Sherwood number alsoincreases with an increase in chemical reactionparameter and Schmidt number
Nomenclature
119880119877 Reference velocity
119892 Gravitational acceleration119862119901 Specific heat at constant pressure
119863 Mass diffusivity120573 Thermal expansion coefficient120573119862 Concentration expansion coefficient
120588 Density120581 Thermal conductivity of fluid120581lowast Mean absorption coefficient
120590 Electrical conductivity of fluid] Kinematic viscosity119902119903 Radiative heat flux
1205901015840 Stefan-Boltzmann constant
119871119877 Reference length
119905119877 Reference time
Gm Modified Grashof numberPr Prandtl numberSc Schmidt number120596 Frequency of oscillation119906 Dimensionless velocity component120579 Dimensionless temperature119862 Dimensionless concentration120583 Viscosity of fluid119905 Time in dimensionless coordinate119877 Radiation parameter119896 Chemical reaction parameter
References
[1] S T Revankar ldquoFree convection effect on a flow past an impul-sively started or oscillating infinite vertical platerdquo MechanicsResearch Communications vol 27 no 2 pp 241ndash246 2000
[2] J Li D B Ingham and I Pop ldquoNatural convection froma vertical flat plate with a surface temperature oscillationrdquoInternational Journal of Heat and Mass Transfer vol 44 no 12pp 2311ndash2322 2001
[3] E L Cussler Diffusion Mass Transfer in Fluid Systems Cam-bridge University Press Cambridge UK 1998
[4] R C Chaudhary and J Arpita ldquoCombined heat and masstransfer effect on MHD free convection flow past an oscillatingplate embedded in porous mediumrdquo Romanian Journal ofPhysics vol 52 pp 505ndash524 2007
[5] R Muthucumaraswamy and A Vijayalakshmi ldquoEffects of heatand mass transfer on flow past an oscillating vertical platewith variable temperaturerdquo International Journal of AppliedMathematics and Mechanics vol 4 pp 59ndash65 2008
[6] R Muthucumaraswamy M Raj Sundar and V S A Subra-manian ldquoUnsteady flow past an accelerated infinite vertical
plate with variable temperature and uniform mass diffusionrdquoInternational Journal of Applied Mathematics and Mechanicsvol 5 pp 51ndash56 2009
[7] U S Rajput and S Kumar ldquoMHD flow past an impulsivelystarted vertical plate with variable temperature and mass diffu-sionrdquoAppliedMathematical Sciences vol 5 no 1ndash4 pp 149ndash1572011
[8] G Juncu ldquoUnsteady forced convection heatmass transfer froma flat platerdquoHeat andMass TransferWaerme- und Stoffuebertra-gung vol 41 no 12 pp 1095ndash1102 2005
[9] U N Das R Deka and V M Soundalgekar ldquoEffects of masstransfer on flowpast an impulsively started infinite vertical platewith constant heat flux and chemical reactionrdquo Forschung imIngenieurwesenEngineering Research vol 60 no 10 pp 284ndash287 1994
[10] R Muthucumaraswamy and S Meenakshisundaram ldquoTheoret-ical study of chemical reaction effects on vertical oscillatingplate with variable temperature Theoretrdquo Journal of AppliedMechanics vol 33 pp 245ndash257 2006
[11] B C Neog and K R Das Rudra ldquoUnsteady Free ConvectionMHD Flow past a vertical plate with variable temperatureand chemical reactionrdquo International Journal of EngineeringResearch amp Technology vol 1 pp 1ndash5 2012
[12] V Rajesh ldquoRadiation effects onMHD free convection flow neara vertical plate with ramped wall temperaturerdquo InternationalJournal of Applied Mathematics and Mechanics vol 6 pp 60ndash77 2010
[13] M Muralidharan and R Muthucumaraswamy ldquoThermal radi-ation on linearly accelerated vertical plate with variable tem-perature and uniform mass fluxrdquo Indian Journal of Science andTechnology vol 3 pp 398ndash401 2010
[14] U S Rajput and S Kumar ldquoRadiation effects on MHD flowpast an impulsively started vertical plate with variable heat andmass transferrdquo International Journal of AppliedMathematics andMechanics vol 8 pp 66ndash85 2012
[15] R C Chaudhary and A Jain ldquoUnsteady free convection flowpast an oscillating plate with constant mass flux in the presenceof radiationrdquoActaTechnicaCSAV vol 52 no 1 pp 93ndash108 2007
[16] A Ogulu and O D Makinde ldquoUnsteady hydromagnetic freeconvection flow of a dissipative and radiating fluid past avertical plate with constant heat fluxrdquo Chemical EngineeringCommunications vol 196 no 4 pp 454ndash462 2009
[17] M Narahari and L Debnath ldquoUnsteady magnetohydrody-namic free convection flow past an accelerated vertical platewith constant heat flux and heat generation or absorptionrdquoZeitschrift fur Angewandte Mathematik und Mechanik AppliedMathematics and Mechanics vol 93 pp 38ndash49 2013
[18] R C Chaudhary and J Arpita ldquoFree convection effects onMHD flow past an infinite vertical accelerated plate embeddedin porous media with constant heat fluxrdquoMatematicas pp 73ndash82 2009
[19] B M Abramowitz and I A StegunHandbook of MathematicalFunctions Dover New York NY USA 1970
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International Journal of
ISRN Chemical Engineering 3
the governing equations (1) to (3) reduce to the followingnondimensional form
Pr120597120579120597119905
= (1 +
4
3119877
)
1205972120579
1205971199102
120597119862
120597119905
=
1
Sc1205972119862
1205971199102minus 119896119862
120597119906
120597119905
=
1205972119906
1205971199102+ 119866120579 + Gm119862
(9)
with the following initial and boundary conditions
119906 = 0 120579 = 0 119862 = 0 forall119910 119905 le 0
119906 = cos120596t 120597120579
120597119910
= minus1 119862 = 1 at 119910 = 0 119905 gt 0
119906 997888rarr 0 120579 997888rarr 0 119862 997888rarr 0 as 119910 997888rarr infin 119905 gt 0
(10)
On taking Laplace-transformof (9) and boundary conditions(10) we get
(1 +
4
3119877
)
1198892120579
1198891199102minus 119901Pr 120579 = 0
1198892120601
1198891199102(119896 + 119901 Sc) = 0
1198892119906
1198891199102minus 119901119906 = minus119866120579 (119910 119901) minus Gm997888
119862
(11)
119906 =
119901
1199012+ 1205962
119889120579
119889119910
= minus
1
119901
119862 =
1
119901
at 119910 = 0 119905 gt 0
119906 997888rarr 0 120579 997888rarr 0 119862 997888rarr 0 as 119910 997888rarr infin 119905 gt 0
(12)
where 119901 is the Laplace-transform parameterSolving (11) with the help of boundary condition (12) we
get
120579 (119910 119901) = radic119867
Prexp (minus119910radic119901Pr119867)
11990132
119862 (119910 119901) =
exp(minus119910radic(119896 Sc + 119901 Sc))
119901
119906 (119910 119901) =
119901 exp (minus119910radic119901)
1199012+ 1205962
+ radic119867
Pr119866 exp (minus119910radic119901)
11990152
((Pr119867) minus 1)
+
Gm exp (minus119910radic119901)
119901 (Sc minus 1) (119901 + (119896 Sc (Sc minus 1)))
minus
Gm119901 (Sc minus 1) (119901 + (119896 Sc (Sc minus 1)))
times exp(minus119910radic119901 Sc + 119896 Sc)
minus radic119867
Pr119866 exp (minus119910radic119901Pr119867)
11990152
(
Pr119867
minus 1)
(13)
On taking inverse Laplace-transform of (13) we get
120579 = radic119905 119867
120587Prerfc(
minus1205782Pr119867
) minus 120578radic119905 erfc(120578radicPr119867
)
119862 =
1
2
exp (2120578radic119896 Sc 119905) erfc (120578radicSc + radic119896119905)
+ exp (minus2120578radic119896 Sc 119905) erfc (120578radicSc minus radic119896119905)
(14)
For Pr = Sc = 1 one has
119906 =
exp (119894120596119905)4
times exp (2120578radic119894120596119905) erfc (120578 + radic119894120596119905)
+ exp (minus2120578radic119894120596119905) erfc (120578 minus radic119894120596119905) +
exp (minus119894120596119905)4
times exp (2120578radicminus119894120596119905) erfc (120578 + radicminus119894120596119905)
+ exp (minus2120578radicminus119894120596119905) erfc (120578 minus radicminus119894120596119905)
+ radic119867
Pr11986611990532
3 ((Pr119867) minus 1)
4
radic120587
(1 + 1205782) exp (minus1205782)
minus 120578 (6 + 41205782) erfc (120578)
+ radic119867
Pr11986611990532
3 (1 minus (Pr119867))
times
4
radic120587
(1 + 1205782 Pr119867
) exp (minus1205782 Pr119867
)
minus120578radicPr119867
(6 + 41205782 Pr119867
) erfc(120578radicPr119867
)
4 ISRN Chemical Engineering
+
Gm119896 Sc
erfc (120578) minus Gm2119896 Sc
exp( 119896 Sc 119905
1 minus Sc)
times
(exp(2120578radic119896 Sc 119905
1 minus Sc) erfc(120578 + radic
119896 Sc 119905
1 minus Sc)
+ exp(minus2120578radic119896 Sc 119905
1 minus Sc) erfc(120578 minus radic
119896 Sc 119905
1 minus Sc)
minus
Gm2119896 Sc
exp( 119896 Sc 119905
1 minus Sc)
times
(exp(2120578radic119896 Sc 119905
1 minus Sc) erfc(120578radicSc + radic
119896119905
1 minus Sc)
+ exp(minus2120578radic119896 Sc 119905
1 minus Sc) erfc(120578radicSc minus radic
119896119905
1 minus Sc)
minus
Gm2119896 Sc
(exp (2120578radic119896 Sc 119905) erfc (120578radicSc + radic119896119905)
+ (exp (minus2120578radic119896 Sc 119905) erfc (120578radicSc minus radic119896119905)
(15)
where119867 = 1 + (43119877) and for Pr = Sc = 1 when 119877 rarr infin
119906 =
exp (119894120596119905)4
exp (2120578radic119894120596119905) erfc (120578 + radic119894120596119905)
+ exp (minus2120578radic119894120596119905) erfc (120578 minus radic119894120596119905)
+
exp (minus119894120596119905)4
times exp (2120578radicminus119894120596119905) erfc (120578 + radicminus119894120596119905)
+ exp (minus2120578radicminus119894120596119905) erfc (120578 minus radicminus119894120596119905)
(16)
In expressions erfc (1199091+ 1198941199101) is complementary error
function of complex argument which can be calculated interms of tabulated functions in Abramowitz and Stegun [19]The tables given in Abramowitz and Stegun [19] do not giveerfc (119909
1+ 1198941199101) directly but an auxiliary function119882
1(1199091+ 1198941199101)
which is defined as
erfc (1199091+ 1198941199101) = 119882
1(minus1199101+ 1198941199091) exp minus(119909
1+ 1198941199101)2
(17)
Some properties of1198821(1199091+ 1198941199101) are
1198821(minus1199091+ 1198941199101) = 119882
2(1199091+ 1198941199101)
1198821(1199091minus 1198941199101) = 2 exp minus(119909
1minus 1198941199101)2
minus1198822(1199091+ 1198941199101)
(18)
where1198822(1199091+ 1198941199101) is complex conjugate of119882
1(1199091+ 1198941199101)
3 Skin Friction
From velocity field skin friction at the plate in nondimen-sional form is expressed as
120591 = minus(
120597119906
120597119910
)
119910=0
120591 =
exp (119894120596119905)2radic119905
radic119894120596119905 erf (radic119894120596119905) +
exp (minus119894120596119905)2radic119905
times radicminus119894120596119905 erf (radicminus119894120596119905) +
119866119905119867
(Pr minus 119867)
radic119867
Pr(1 minus radic
Pr119867
)
+
1
radic120587119905
minus
Gm119896 Sc
exp( 119896 Sc1199051 minus Sc
) erf (radic119896 Sc1199051 minus Sc
)radic119896 Sc1 minus Sc
+
Gm119896 Sc
exp( 119896 Sc1199051 minus Sc
) erf (radic119896119905
1 minus Sc)
times radic119896 Sc1 minus Sc
minus
Gm119896 Sc
radic119896 Sc erf (radic119896119905)
(19)
4 Nusselt Number
From temperature field the rate of heat transfer in nondimen-sional form is expressed as
Nu = minus
1
120579 (0)
120597120579
120597119910
10038161003816100381610038161003816100381610038161003816119910=0
= +
1
120579 (0)
= radic120587Pr119905119867
(20)
5 Sherwood Number
From the concentration field the rate of concentrationtransfer expressed in nondimensional form is given by
Sh = minus
120597120593
120597119910
10038161003816100381610038161003816100381610038161003816119910=0
= radic119896 Sc2
(2 +
1
2
radicSc119905
exp (minus119896119905)) (21)
6 Discussion
In order to get physical insight into the problem the valuesof Schmidt number are chosen to represent the presence ofspecies by hydrogen (022) water vapour (060) ammonia(078) and Carbon dioxide (096) at 25∘C temperature and1 atmospheric pressure The values of Pr are chosen 0717 which represent air and water respectively at 20∘C tem-perature and 1 atmospheric pressure The values of radia-tion parameter and chemical reaction parameter are chosenarbitrarily Only real values of solutions are considered forplotting graphs
Figure 1 reveals temperature profiles against 120578 (distancefrom the plate) It is obvious from the figure that themagnitude of temperature is maximum at the plate andthen decays to zero asymptotically Further the magnitudeof the temperature for air is greater than that of water
ISRN Chemical Engineering 5
0
01
02
03
04
05
06
07
0 05 1 15 2 25
minus01
4 02 7
15 02 7
4 06 7
4 02 071
15 02 071
4 06 071
R t Pr R t Pr
120578
120579
Figure 1 Temperature profile
This is due to the fact that thermal conductivity of fluiddecreases with increasing Pr resulting in a decrease in ther-mal boundary layer thickness It is also seen that it decreasessteeply for Pr = 7 than that of Pr = 071 Furthermore itis noticed that an increase in radiation parameter from 4 to15 the temperature marginally decreases It is obvious sincethe radiation parameter defines the relative contribution ofconduction heat transfer to the thermal radiation transferFurther it increases with an increase in time
The species concentration profile against 120578 is plotted inFigure 2 It is observed that the concentration at all points inthe flow field decreases with 120578 and tends to zero as 120578 rarr infinFurthermore an increase in the value of Sc leads to a fall in theconcentration Physically it is true since increase of Sc meansdecrease of molecular diffusivity which results in decrease ofconcentration boundary layer Hence the concentration ofspecies is higher for small values of Sc It is also observed thatconcentration decreases with an increase in time
Figure 3 elucidates the effect of 120596119905 and Pr on the velocityIt is evident from Figure 3 that the velocity increases andattains itsmaximumvalue in the vicinity of the plate (120578 le 05)and then tends to zero asymptotically It is also noted thatboth the velocity and the penetration for Pr = 071 is higherthan those of Pr = 7 Physically it is possible because fluidswith high Prandtl number have high viscosity and hencemove slowly It is also seen that the velocity decreases withan increase in 120596119905 Figure 4 illustrates the influences of Sc119905 and Pr on the velocity It is noticed that the maximumvelocity attains near the plate then decreases and vanishesfar away from the plate When Pr = 7 and 071 it decreasescontinuously to asymptotic value Further they also increaseswith an increase in time at each point in the flow field forhydrogen gas Moreover the effect of time on the velocity is
0
02
04
06
08
1
12
0 1 2 3 4 5 6
Sc
Sc
minus02
022 02
060 02
022 10
t
120578
Figure 2 Concentration profile for 119896 = 02
more dominant than other parameters Finally It is observedthat the velocity for hydrogen (Sc = 022) is higher thanthat of water vapour (Sc = 060) and carbon dioxide (Sc =
096) for both air and water Physically it is possible sincean increase in Sc means increase in kinematic viscosity orviscosity of fluid due to which the velocity of fluid decreasesFigure 5 illustrates the influences of Gm 119866 and Pr on thevelocityThe figure reveals that the maximum velocity attainsnear the plate then decreases and vanish far away from theplate Moreover It is observed that an increase in119866 Gm leadsto an increase in velocity for both Pr = 7 and Pr = 071
when Hydrogen gas is present in the flow The reason is thatthe values of Grashof number and modified Grashof numberhave the tendency to increase the mass buoyancy effect Theincrease in velocity due to increase in 119866 and Gm is nearerthe plate than away from the plate Figure 6 elucidates theeffect of 119896 Pr on velocity profile It is found that the velocitydecreases with an increase in chemical reaction parameter 119896for both Pr = 7 and 071 It is due to the fact that increasein chemical reaction parameter 119896 gives rise to an increasein viscosity of fluid which means velocity boundary layerthickness decreases
Figure 7 elucidates the effects of parameters Sc 119896 119866and 120596119905 on skin friction at the plate at different values of119905 It is noticed that skin friction decreases sharply with anincrease in time and for 120596119905 = 0 it becomes negative whichmeans separation of boundary later occurswhich is for highervalues of 119905 (119905 gt 03) The figure depicts that skin frictionis higher for Sc = 096 in comparison to Sc = 022Physically it is correct since an increase in Sc serves toincrease momentum boundary layer thickness Further itdecreases with an increase in value of Gm and marginallyincreases with an increase in 119896 The increase of 120596119905 gives riseto an increase in the value of skin friction
6 ISRN Chemical Engineering
0 071
0 7
120596t Pr
6 071pi
76pi72pi
2 071pi
u
Figure 3 Velocity profile for Sc = 022 119905 = 02 119877 = 4 119896 = 02119866 = 5 and Gm = 2
0
02
04
06
08
1
12
14
16
0 1 2 3 4
7
7
7
7
Pr
u
120578
022
Sc022 02
060 02
096 02
t
minus02
04
071
071
071
071
Pr
022
Sc022 02
060 02
096 02
t
04
Figure 4 Velocity profile 120596119905 = 1205874 119877 = 4 119896 = 02 119866 = 5 andGm = 2
0
02
04
06
08
1
12
14
0 1 2 3 4
u
120578minus02
2 5 7
2 10 7
4 5 7
2 5 071
2 10 071
4 5 071
Gm G Pr Gm G Pr
Figure 5 Velocity profile for 120596119905 = 1205874 Sc = 022 119905 = 02 119877 = 4and 119896 = 02
0
02
04
06
08
1
12
0 05 1 15 2 25 3 35
u
120578minus02
02 7
15 7
02 071
15 071
k Pr k Pr
Figure 6 Velocity profile for 120596119905 = 1205874 Sc = 022 119905 = 02 119891 = 4119866 = 5 and Gm = 2
Figure 8 exhibits the Nusselt number against time It isconcluded from the figure that there is an increase in it withan increase in the value of radiation parameter Furthermorethe value of Nusselt number for water is greater than airIt is consistent with the fact that smaller values of Pr areequivalent to increasing thermal conductivities and therefore
ISRN Chemical Engineering 7
0
05
1
15
2
25
3
35
4
45
5
0 01 02 03 04
Sc Gm
120591
t
1205874
1205874
1205874
1205874
120596tk Sc Gm 120596tk022
022
022
022096
02
02
02
02
15
2
2
2
2
3
0
minus05
Figure 7 Skin friction for 119877 = 4 119866 = 5 Pr = 7
heat is able to diffuse away from the plate more rapidly thanhigher values of Pr hence the rate of heat transfer is reduced
Figure 9 depicts the effect of chemical reaction parameter119896 and Schmidt number Sc on Sherwood number It isobserved that Sherwood number increases with an increasein 119896 and Sc Since increase in Sc means decrease in moleculardiffusivity this in turn gives rise to increase in Sherwoodnumber as Sherwood number is the ratio of convective anddiffusive mass transfer coefficient Chemical reaction param-eter increases the interfacial mass transfer so Sherwoodnumber increases with an increase in 119896
7 Conclusion
A semi-analytical study has been carried out for flow pasta vertical surface in the presence of radiation and chemicalreaction when heat is supplied to the plate at constantrate The dimensionless governing equations are solved byusing Laplace transform technique Numerical evaluations ofclosed form solutions were performed and some graphicalresults were obtained to illustrate the details of flow heat andmass transfer characteristics and their dependence on somephysical parameters The physical aspects of the problem arealso discussed The results of the flow problem indicates thefollowing
