Finite Difference Solutions of MHD Natural Convective ... · with dissociating fluids. The possible...

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 3 (2019) pp. 703-716 © Research India Publications. http://www.ripublication.com

703

Finite Difference Solutions of MHD Natural Convective Rivlin-Ericksen

Fluid Flow Past a Vertically Inclined Porous Plate in Presence of Thermal

Diffusion, Diffusion Thermo, Heat and Mass Transfer Effects

V. Omeshwar Reddy*, A.Neelima1 and S. Thiagarajan2

* Asst. Professor, Department of Mathematics, TKR College of Engineering and Technology, Hyderabad, 500097, Telangana, India. 1 Asst. Professor, Department of Mathematics, MVSR Engineering College, Hyderabad, 501510, