Research Article Radiation and Chemical Reaction Effects...

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


(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

)


+

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)


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



(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


) 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


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


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


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


(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

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



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)



Pr120597120579120597119905

= (1 +

4

3119877

)

1205972120579

1205971199102

120597119862

120597119905

=

1

Sc1205972119862

1205971199102minus 119896119862

120597119906

120597119905

=

1205972119906

1205971199102+ 119866120579 + Gm119862

(9)


119906 = 0 120579 = 0 119862 = 0 forall119910 119905 le 0

119906 = cos120596t 120597120579

120597119910

= minus1 119862 = 1 at 119910 = 0 119905 gt 0


(10)


(1 +

4

3119877

)

1198892120579

1198891199102minus 119901Pr 120579 = 0

1198892120601

1198891199102(119896 + 119901 Sc) = 0

1198892119906


119862

(11)

119906 =

119901

1199012+ 1205962

119889120579

119889119910

= minus

1

119901

119862 =

1

119901

at 119910 = 0 119905 gt 0


(12)


get

120579 (119910 119901) = radic119867


11990132

119862 (119910 119901) =


119901

119906 (119910 119901) =

119901 exp (minus119910radic119901)

1199012+ 1205962

+ radic119867


11990152

((Pr119867) minus 1)

+



minus



minus radic119867


11990152

(

Pr119867

minus 1)

(13)


120579 = radic119905 119867

120587Prerfc(

minus1205782Pr119867


)

119862 =

1

2



(14)


119906 =

exp (119894120596119905)4



exp (minus119894120596119905)4



+ 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

)


(6 + 41205782 Pr119867


)


+

Gm119896 Sc


exp( 119896 Sc 119905

1 minus Sc)

times

(exp(2120578radic119896 Sc 119905


119896 Sc 119905

1 minus Sc)



119896 Sc 119905

1 minus Sc)

minus

Gm2119896 Sc

exp( 119896 Sc 119905

1 minus Sc)

times

(exp(2120578radic119896 Sc 119905


119896119905

1 minus Sc)



119896119905

1 minus Sc)

minus

Gm2119896 Sc



(15)


119906 =

exp (119894120596119905)4



+

exp (minus119894120596119905)4



(16)




1(1199091+ 1198941199101)

which is defined as

erfc (1199091+ 1198941199101) = 119882

1(minus1199101+ 1198941199091) exp minus(119909

1+ 1198941199101)2

(17)


1198821(minus1199091+ 1198941199101) = 119882

2(1199091+ 1198941199101)

1198821(1199091minus 1198941199101) = 2 exp minus(119909

1minus 1198941199101)2

minus1198822(1199091+ 1198941199101)

(18)


1(1199091+ 1198941199101)

3 Skin Friction


120591 = minus(

120597119906

120597119910

)

119910=0

120591 =

exp (119894120596119905)2radic119905

radic119894120596119905 erf (radic119894120596119905) +

exp (minus119894120596119905)2radic119905


119866119905119867

(Pr minus 119867)

radic119867

Pr(1 minus radic

Pr119867

)

+

1

radic120587119905

minus

Gm119896 Sc




+

Gm119896 Sc


) erf (radic119896119905

1 minus Sc)


minus

Gm119896 Sc


(19)

4 Nusselt Number


Nu = minus

1

120579 (0)

120597120579

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= +

1

120579 (0)

= radic120587Pr119905119867

(20)

5 Sherwood Number


Sh = minus

120597120593

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= radic119896 Sc2

(2 +

1

2

radicSc119905

exp (minus119896119905)) (21)

6 Discussion




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





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







0 071

0 7

120596t Pr

6 071pi

76pi72pi

2 071pi

u


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


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


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




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




7 Conclusion




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


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





Nomenclature










References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Pr120597120579120597119905

= (1 +

4

3119877

)

1205972120579

1205971199102

120597119862

120597119905

=

1

Sc1205972119862

1205971199102minus 119896119862

120597119906

120597119905

=

1205972119906

1205971199102+ 119866120579 + Gm119862

(9)


119906 = 0 120579 = 0 119862 = 0 forall119910 119905 le 0

119906 = cos120596t 120597120579

120597119910

= minus1 119862 = 1 at 119910 = 0 119905 gt 0


(10)


(1 +

4

3119877

)

1198892120579

1198891199102minus 119901Pr 120579 = 0

1198892120601

1198891199102(119896 + 119901 Sc) = 0

1198892119906


119862

(11)

119906 =

119901

1199012+ 1205962

119889120579

119889119910

= minus

1

119901

119862 =

1

119901

at 119910 = 0 119905 gt 0


(12)


get

120579 (119910 119901) = radic119867


11990132

119862 (119910 119901) =


119901

119906 (119910 119901) =

119901 exp (minus119910radic119901)

1199012+ 1205962

+ radic119867


11990152

((Pr119867) minus 1)

+



minus



minus radic119867


11990152

(

Pr119867

minus 1)

(13)