(1) Increasing radiation parameter and Prandtl Numberthe temperature decreases whereas it increases withincreasing time
(2) Concentration profile decreases with an increase intime and Schmidt number
0
1
2
3
4
5
6
0 02 04 06 08 1 12
Nu
02 7
04 7
02 071
04 071
R
t
Pr R Pr
Figure 8 Nusselt number
0
005
01
015
02
025
03
035
04
045
0 2 4 6 8 10
Sh
Sck
022
060
022
02
02
04
t
Figure 9 Sherwood number
(3) Fluid velocity decreases with an increase in Schmidtnumber phase angle Prandtl number and chemicalreaction parameter but increases with an increasein Grashof number modified Grashof number andtime
8 ISRN Chemical Engineering
(4) There is a rise in skin frictionwith increasing Schmidtnumber chemical reaction parameter and phaseangle and fall with an increase in value of modifiedGrashof number The rate of heat transfer increaseswith an increase in the value of radiation param-eter and Prandtl number Sherwood number alsoincreases with an increase in chemical reactionparameter and Schmidt number
Nomenclature
119880119877 Reference velocity
119892 Gravitational acceleration119862119901 Specific heat at constant pressure
119863 Mass diffusivity120573 Thermal expansion coefficient120573119862 Concentration expansion coefficient
120588 Density120581 Thermal conductivity of fluid120581lowast Mean absorption coefficient
120590 Electrical conductivity of fluid] Kinematic viscosity119902119903 Radiative heat flux
1205901015840 Stefan-Boltzmann constant
119871119877 Reference length
119905119877 Reference time
Gm Modified Grashof numberPr Prandtl numberSc Schmidt number120596 Frequency of oscillation119906 Dimensionless velocity component120579 Dimensionless temperature119862 Dimensionless concentration120583 Viscosity of fluid119905 Time in dimensionless coordinate119877 Radiation parameter119896 Chemical reaction parameter
References
[1] S T Revankar ldquoFree convection effect on a flow past an impul-sively started or oscillating infinite vertical platerdquo MechanicsResearch Communications vol 27 no 2 pp 241ndash246 2000
[2] J Li D B Ingham and I Pop ldquoNatural convection froma vertical flat plate with a surface temperature oscillationrdquoInternational Journal of Heat and Mass Transfer vol 44 no 12pp 2311ndash2322 2001
[3] E L Cussler Diffusion Mass Transfer in Fluid Systems Cam-bridge University Press Cambridge UK 1998
[4] R C Chaudhary and J Arpita ldquoCombined heat and masstransfer effect on MHD free convection flow past an oscillatingplate embedded in porous mediumrdquo Romanian Journal ofPhysics vol 52 pp 505ndash524 2007
[5] R Muthucumaraswamy and A Vijayalakshmi ldquoEffects of heatand mass transfer on flow past an oscillating vertical platewith variable temperaturerdquo International Journal of AppliedMathematics and Mechanics vol 4 pp 59ndash65 2008
[6] R Muthucumaraswamy M Raj Sundar and V S A Subra-manian ldquoUnsteady flow past an accelerated infinite vertical
plate with variable temperature and uniform mass diffusionrdquoInternational Journal of Applied Mathematics and Mechanicsvol 5 pp 51ndash56 2009
[7] U S Rajput and S Kumar ldquoMHD flow past an impulsivelystarted vertical plate with variable temperature and mass diffu-sionrdquoAppliedMathematical Sciences vol 5 no 1ndash4 pp 149ndash1572011
[8] G Juncu ldquoUnsteady forced convection heatmass transfer froma flat platerdquoHeat andMass TransferWaerme- und Stoffuebertra-gung vol 41 no 12 pp 1095ndash1102 2005
[9] U N Das R Deka and V M Soundalgekar ldquoEffects of masstransfer on flowpast an impulsively started infinite vertical platewith constant heat flux and chemical reactionrdquo Forschung imIngenieurwesenEngineering Research vol 60 no 10 pp 284ndash287 1994
[10] R Muthucumaraswamy and S Meenakshisundaram ldquoTheoret-ical study of chemical reaction effects on vertical oscillatingplate with variable temperature Theoretrdquo Journal of AppliedMechanics vol 33 pp 245ndash257 2006
[11] B C Neog and K R Das Rudra ldquoUnsteady Free ConvectionMHD Flow past a vertical plate with variable temperatureand chemical reactionrdquo International Journal of EngineeringResearch amp Technology vol 1 pp 1ndash5 2012
[12] V Rajesh ldquoRadiation effects onMHD free convection flow neara vertical plate with ramped wall temperaturerdquo InternationalJournal of Applied Mathematics and Mechanics vol 6 pp 60ndash77 2010
[13] M Muralidharan and R Muthucumaraswamy ldquoThermal radi-ation on linearly accelerated vertical plate with variable tem-perature and uniform mass fluxrdquo Indian Journal of Science andTechnology vol 3 pp 398ndash401 2010
[14] U S Rajput and S Kumar ldquoRadiation effects on MHD flowpast an impulsively started vertical plate with variable heat andmass transferrdquo International Journal of AppliedMathematics andMechanics vol 8 pp 66ndash85 2012
[15] R C Chaudhary and A Jain ldquoUnsteady free convection flowpast an oscillating plate with constant mass flux in the presenceof radiationrdquoActaTechnicaCSAV vol 52 no 1 pp 93ndash108 2007
[16] A Ogulu and O D Makinde ldquoUnsteady hydromagnetic freeconvection flow of a dissipative and radiating fluid past avertical plate with constant heat fluxrdquo Chemical EngineeringCommunications vol 196 no 4 pp 454ndash462 2009
[17] M Narahari and L Debnath ldquoUnsteady magnetohydrody-namic free convection flow past an accelerated vertical platewith constant heat flux and heat generation or absorptionrdquoZeitschrift fur Angewandte Mathematik und Mechanik AppliedMathematics and Mechanics vol 93 pp 38ndash49 2013
[18] R C Chaudhary and J Arpita ldquoFree convection effects onMHD flow past an infinite vertical accelerated plate embeddedin porous media with constant heat fluxrdquoMatematicas pp 73ndash82 2009
[19] B M Abramowitz and I A StegunHandbook of MathematicalFunctions Dover New York NY USA 1970
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
4 ISRN Chemical Engineering
+
Gm119896 Sc
erfc (120578) minus Gm2119896 Sc
exp( 119896 Sc 119905
1 minus Sc)
times
(exp(2120578radic119896 Sc 119905
1 minus Sc) erfc(120578 + radic
119896 Sc 119905
1 minus Sc)
+ exp(minus2120578radic119896 Sc 119905
1 minus Sc) erfc(120578 minus radic
119896 Sc 119905
1 minus Sc)
minus
Gm2119896 Sc
exp( 119896 Sc 119905
1 minus Sc)
times
(exp(2120578radic119896 Sc 119905
1 minus Sc) erfc(120578radicSc + radic
119896119905
1 minus Sc)
+ exp(minus2120578radic119896 Sc 119905
1 minus Sc) erfc(120578radicSc minus radic
119896119905
1 minus Sc)
minus
Gm2119896 Sc
(exp (2120578radic119896 Sc 119905) erfc (120578radicSc + radic119896119905)
+ (exp (minus2120578radic119896 Sc 119905) erfc (120578radicSc minus radic119896119905)
(15)
where119867 = 1 + (43119877) and for Pr = Sc = 1 when 119877 rarr infin
119906 =
exp (119894120596119905)4
exp (2120578radic119894120596119905) erfc (120578 + radic119894120596119905)
+ exp (minus2120578radic119894120596119905) erfc (120578 minus radic119894120596119905)
+
exp (minus119894120596119905)4
times exp (2120578radicminus119894120596119905) erfc (120578 + radicminus119894120596119905)
+ exp (minus2120578radicminus119894120596119905) erfc (120578 minus radicminus119894120596119905)
(16)
In expressions erfc (1199091+ 1198941199101) is complementary error
function of complex argument which can be calculated interms of tabulated functions in Abramowitz and Stegun [19]The tables given in Abramowitz and Stegun [19] do not giveerfc (119909
1+ 1198941199101) directly but an auxiliary function119882
1(1199091+ 1198941199101)
which is defined as
erfc (1199091+ 1198941199101) = 119882
1(minus1199101+ 1198941199091) exp minus(119909
1+ 1198941199101)2
(17)
Some properties of1198821(1199091+ 1198941199101) are
1198821(minus1199091+ 1198941199101) = 119882
2(1199091+ 1198941199101)
1198821(1199091minus 1198941199101) = 2 exp minus(119909
1minus 1198941199101)2
minus1198822(1199091+ 1198941199101)
(18)
where1198822(1199091+ 1198941199101) is complex conjugate of119882
1(1199091+ 1198941199101)
3 Skin Friction
From velocity field skin friction at the plate in nondimen-sional form is expressed as
120591 = minus(
120597119906
120597119910
)
119910=0
120591 =
exp (119894120596119905)2radic119905
radic119894120596119905 erf (radic119894120596119905) +
exp (minus119894120596119905)2radic119905
times radicminus119894120596119905 erf (radicminus119894120596119905) +
119866119905119867
(Pr minus 119867)
radic119867
Pr(1 minus radic
Pr119867
)
+
1
radic120587119905
minus
Gm119896 Sc
exp( 119896 Sc1199051 minus Sc
) erf (radic119896 Sc1199051 minus Sc
)radic119896 Sc1 minus Sc
+
Gm119896 Sc
exp( 119896 Sc1199051 minus Sc
) erf (radic119896119905
1 minus Sc)
times radic119896 Sc1 minus Sc
minus
Gm119896 Sc
radic119896 Sc erf (radic119896119905)
(19)
4 Nusselt Number
From temperature field the rate of heat transfer in nondimen-sional form is expressed as
Nu = minus
1
120579 (0)
120597120579
120597119910
10038161003816100381610038161003816100381610038161003816119910=0
= +
1
120579 (0)
= radic120587Pr119905119867
(20)
5 Sherwood Number
From the concentration field the rate of concentrationtransfer expressed in nondimensional form is given by
Sh = minus
120597120593
120597119910
10038161003816100381610038161003816100381610038161003816119910=0
= radic119896 Sc2
(2 +
1
2
radicSc119905
exp (minus119896119905)) (21)
6 Discussion
In order to get physical insight into the problem the valuesof Schmidt number are chosen to represent the presence ofspecies by hydrogen (022) water vapour (060) ammonia(078) and Carbon dioxide (096) at 25∘C temperature and1 atmospheric pressure The values of Pr are chosen 0717 which represent air and water respectively at 20∘C tem-perature and 1 atmospheric pressure The values of radia-tion parameter and chemical reaction parameter are chosenarbitrarily Only real values of solutions are considered forplotting graphs
Figure 1 reveals temperature profiles against 120578 (distancefrom the plate) It is obvious from the figure that themagnitude of temperature is maximum at the plate andthen decays to zero asymptotically Further the magnitudeof the temperature for air is greater than that of water
ISRN Chemical Engineering 5
0
01
02
03
04
05
06
07
0 05 1 15 2 25
minus01
4 02 7
15 02 7
4 06 7
4 02 071
15 02 071
4 06 071
R t Pr R t Pr
120578
120579
Figure 1 Temperature profile
This is due to the fact that thermal conductivity of fluiddecreases with increasing Pr resulting in a decrease in ther-mal boundary layer thickness It is also seen that it decreasessteeply for Pr = 7 than that of Pr = 071 Furthermore itis noticed that an increase in radiation parameter from 4 to15 the temperature marginally decreases It is obvious sincethe radiation parameter defines the relative contribution ofconduction heat transfer to the thermal radiation transferFurther it increases with an increase in time
The species concentration profile against 120578 is plotted inFigure 2 It is observed that the concentration at all points inthe flow field decreases with 120578 and tends to zero as 120578 rarr infinFurthermore an increase in the value of Sc leads to a fall in theconcentration Physically it is true since increase of Sc meansdecrease of molecular diffusivity which results in decrease ofconcentration boundary layer Hence the concentration ofspecies is higher for small values of Sc It is also observed thatconcentration decreases with an increase in time
Figure 3 elucidates the effect of 120596119905 and Pr on the velocityIt is evident from Figure 3 that the velocity increases andattains itsmaximumvalue in the vicinity of the plate (120578 le 05)and then tends to zero asymptotically It is also noted thatboth the velocity and the penetration for Pr = 071 is higherthan those of Pr = 7 Physically it is possible because fluidswith high Prandtl number have high viscosity and hencemove slowly It is also seen that the velocity decreases withan increase in 120596119905 Figure 4 illustrates the influences of Sc119905 and Pr on the velocity It is noticed that the maximumvelocity attains near the plate then decreases and vanishesfar away from the plate When Pr = 7 and 071 it decreasescontinuously to asymptotic value Further they also increaseswith an increase in time at each point in the flow field forhydrogen gas Moreover the effect of time on the velocity is
0
02
04
06
08
1
12
0 1 2 3 4 5 6
Sc
Sc
minus02
022 02
060 02
022 10
t
120578
Figure 2 Concentration profile for 119896 = 02
more dominant than other parameters Finally It is observedthat the velocity for hydrogen (Sc = 022) is higher thanthat of water vapour (Sc = 060) and carbon dioxide (Sc =
096) for both air and water Physically it is possible sincean increase in Sc means increase in kinematic viscosity orviscosity of fluid due to which the velocity of fluid decreasesFigure 5 illustrates the influences of Gm 119866 and Pr on thevelocityThe figure reveals that the maximum velocity attainsnear the plate then decreases and vanish far away from theplate Moreover It is observed that an increase in119866 Gm leadsto an increase in velocity for both Pr = 7 and Pr = 071
when Hydrogen gas is present in the flow The reason is thatthe values of Grashof number and modified Grashof numberhave the tendency to increase the mass buoyancy effect Theincrease in velocity due to increase in 119866 and Gm is nearerthe plate than away from the plate Figure 6 elucidates theeffect of 119896 Pr on velocity profile It is found that the velocitydecreases with an increase in chemical reaction parameter 119896for both Pr = 7 and 071 It is due to the fact that increasein chemical reaction parameter 119896 gives rise to an increasein viscosity of fluid which means velocity boundary layerthickness decreases
Figure 7 elucidates the effects of parameters Sc 119896 119866and 120596119905 on skin friction at the plate at different values of119905 It is noticed that skin friction decreases sharply with anincrease in time and for 120596119905 = 0 it becomes negative whichmeans separation of boundary later occurswhich is for highervalues of 119905 (119905 gt 03) The figure depicts that skin frictionis higher for Sc = 096 in comparison to Sc = 022Physically it is correct since an increase in Sc serves toincrease momentum boundary layer thickness Further itdecreases with an increase in value of Gm and marginallyincreases with an increase in 119896 The increase of 120596119905 gives riseto an increase in the value of skin friction
6 ISRN Chemical Engineering
0 071
0 7
120596t Pr
6 071pi
76pi72pi
2 071pi
u
Figure 3 Velocity profile for Sc = 022 119905 = 02 119877 = 4 119896 = 02119866 = 5 and Gm = 2
0
02
04
06
08
1
12
14
16
0 1 2 3 4
7
7
7
7
Pr
u
120578
022
Sc022 02
060 02
096 02
t
minus02
04
071
071
071
071
Pr
022
Sc022 02
060 02
096 02
t
04
Figure 4 Velocity profile 120596119905 = 1205874 119877 = 4 119896 = 02 119866 = 5 andGm = 2
0
02
04
06
08
1
12
14
0 1 2 3 4
u
120578minus02
2 5 7
2 10 7
4 5 7
2 5 071
2 10 071
4 5 071
Gm G Pr Gm G Pr
Figure 5 Velocity profile for 120596119905 = 1205874 Sc = 022 119905 = 02 119877 = 4and 119896 = 02
0
02
04
06
08
1
12
0 05 1 15 2 25 3 35
u
120578minus02
02 7
15 7
02 071
15 071
k Pr k Pr
Figure 6 Velocity profile for 120596119905 = 1205874 Sc = 022 119905 = 02 119891 = 4119866 = 5 and Gm = 2
Figure 8 exhibits the Nusselt number against time It isconcluded from the figure that there is an increase in it withan increase in the value of radiation parameter Furthermorethe value of Nusselt number for water is greater than airIt is consistent with the fact that smaller values of Pr areequivalent to increasing thermal conductivities and therefore
ISRN Chemical Engineering 7
0
05
1
15
2
25
3
35
4
45
5
0 01 02 03 04
Sc Gm
120591
t
1205874
1205874
1205874
1205874
120596tk Sc Gm 120596tk022
022
022
022096
02
02
02
02
15
2
2
2
2
3
0
minus05
Figure 7 Skin friction for 119877 = 4 119866 = 5 Pr = 7
heat is able to diffuse away from the plate more rapidly thanhigher values of Pr hence the rate of heat transfer is reduced
Figure 9 depicts the effect of chemical reaction parameter119896 and Schmidt number Sc on Sherwood number It isobserved that Sherwood number increases with an increasein 119896 and Sc Since increase in Sc means decrease in moleculardiffusivity this in turn gives rise to increase in Sherwoodnumber as Sherwood number is the ratio of convective anddiffusive mass transfer coefficient Chemical reaction param-eter increases the interfacial mass transfer so Sherwoodnumber increases with an increase in 119896
7 Conclusion
A semi-analytical study has been carried out for flow pasta vertical surface in the presence of radiation and chemicalreaction when heat is supplied to the plate at constantrate The dimensionless governing equations are solved byusing Laplace transform technique Numerical evaluations ofclosed form solutions were performed and some graphicalresults were obtained to illustrate the details of flow heat andmass transfer characteristics and their dependence on somephysical parameters The physical aspects of the problem arealso discussed The results of the flow problem indicates thefollowing
(1) Increasing radiation parameter and Prandtl Numberthe temperature decreases whereas it increases withincreasing time
(2) Concentration profile decreases with an increase intime and Schmidt number
0
1
2
3
4
5
6
0 02 04 06 08 1 12
Nu
02 7
04 7
02 071
04 071
R
t
Pr R Pr
Figure 8 Nusselt number
0
005
01
015
02
025
03
035
04
045
0 2 4 6 8 10
Sh
Sck
022
060
022
02
02
04
t
Figure 9 Sherwood number
(3) Fluid velocity decreases with an increase in Schmidtnumber phase angle Prandtl number and chemicalreaction parameter but increases with an increasein Grashof number modified Grashof number andtime
8 ISRN Chemical Engineering
(4) There is a rise in skin frictionwith increasing Schmidtnumber chemical reaction parameter and phaseangle and fall with an increase in value of modifiedGrashof number The rate of heat transfer increaseswith an increase in the value of radiation param-eter and Prandtl number Sherwood number alsoincreases with an increase in chemical reactionparameter and Schmidt number
Nomenclature
119880119877 Reference velocity
119892 Gravitational acceleration119862119901 Specific heat at constant pressure
119863 Mass diffusivity120573 Thermal expansion coefficient120573119862 Concentration expansion coefficient
120588 Density120581 Thermal conductivity of fluid120581lowast Mean absorption coefficient
120590 Electrical conductivity of fluid] Kinematic viscosity119902119903 Radiative heat flux
1205901015840 Stefan-Boltzmann constant
119871119877 Reference length
119905119877 Reference time
Gm Modified Grashof numberPr Prandtl numberSc Schmidt number120596 Frequency of oscillation119906 Dimensionless velocity component120579 Dimensionless temperature119862 Dimensionless concentration120583 Viscosity of fluid119905 Time in dimensionless coordinate119877 Radiation parameter119896 Chemical reaction parameter