120579 = radic119905 119867

120587Prerfc(

minus1205782Pr119867


)

119862 =

1

2



(14)


119906 =

exp (119894120596119905)4



exp (minus119894120596119905)4



+ 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

)


(6 + 41205782 Pr119867


)


+

Gm119896 Sc


exp( 119896 Sc 119905

1 minus Sc)

times

(exp(2120578radic119896 Sc 119905


119896 Sc 119905

1 minus Sc)



119896 Sc 119905

1 minus Sc)

minus

Gm2119896 Sc

exp( 119896 Sc 119905

1 minus Sc)

times

(exp(2120578radic119896 Sc 119905


119896119905

1 minus Sc)



119896119905

1 minus Sc)

minus

Gm2119896 Sc



(15)


119906 =

exp (119894120596119905)4



+

exp (minus119894120596119905)4



(16)




1(1199091+ 1198941199101)

which is defined as

erfc (1199091+ 1198941199101) = 119882

1(minus1199101+ 1198941199091) exp minus(119909

1+ 1198941199101)2

(17)


1198821(minus1199091+ 1198941199101) = 119882

2(1199091+ 1198941199101)

1198821(1199091minus 1198941199101) = 2 exp minus(119909

1minus 1198941199101)2

minus1198822(1199091+ 1198941199101)

(18)


1(1199091+ 1198941199101)

3 Skin Friction


120591 = minus(

120597119906

120597119910

)

119910=0

120591 =

exp (119894120596119905)2radic119905

radic119894120596119905 erf (radic119894120596119905) +

exp (minus119894120596119905)2radic119905


119866119905119867

(Pr minus 119867)

radic119867

Pr(1 minus radic

Pr119867

)

+

1

radic120587119905

minus

Gm119896 Sc




+

Gm119896 Sc


) erf (radic119896119905

1 minus Sc)


minus

Gm119896 Sc


(19)

4 Nusselt Number


Nu = minus

1

120579 (0)

120597120579

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= +

1

120579 (0)

= radic120587Pr119905119867

(20)

5 Sherwood Number


Sh = minus

120597120593

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= radic119896 Sc2

(2 +

1

2

radicSc119905

exp (minus119896119905)) (21)

6 Discussion




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





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







0 071

0 7

120596t Pr

6 071pi

76pi72pi

2 071pi

u


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


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


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




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




7 Conclusion




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


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





Nomenclature










References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










+

Gm119896 Sc


exp( 119896 Sc 119905

1 minus Sc)

times

(exp(2120578radic119896 Sc 119905


119896 Sc 119905

1 minus Sc)



119896 Sc 119905

1 minus Sc)

minus

Gm2119896 Sc

exp( 119896 Sc 119905

1 minus Sc)

times

(exp(2120578radic119896 Sc 119905


119896119905

1 minus Sc)



119896119905

1 minus Sc)

minus

Gm2119896 Sc



(15)


119906 =

exp (119894120596119905)4



+

exp (minus119894120596119905)4



(16)




1(1199091+ 1198941199101)

which is defined as

erfc (1199091+ 1198941199101) = 119882

1(minus1199101+ 1198941199091) exp minus(119909

1+ 1198941199101)2

(17)


1198821(minus1199091+ 1198941199101) = 119882

2(1199091+ 1198941199101)

1198821(1199091minus 1198941199101) = 2 exp minus(119909

1minus 1198941199101)2

minus1198822(1199091+ 1198941199101)

(18)


1(1199091+ 1198941199101)

3 Skin Friction


120591 = minus(

120597119906

120597119910

)

119910=0

120591 =

exp (119894120596119905)2radic119905

radic119894120596119905 erf (radic119894120596119905) +

exp (minus119894120596119905)2radic119905


119866119905119867

(Pr minus 119867)

radic119867

Pr(1 minus radic

Pr119867

)

+

1

radic120587119905

minus

Gm119896 Sc




+

Gm119896 Sc


) erf (radic119896119905

1 minus Sc)


minus

Gm119896 Sc


(19)

4 Nusselt Number


Nu = minus

1

120579 (0)

120597120579

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= +

1

120579 (0)

= radic120587Pr119905119867

(20)

5 Sherwood Number


Sh = minus

120597120593

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= radic119896 Sc2

(2 +

1

2

radicSc119905

exp (minus119896119905)) (21)

6 Discussion




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





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







0 071

0 7

120596t Pr

6 071pi

76pi72pi

2 071pi

u


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


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


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




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




7 Conclusion




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


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





Nomenclature










References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










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





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







0 071

0 7

120596t Pr

6 071pi

76pi72pi

2 071pi

u


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


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


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




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




7 Conclusion




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


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





Nomenclature










References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 071

0 7

120596t Pr

6 071pi

76pi72pi

2 071pi

u


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


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


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




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




7 Conclusion




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


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





Nomenclature










References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










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




7 Conclusion




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


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





Nomenclature










References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Nomenclature










References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Radiation and Chemical Reaction Effects...

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