References
[1] S T Revankar ldquoFree convection effect on a flow past an impul-sively started or oscillating infinite vertical platerdquo MechanicsResearch Communications vol 27 no 2 pp 241ndash246 2000
[2] J Li D B Ingham and I Pop ldquoNatural convection froma vertical flat plate with a surface temperature oscillationrdquoInternational Journal of Heat and Mass Transfer vol 44 no 12pp 2311ndash2322 2001
[3] E L Cussler Diffusion Mass Transfer in Fluid Systems Cam-bridge University Press Cambridge UK 1998
[4] R C Chaudhary and J Arpita ldquoCombined heat and masstransfer effect on MHD free convection flow past an oscillatingplate embedded in porous mediumrdquo Romanian Journal ofPhysics vol 52 pp 505ndash524 2007
[5] R Muthucumaraswamy and A Vijayalakshmi ldquoEffects of heatand mass transfer on flow past an oscillating vertical platewith variable temperaturerdquo International Journal of AppliedMathematics and Mechanics vol 4 pp 59ndash65 2008
[6] R Muthucumaraswamy M Raj Sundar and V S A Subra-manian ldquoUnsteady flow past an accelerated infinite vertical
plate with variable temperature and uniform mass diffusionrdquoInternational Journal of Applied Mathematics and Mechanicsvol 5 pp 51ndash56 2009
[7] U S Rajput and S Kumar ldquoMHD flow past an impulsivelystarted vertical plate with variable temperature and mass diffu-sionrdquoAppliedMathematical Sciences vol 5 no 1ndash4 pp 149ndash1572011
[8] G Juncu ldquoUnsteady forced convection heatmass transfer froma flat platerdquoHeat andMass TransferWaerme- und Stoffuebertra-gung vol 41 no 12 pp 1095ndash1102 2005
[9] U N Das R Deka and V M Soundalgekar ldquoEffects of masstransfer on flowpast an impulsively started infinite vertical platewith constant heat flux and chemical reactionrdquo Forschung imIngenieurwesenEngineering Research vol 60 no 10 pp 284ndash287 1994
[10] R Muthucumaraswamy and S Meenakshisundaram ldquoTheoret-ical study of chemical reaction effects on vertical oscillatingplate with variable temperature Theoretrdquo Journal of AppliedMechanics vol 33 pp 245ndash257 2006
[11] B C Neog and K R Das Rudra ldquoUnsteady Free ConvectionMHD Flow past a vertical plate with variable temperatureand chemical reactionrdquo International Journal of EngineeringResearch amp Technology vol 1 pp 1ndash5 2012
[12] V Rajesh ldquoRadiation effects onMHD free convection flow neara vertical plate with ramped wall temperaturerdquo InternationalJournal of Applied Mathematics and Mechanics vol 6 pp 60ndash77 2010
[13] M Muralidharan and R Muthucumaraswamy ldquoThermal radi-ation on linearly accelerated vertical plate with variable tem-perature and uniform mass fluxrdquo Indian Journal of Science andTechnology vol 3 pp 398ndash401 2010
[14] U S Rajput and S Kumar ldquoRadiation effects on MHD flowpast an impulsively started vertical plate with variable heat andmass transferrdquo International Journal of AppliedMathematics andMechanics vol 8 pp 66ndash85 2012
[15] R C Chaudhary and A Jain ldquoUnsteady free convection flowpast an oscillating plate with constant mass flux in the presenceof radiationrdquoActaTechnicaCSAV vol 52 no 1 pp 93ndash108 2007
[16] A Ogulu and O D Makinde ldquoUnsteady hydromagnetic freeconvection flow of a dissipative and radiating fluid past avertical plate with constant heat fluxrdquo Chemical EngineeringCommunications vol 196 no 4 pp 454ndash462 2009
[17] M Narahari and L Debnath ldquoUnsteady magnetohydrody-namic free convection flow past an accelerated vertical platewith constant heat flux and heat generation or absorptionrdquoZeitschrift fur Angewandte Mathematik und Mechanik AppliedMathematics and Mechanics vol 93 pp 38ndash49 2013
[18] R C Chaudhary and J Arpita ldquoFree convection effects onMHD flow past an infinite vertical accelerated plate embeddedin porous media with constant heat fluxrdquoMatematicas pp 73ndash82 2009
[19] B M Abramowitz and I A StegunHandbook of MathematicalFunctions Dover New York NY USA 1970
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
ISRN Chemical Engineering 5
0
01
02
03
04
05
06
07
0 05 1 15 2 25
minus01
4 02 7
15 02 7
4 06 7
4 02 071
15 02 071
4 06 071
R t Pr R t Pr
120578
120579
Figure 1 Temperature profile
This is due to the fact that thermal conductivity of fluiddecreases with increasing Pr resulting in a decrease in ther-mal boundary layer thickness It is also seen that it decreasessteeply for Pr = 7 than that of Pr = 071 Furthermore itis noticed that an increase in radiation parameter from 4 to15 the temperature marginally decreases It is obvious sincethe radiation parameter defines the relative contribution ofconduction heat transfer to the thermal radiation transferFurther it increases with an increase in time
The species concentration profile against 120578 is plotted inFigure 2 It is observed that the concentration at all points inthe flow field decreases with 120578 and tends to zero as 120578 rarr infinFurthermore an increase in the value of Sc leads to a fall in theconcentration Physically it is true since increase of Sc meansdecrease of molecular diffusivity which results in decrease ofconcentration boundary layer Hence the concentration ofspecies is higher for small values of Sc It is also observed thatconcentration decreases with an increase in time
Figure 3 elucidates the effect of 120596119905 and Pr on the velocityIt is evident from Figure 3 that the velocity increases andattains itsmaximumvalue in the vicinity of the plate (120578 le 05)and then tends to zero asymptotically It is also noted thatboth the velocity and the penetration for Pr = 071 is higherthan those of Pr = 7 Physically it is possible because fluidswith high Prandtl number have high viscosity and hencemove slowly It is also seen that the velocity decreases withan increase in 120596119905 Figure 4 illustrates the influences of Sc119905 and Pr on the velocity It is noticed that the maximumvelocity attains near the plate then decreases and vanishesfar away from the plate When Pr = 7 and 071 it decreasescontinuously to asymptotic value Further they also increaseswith an increase in time at each point in the flow field forhydrogen gas Moreover the effect of time on the velocity is
0
02
04
06
08
1
12
0 1 2 3 4 5 6
Sc
Sc
minus02
022 02
060 02
022 10
t
120578
Figure 2 Concentration profile for 119896 = 02
more dominant than other parameters Finally It is observedthat the velocity for hydrogen (Sc = 022) is higher thanthat of water vapour (Sc = 060) and carbon dioxide (Sc =
096) for both air and water Physically it is possible sincean increase in Sc means increase in kinematic viscosity orviscosity of fluid due to which the velocity of fluid decreasesFigure 5 illustrates the influences of Gm 119866 and Pr on thevelocityThe figure reveals that the maximum velocity attainsnear the plate then decreases and vanish far away from theplate Moreover It is observed that an increase in119866 Gm leadsto an increase in velocity for both Pr = 7 and Pr = 071
when Hydrogen gas is present in the flow The reason is thatthe values of Grashof number and modified Grashof numberhave the tendency to increase the mass buoyancy effect Theincrease in velocity due to increase in 119866 and Gm is nearerthe plate than away from the plate Figure 6 elucidates theeffect of 119896 Pr on velocity profile It is found that the velocitydecreases with an increase in chemical reaction parameter 119896for both Pr = 7 and 071 It is due to the fact that increasein chemical reaction parameter 119896 gives rise to an increasein viscosity of fluid which means velocity boundary layerthickness decreases
Figure 7 elucidates the effects of parameters Sc 119896 119866and 120596119905 on skin friction at the plate at different values of119905 It is noticed that skin friction decreases sharply with anincrease in time and for 120596119905 = 0 it becomes negative whichmeans separation of boundary later occurswhich is for highervalues of 119905 (119905 gt 03) The figure depicts that skin frictionis higher for Sc = 096 in comparison to Sc = 022Physically it is correct since an increase in Sc serves toincrease momentum boundary layer thickness Further itdecreases with an increase in value of Gm and marginallyincreases with an increase in 119896 The increase of 120596119905 gives riseto an increase in the value of skin friction
6 ISRN Chemical Engineering
0 071
0 7
120596t Pr
6 071pi
76pi72pi
2 071pi
u
Figure 3 Velocity profile for Sc = 022 119905 = 02 119877 = 4 119896 = 02119866 = 5 and Gm = 2
0
02
04
06
08
1
12
14
16
0 1 2 3 4
7
7
7
7
Pr
u
120578
022
Sc022 02
060 02
096 02
t
minus02
04
071
071
071
071
Pr
022
Sc022 02
060 02
096 02
t
04
Figure 4 Velocity profile 120596119905 = 1205874 119877 = 4 119896 = 02 119866 = 5 andGm = 2
0
02
04
06
08
1
12
14
0 1 2 3 4
u
120578minus02
2 5 7
2 10 7
4 5 7
2 5 071
2 10 071
4 5 071
Gm G Pr Gm G Pr
Figure 5 Velocity profile for 120596119905 = 1205874 Sc = 022 119905 = 02 119877 = 4and 119896 = 02
0
02
04
06
08
1
12
0 05 1 15 2 25 3 35
u
120578minus02
02 7
15 7
02 071
15 071
k Pr k Pr
Figure 6 Velocity profile for 120596119905 = 1205874 Sc = 022 119905 = 02 119891 = 4119866 = 5 and Gm = 2
Figure 8 exhibits the Nusselt number against time It isconcluded from the figure that there is an increase in it withan increase in the value of radiation parameter Furthermorethe value of Nusselt number for water is greater than airIt is consistent with the fact that smaller values of Pr areequivalent to increasing thermal conductivities and therefore
ISRN Chemical Engineering 7
0
05
1
15
2
25
3
35
4
45
5
0 01 02 03 04
Sc Gm
120591
t
1205874
1205874
1205874
1205874
120596tk Sc Gm 120596tk022
022
022
022096
02
02
02
02
15
2
2
2
2
3
0
minus05
Figure 7 Skin friction for 119877 = 4 119866 = 5 Pr = 7
heat is able to diffuse away from the plate more rapidly thanhigher values of Pr hence the rate of heat transfer is reduced
Figure 9 depicts the effect of chemical reaction parameter119896 and Schmidt number Sc on Sherwood number It isobserved that Sherwood number increases with an increasein 119896 and Sc Since increase in Sc means decrease in moleculardiffusivity this in turn gives rise to increase in Sherwoodnumber as Sherwood number is the ratio of convective anddiffusive mass transfer coefficient Chemical reaction param-eter increases the interfacial mass transfer so Sherwoodnumber increases with an increase in 119896
7 Conclusion
A semi-analytical study has been carried out for flow pasta vertical surface in the presence of radiation and chemicalreaction when heat is supplied to the plate at constantrate The dimensionless governing equations are solved byusing Laplace transform technique Numerical evaluations ofclosed form solutions were performed and some graphicalresults were obtained to illustrate the details of flow heat andmass transfer characteristics and their dependence on somephysical parameters The physical aspects of the problem arealso discussed The results of the flow problem indicates thefollowing
(1) Increasing radiation parameter and Prandtl Numberthe temperature decreases whereas it increases withincreasing time
(2) Concentration profile decreases with an increase intime and Schmidt number
0
1
2
3
4
5
6
0 02 04 06 08 1 12
Nu
02 7
04 7
02 071
04 071
R
t
Pr R Pr
Figure 8 Nusselt number
0
005
01
015
02
025
03
035
04
045
0 2 4 6 8 10
Sh
Sck
022
060
022
02
02
04
t
Figure 9 Sherwood number
(3) Fluid velocity decreases with an increase in Schmidtnumber phase angle Prandtl number and chemicalreaction parameter but increases with an increasein Grashof number modified Grashof number andtime
8 ISRN Chemical Engineering
(4) There is a rise in skin frictionwith increasing Schmidtnumber chemical reaction parameter and phaseangle and fall with an increase in value of modifiedGrashof number The rate of heat transfer increaseswith an increase in the value of radiation param-eter and Prandtl number Sherwood number alsoincreases with an increase in chemical reactionparameter and Schmidt number
Nomenclature
119880119877 Reference velocity
119892 Gravitational acceleration119862119901 Specific heat at constant pressure
119863 Mass diffusivity120573 Thermal expansion coefficient120573119862 Concentration expansion coefficient
120588 Density120581 Thermal conductivity of fluid120581lowast Mean absorption coefficient
120590 Electrical conductivity of fluid] Kinematic viscosity119902119903 Radiative heat flux
1205901015840 Stefan-Boltzmann constant
119871119877 Reference length
119905119877 Reference time
Gm Modified Grashof numberPr Prandtl numberSc Schmidt number120596 Frequency of oscillation119906 Dimensionless velocity component120579 Dimensionless temperature119862 Dimensionless concentration120583 Viscosity of fluid119905 Time in dimensionless coordinate119877 Radiation parameter119896 Chemical reaction parameter
References
[1] S T Revankar ldquoFree convection effect on a flow past an impul-sively started or oscillating infinite vertical platerdquo MechanicsResearch Communications vol 27 no 2 pp 241ndash246 2000
[2] J Li D B Ingham and I Pop ldquoNatural convection froma vertical flat plate with a surface temperature oscillationrdquoInternational Journal of Heat and Mass Transfer vol 44 no 12pp 2311ndash2322 2001
[3] E L Cussler Diffusion Mass Transfer in Fluid Systems Cam-bridge University Press Cambridge UK 1998
[4] R C Chaudhary and J Arpita ldquoCombined heat and masstransfer effect on MHD free convection flow past an oscillatingplate embedded in porous mediumrdquo Romanian Journal ofPhysics vol 52 pp 505ndash524 2007
[5] R Muthucumaraswamy and A Vijayalakshmi ldquoEffects of heatand mass transfer on flow past an oscillating vertical platewith variable temperaturerdquo International Journal of AppliedMathematics and Mechanics vol 4 pp 59ndash65 2008
[6] R Muthucumaraswamy M Raj Sundar and V S A Subra-manian ldquoUnsteady flow past an accelerated infinite vertical
plate with variable temperature and uniform mass diffusionrdquoInternational Journal of Applied Mathematics and Mechanicsvol 5 pp 51ndash56 2009
[7] U S Rajput and S Kumar ldquoMHD flow past an impulsivelystarted vertical plate with variable temperature and mass diffu-sionrdquoAppliedMathematical Sciences vol 5 no 1ndash4 pp 149ndash1572011
[8] G Juncu ldquoUnsteady forced convection heatmass transfer froma flat platerdquoHeat andMass TransferWaerme- und Stoffuebertra-gung vol 41 no 12 pp 1095ndash1102 2005
[9] U N Das R Deka and V M Soundalgekar ldquoEffects of masstransfer on flowpast an impulsively started infinite vertical platewith constant heat flux and chemical reactionrdquo Forschung imIngenieurwesenEngineering Research vol 60 no 10 pp 284ndash287 1994
[10] R Muthucumaraswamy and S Meenakshisundaram ldquoTheoret-ical study of chemical reaction effects on vertical oscillatingplate with variable temperature Theoretrdquo Journal of AppliedMechanics vol 33 pp 245ndash257 2006
[11] B C Neog and K R Das Rudra ldquoUnsteady Free ConvectionMHD Flow past a vertical plate with variable temperatureand chemical reactionrdquo International Journal of EngineeringResearch amp Technology vol 1 pp 1ndash5 2012
[12] V Rajesh ldquoRadiation effects onMHD free convection flow neara vertical plate with ramped wall temperaturerdquo InternationalJournal of Applied Mathematics and Mechanics vol 6 pp 60ndash77 2010
[13] M Muralidharan and R Muthucumaraswamy ldquoThermal radi-ation on linearly accelerated vertical plate with variable tem-perature and uniform mass fluxrdquo Indian Journal of Science andTechnology vol 3 pp 398ndash401 2010
[14] U S Rajput and S Kumar ldquoRadiation effects on MHD flowpast an impulsively started vertical plate with variable heat andmass transferrdquo International Journal of AppliedMathematics andMechanics vol 8 pp 66ndash85 2012
[15] R C Chaudhary and A Jain ldquoUnsteady free convection flowpast an oscillating plate with constant mass flux in the presenceof radiationrdquoActaTechnicaCSAV vol 52 no 1 pp 93ndash108 2007
[16] A Ogulu and O D Makinde ldquoUnsteady hydromagnetic freeconvection flow of a dissipative and radiating fluid past avertical plate with constant heat fluxrdquo Chemical EngineeringCommunications vol 196 no 4 pp 454ndash462 2009
[17] M Narahari and L Debnath ldquoUnsteady magnetohydrody-namic free convection flow past an accelerated vertical platewith constant heat flux and heat generation or absorptionrdquoZeitschrift fur Angewandte Mathematik und Mechanik AppliedMathematics and Mechanics vol 93 pp 38ndash49 2013
[18] R C Chaudhary and J Arpita ldquoFree convection effects onMHD flow past an infinite vertical accelerated plate embeddedin porous media with constant heat fluxrdquoMatematicas pp 73ndash82 2009
[19] B M Abramowitz and I A StegunHandbook of MathematicalFunctions Dover New York NY USA 1970
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 ISRN Chemical Engineering
0 071
0 7
120596t Pr
6 071pi
76pi72pi
2 071pi
u
Figure 3 Velocity profile for Sc = 022 119905 = 02 119877 = 4 119896 = 02119866 = 5 and Gm = 2
0
02
04
06
08
1
12
14
16
0 1 2 3 4
7
7
7
7
Pr
u
120578
022
Sc022 02
060 02
096 02
t
minus02
04
071
071
071
071
Pr
022
Sc022 02
060 02
096 02
t
04
Figure 4 Velocity profile 120596119905 = 1205874 119877 = 4 119896 = 02 119866 = 5 andGm = 2
0
02
04
06
08
1
12
14
0 1 2 3 4
u
120578minus02
2 5 7
2 10 7
4 5 7
2 5 071
2 10 071
4 5 071
Gm G Pr Gm G Pr
Figure 5 Velocity profile for 120596119905 = 1205874 Sc = 022 119905 = 02 119877 = 4and 119896 = 02
0
02
04
06
08
1
12
0 05 1 15 2 25 3 35
u
120578minus02
02 7
15 7
02 071
15 071
k Pr k Pr
Figure 6 Velocity profile for 120596119905 = 1205874 Sc = 022 119905 = 02 119891 = 4119866 = 5 and Gm = 2
Figure 8 exhibits the Nusselt number against time It isconcluded from the figure that there is an increase in it withan increase in the value of radiation parameter Furthermorethe value of Nusselt number for water is greater than airIt is consistent with the fact that smaller values of Pr areequivalent to increasing thermal conductivities and therefore
ISRN Chemical Engineering 7
0
05
1
15
2
25
3
35
4
45
5
0 01 02 03 04
Sc Gm
120591
t
1205874
1205874
1205874
1205874
120596tk Sc Gm 120596tk022
022
022
022096
02
02
02
02
15
2
2
2
2
3
0
minus05
Figure 7 Skin friction for 119877 = 4 119866 = 5 Pr = 7
heat is able to diffuse away from the plate more rapidly thanhigher values of Pr hence the rate of heat transfer is reduced
Figure 9 depicts the effect of chemical reaction parameter119896 and Schmidt number Sc on Sherwood number It isobserved that Sherwood number increases with an increasein 119896 and Sc Since increase in Sc means decrease in moleculardiffusivity this in turn gives rise to increase in Sherwoodnumber as Sherwood number is the ratio of convective anddiffusive mass transfer coefficient Chemical reaction param-eter increases the interfacial mass transfer so Sherwoodnumber increases with an increase in 119896
7 Conclusion
A semi-analytical study has been carried out for flow pasta vertical surface in the presence of radiation and chemicalreaction when heat is supplied to the plate at constantrate The dimensionless governing equations are solved byusing Laplace transform technique Numerical evaluations ofclosed form solutions were performed and some graphicalresults were obtained to illustrate the details of flow heat andmass transfer characteristics and their dependence on somephysical parameters The physical aspects of the problem arealso discussed The results of the flow problem indicates thefollowing
(1) Increasing radiation parameter and Prandtl Numberthe temperature decreases whereas it increases withincreasing time
(2) Concentration profile decreases with an increase intime and Schmidt number
0
1
2
3
4
5
6
0 02 04 06 08 1 12
Nu
02 7
04 7
02 071
04 071
R
t
Pr R Pr
Figure 8 Nusselt number
0
005
01
015
02
025
03
035
04
045
0 2 4 6 8 10
Sh
Sck
022
060
022
02
02
04
t
Figure 9 Sherwood number
(3) Fluid velocity decreases with an increase in Schmidtnumber phase angle Prandtl number and chemicalreaction parameter but increases with an increasein Grashof number modified Grashof number andtime
8 ISRN Chemical Engineering
(4) There is a rise in skin frictionwith increasing Schmidtnumber chemical reaction parameter and phaseangle and fall with an increase in value of modifiedGrashof number The rate of heat transfer increaseswith an increase in the value of radiation param-eter and Prandtl number Sherwood number alsoincreases with an increase in chemical reactionparameter and Schmidt number
Nomenclature
119880119877 Reference velocity
119892 Gravitational acceleration119862119901 Specific heat at constant pressure
119863 Mass diffusivity120573 Thermal expansion coefficient120573119862 Concentration expansion coefficient
120588 Density120581 Thermal conductivity of fluid120581lowast Mean absorption coefficient
120590 Electrical conductivity of fluid] Kinematic viscosity119902119903 Radiative heat flux
1205901015840 Stefan-Boltzmann constant
119871119877 Reference length
119905119877 Reference time
Gm Modified Grashof numberPr Prandtl numberSc Schmidt number120596 Frequency of oscillation119906 Dimensionless velocity component120579 Dimensionless temperature119862 Dimensionless concentration120583 Viscosity of fluid119905 Time in dimensionless coordinate119877 Radiation parameter119896 Chemical reaction parameter
References
[1] S T Revankar ldquoFree convection effect on a flow past an impul-sively started or oscillating infinite vertical platerdquo MechanicsResearch Communications vol 27 no 2 pp 241ndash246 2000
[2] J Li D B Ingham and I Pop ldquoNatural convection froma vertical flat plate with a surface temperature oscillationrdquoInternational Journal of Heat and Mass Transfer vol 44 no 12pp 2311ndash2322 2001
[3] E L Cussler Diffusion Mass Transfer in Fluid Systems Cam-bridge University Press Cambridge UK 1998
[4] R C Chaudhary and J Arpita ldquoCombined heat and masstransfer effect on MHD free convection flow past an oscillatingplate embedded in porous mediumrdquo Romanian Journal ofPhysics vol 52 pp 505ndash524 2007
[5] R Muthucumaraswamy and A Vijayalakshmi ldquoEffects of heatand mass transfer on flow past an oscillating vertical platewith variable temperaturerdquo International Journal of AppliedMathematics and Mechanics vol 4 pp 59ndash65 2008
[6] R Muthucumaraswamy M Raj Sundar and V S A Subra-manian ldquoUnsteady flow past an accelerated infinite vertical
plate with variable temperature and uniform mass diffusionrdquoInternational Journal of Applied Mathematics and Mechanicsvol 5 pp 51ndash56 2009
[7] U S Rajput and S Kumar ldquoMHD flow past an impulsivelystarted vertical plate with variable temperature and mass diffu-sionrdquoAppliedMathematical Sciences vol 5 no 1ndash4 pp 149ndash1572011
[8] G Juncu ldquoUnsteady forced convection heatmass transfer froma flat platerdquoHeat andMass TransferWaerme- und Stoffuebertra-gung vol 41 no 12 pp 1095ndash1102 2005
[9] U N Das R Deka and V M Soundalgekar ldquoEffects of masstransfer on flowpast an impulsively started infinite vertical platewith constant heat flux and chemical reactionrdquo Forschung imIngenieurwesenEngineering Research vol 60 no 10 pp 284ndash287 1994
[10] R Muthucumaraswamy and S Meenakshisundaram ldquoTheoret-ical study of chemical reaction effects on vertical oscillatingplate with variable temperature Theoretrdquo Journal of AppliedMechanics vol 33 pp 245ndash257 2006
[11] B C Neog and K R Das Rudra ldquoUnsteady Free ConvectionMHD Flow past a vertical plate with variable temperatureand chemical reactionrdquo International Journal of EngineeringResearch amp Technology vol 1 pp 1ndash5 2012
[12] V Rajesh ldquoRadiation effects onMHD free convection flow neara vertical plate with ramped wall temperaturerdquo InternationalJournal of Applied Mathematics and Mechanics vol 6 pp 60ndash77 2010
[13] M Muralidharan and R Muthucumaraswamy ldquoThermal radi-ation on linearly accelerated vertical plate with variable tem-perature and uniform mass fluxrdquo Indian Journal of Science andTechnology vol 3 pp 398ndash401 2010
[14] U S Rajput and S Kumar ldquoRadiation effects on MHD flowpast an impulsively started vertical plate with variable heat andmass transferrdquo International Journal of AppliedMathematics andMechanics vol 8 pp 66ndash85 2012
[15] R C Chaudhary and A Jain ldquoUnsteady free convection flowpast an oscillating plate with constant mass flux in the presenceof radiationrdquoActaTechnicaCSAV vol 52 no 1 pp 93ndash108 2007
[16] A Ogulu and O D Makinde ldquoUnsteady hydromagnetic freeconvection flow of a dissipative and radiating fluid past avertical plate with constant heat fluxrdquo Chemical EngineeringCommunications vol 196 no 4 pp 454ndash462 2009
[17] M Narahari and L Debnath ldquoUnsteady magnetohydrody-namic free convection flow past an accelerated vertical platewith constant heat flux and heat generation or absorptionrdquoZeitschrift fur Angewandte Mathematik und Mechanik AppliedMathematics and Mechanics vol 93 pp 38ndash49 2013
[18] R C Chaudhary and J Arpita ldquoFree convection effects onMHD flow past an infinite vertical accelerated plate embeddedin porous media with constant heat fluxrdquoMatematicas pp 73ndash82 2009
[19] B M Abramowitz and I A StegunHandbook of MathematicalFunctions Dover New York NY USA 1970
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
ISRN Chemical Engineering 7
0
05
1
15
2
25
3
35
4
45
5
0 01 02 03 04
Sc Gm
120591
t
1205874
1205874
1205874
1205874
120596tk Sc Gm 120596tk022
022
022
022096
02
02
02
02
15
2
2
2
2
3
0
minus05
Figure 7 Skin friction for 119877 = 4 119866 = 5 Pr = 7
heat is able to diffuse away from the plate more rapidly thanhigher values of Pr hence the rate of heat transfer is reduced
Figure 9 depicts the effect of chemical reaction parameter119896 and Schmidt number Sc on Sherwood number It isobserved that Sherwood number increases with an increasein 119896 and Sc Since increase in Sc means decrease in moleculardiffusivity this in turn gives rise to increase in Sherwoodnumber as Sherwood number is the ratio of convective anddiffusive mass transfer coefficient Chemical reaction param-eter increases the interfacial mass transfer so Sherwoodnumber increases with an increase in 119896
7 Conclusion
A semi-analytical study has been carried out for flow pasta vertical surface in the presence of radiation and chemicalreaction when heat is supplied to the plate at constantrate The dimensionless governing equations are solved byusing Laplace transform technique Numerical evaluations ofclosed form solutions were performed and some graphicalresults were obtained to illustrate the details of flow heat andmass transfer characteristics and their dependence on somephysical parameters The physical aspects of the problem arealso discussed The results of the flow problem indicates thefollowing
(1) Increasing radiation parameter and Prandtl Numberthe temperature decreases whereas it increases withincreasing time
(2) Concentration profile decreases with an increase intime and Schmidt number
0
1
2
3
4
5
6
0 02 04 06 08 1 12
Nu
02 7
04 7
02 071
04 071
R
t
Pr R Pr
Figure 8 Nusselt number
0
005
01
015
02
025
03
035
04
045
0 2 4 6 8 10
Sh
Sck
022
060
022
02
02
04
t
Figure 9 Sherwood number
(3) Fluid velocity decreases with an increase in Schmidtnumber phase angle Prandtl number and chemicalreaction parameter but increases with an increasein Grashof number modified Grashof number andtime
8 ISRN Chemical Engineering
(4) There is a rise in skin frictionwith increasing Schmidtnumber chemical reaction parameter and phaseangle and fall with an increase in value of modifiedGrashof number The rate of heat transfer increaseswith an increase in the value of radiation param-eter and Prandtl number Sherwood number alsoincreases with an increase in chemical reactionparameter and Schmidt number
Nomenclature
119880119877 Reference velocity
119892 Gravitational acceleration119862119901 Specific heat at constant pressure
119863 Mass diffusivity120573 Thermal expansion coefficient120573119862 Concentration expansion coefficient
120588 Density120581 Thermal conductivity of fluid120581lowast Mean absorption coefficient
120590 Electrical conductivity of fluid] Kinematic viscosity119902119903 Radiative heat flux
1205901015840 Stefan-Boltzmann constant
119871119877 Reference length
119905119877 Reference time
Gm Modified Grashof numberPr Prandtl numberSc Schmidt number120596 Frequency of oscillation119906 Dimensionless velocity component120579 Dimensionless temperature119862 Dimensionless concentration120583 Viscosity of fluid119905 Time in dimensionless coordinate119877 Radiation parameter119896 Chemical reaction parameter
References
[1] S T Revankar ldquoFree convection effect on a flow past an impul-sively started or oscillating infinite vertical platerdquo MechanicsResearch Communications vol 27 no 2 pp 241ndash246 2000
[2] J Li D B Ingham and I Pop ldquoNatural convection froma vertical flat plate with a surface temperature oscillationrdquoInternational Journal of Heat and Mass Transfer vol 44 no 12pp 2311ndash2322 2001
[3] E L Cussler Diffusion Mass Transfer in Fluid Systems Cam-bridge University Press Cambridge UK 1998
[4] R C Chaudhary and J Arpita ldquoCombined heat and masstransfer effect on MHD free convection flow past an oscillatingplate embedded in porous mediumrdquo Romanian Journal ofPhysics vol 52 pp 505ndash524 2007
[5] R Muthucumaraswamy and A Vijayalakshmi ldquoEffects of heatand mass transfer on flow past an oscillating vertical platewith variable temperaturerdquo International Journal of AppliedMathematics and Mechanics vol 4 pp 59ndash65 2008
[6] R Muthucumaraswamy M Raj Sundar and V S A Subra-manian ldquoUnsteady flow past an accelerated infinite vertical
plate with variable temperature and uniform mass diffusionrdquoInternational Journal of Applied Mathematics and Mechanicsvol 5 pp 51ndash56 2009
[7] U S Rajput and S Kumar ldquoMHD flow past an impulsivelystarted vertical plate with variable temperature and mass diffu-sionrdquoAppliedMathematical Sciences vol 5 no 1ndash4 pp 149ndash1572011
[8] G Juncu ldquoUnsteady forced convection heatmass transfer froma flat platerdquoHeat andMass TransferWaerme- und Stoffuebertra-gung vol 41 no 12 pp 1095ndash1102 2005
[9] U N Das R Deka and V M Soundalgekar ldquoEffects of masstransfer on flowpast an impulsively started infinite vertical platewith constant heat flux and chemical reactionrdquo Forschung imIngenieurwesenEngineering Research vol 60 no 10 pp 284ndash287 1994
[10] R Muthucumaraswamy and S Meenakshisundaram ldquoTheoret-ical study of chemical reaction effects on vertical oscillatingplate with variable temperature Theoretrdquo Journal of AppliedMechanics vol 33 pp 245ndash257 2006
[11] B C Neog and K R Das Rudra ldquoUnsteady Free ConvectionMHD Flow past a vertical plate with variable temperatureand chemical reactionrdquo International Journal of EngineeringResearch amp Technology vol 1 pp 1ndash5 2012
[12] V Rajesh ldquoRadiation effects onMHD free convection flow neara vertical plate with ramped wall temperaturerdquo InternationalJournal of Applied Mathematics and Mechanics vol 6 pp 60ndash77 2010
[13] M Muralidharan and R Muthucumaraswamy ldquoThermal radi-ation on linearly accelerated vertical plate with variable tem-perature and uniform mass fluxrdquo Indian Journal of Science andTechnology vol 3 pp 398ndash401 2010
[14] U S Rajput and S Kumar ldquoRadiation effects on MHD flowpast an impulsively started vertical plate with variable heat andmass transferrdquo International Journal of AppliedMathematics andMechanics vol 8 pp 66ndash85 2012
[15] R C Chaudhary and A Jain ldquoUnsteady free convection flowpast an oscillating plate with constant mass flux in the presenceof radiationrdquoActaTechnicaCSAV vol 52 no 1 pp 93ndash108 2007
[16] A Ogulu and O D Makinde ldquoUnsteady hydromagnetic freeconvection flow of a dissipative and radiating fluid past avertical plate with constant heat fluxrdquo Chemical EngineeringCommunications vol 196 no 4 pp 454ndash462 2009
[17] M Narahari and L Debnath ldquoUnsteady magnetohydrody-namic free convection flow past an accelerated vertical platewith constant heat flux and heat generation or absorptionrdquoZeitschrift fur Angewandte Mathematik und Mechanik AppliedMathematics and Mechanics vol 93 pp 38ndash49 2013
[18] R C Chaudhary and J Arpita ldquoFree convection effects onMHD flow past an infinite vertical accelerated plate embeddedin porous media with constant heat fluxrdquoMatematicas pp 73ndash82 2009
[19] B M Abramowitz and I A StegunHandbook of MathematicalFunctions Dover New York NY USA 1970
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
8 ISRN Chemical Engineering
(4) There is a rise in skin frictionwith increasing Schmidtnumber chemical reaction parameter and phaseangle and fall with an increase in value of modifiedGrashof number The rate of heat transfer increaseswith an increase in the value of radiation param-eter and Prandtl number Sherwood number alsoincreases with an increase in chemical reactionparameter and Schmidt number
Nomenclature
119880119877 Reference velocity
119892 Gravitational acceleration119862119901 Specific heat at constant pressure
119863 Mass diffusivity120573 Thermal expansion coefficient120573119862 Concentration expansion coefficient
120588 Density120581 Thermal conductivity of fluid120581lowast Mean absorption coefficient
120590 Electrical conductivity of fluid] Kinematic viscosity119902119903 Radiative heat flux
1205901015840 Stefan-Boltzmann constant
119871119877 Reference length
119905119877 Reference time
Gm Modified Grashof numberPr Prandtl numberSc Schmidt number120596 Frequency of oscillation119906 Dimensionless velocity component120579 Dimensionless temperature119862 Dimensionless concentration120583 Viscosity of fluid119905 Time in dimensionless coordinate119877 Radiation parameter119896 Chemical reaction parameter
References
[1] S T Revankar ldquoFree convection effect on a flow past an impul-sively started or oscillating infinite vertical platerdquo MechanicsResearch Communications vol 27 no 2 pp 241ndash246 2000
[2] J Li D B Ingham and I Pop ldquoNatural convection froma vertical flat plate with a surface temperature oscillationrdquoInternational Journal of Heat and Mass Transfer vol 44 no 12pp 2311ndash2322 2001
[3] E L Cussler Diffusion Mass Transfer in Fluid Systems Cam-bridge University Press Cambridge UK 1998
[4] R C Chaudhary and J Arpita ldquoCombined heat and masstransfer effect on MHD free convection flow past an oscillatingplate embedded in porous mediumrdquo Romanian Journal ofPhysics vol 52 pp 505ndash524 2007
[5] R Muthucumaraswamy and A Vijayalakshmi ldquoEffects of heatand mass transfer on flow past an oscillating vertical platewith variable temperaturerdquo International Journal of AppliedMathematics and Mechanics vol 4 pp 59ndash65 2008
[6] R Muthucumaraswamy M Raj Sundar and V S A Subra-manian ldquoUnsteady flow past an accelerated infinite vertical
plate with variable temperature and uniform mass diffusionrdquoInternational Journal of Applied Mathematics and Mechanicsvol 5 pp 51ndash56 2009
[7] U S Rajput and S Kumar ldquoMHD flow past an impulsivelystarted vertical plate with variable temperature and mass diffu-sionrdquoAppliedMathematical Sciences vol 5 no 1ndash4 pp 149ndash1572011
[8] G Juncu ldquoUnsteady forced convection heatmass transfer froma flat platerdquoHeat andMass TransferWaerme- und Stoffuebertra-gung vol 41 no 12 pp 1095ndash1102 2005
[9] U N Das R Deka and V M Soundalgekar ldquoEffects of masstransfer on flowpast an impulsively started infinite vertical platewith constant heat flux and chemical reactionrdquo Forschung imIngenieurwesenEngineering Research vol 60 no 10 pp 284ndash287 1994
[10] R Muthucumaraswamy and S Meenakshisundaram ldquoTheoret-ical study of chemical reaction effects on vertical oscillatingplate with variable temperature Theoretrdquo Journal of AppliedMechanics vol 33 pp 245ndash257 2006
[11] B C Neog and K R Das Rudra ldquoUnsteady Free ConvectionMHD Flow past a vertical plate with variable temperatureand chemical reactionrdquo International Journal of EngineeringResearch amp Technology vol 1 pp 1ndash5 2012
[12] V Rajesh ldquoRadiation effects onMHD free convection flow neara vertical plate with ramped wall temperaturerdquo InternationalJournal of Applied Mathematics and Mechanics vol 6 pp 60ndash77 2010
[13] M Muralidharan and R Muthucumaraswamy ldquoThermal radi-ation on linearly accelerated vertical plate with variable tem-perature and uniform mass fluxrdquo Indian Journal of Science andTechnology vol 3 pp 398ndash401 2010
[14] U S Rajput and S Kumar ldquoRadiation effects on MHD flowpast an impulsively started vertical plate with variable heat andmass transferrdquo International Journal of AppliedMathematics andMechanics vol 8 pp 66ndash85 2012
[15] R C Chaudhary and A Jain ldquoUnsteady free convection flowpast an oscillating plate with constant mass flux in the presenceof radiationrdquoActaTechnicaCSAV vol 52 no 1 pp 93ndash108 2007
[16] A Ogulu and O D Makinde ldquoUnsteady hydromagnetic freeconvection flow of a dissipative and radiating fluid past avertical plate with constant heat fluxrdquo Chemical EngineeringCommunications vol 196 no 4 pp 454ndash462 2009
[17] M Narahari and L Debnath ldquoUnsteady magnetohydrody-namic free convection flow past an accelerated vertical platewith constant heat flux and heat generation or absorptionrdquoZeitschrift fur Angewandte Mathematik und Mechanik AppliedMathematics and Mechanics vol 93 pp 38ndash49 2013
[18] R C Chaudhary and J Arpita ldquoFree convection effects onMHD flow past an infinite vertical accelerated plate embeddedin porous media with constant heat fluxrdquoMatematicas pp 73ndash82 2009
[19] B M Abramowitz and I A StegunHandbook of MathematicalFunctions Dover New York NY USA 1970
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Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of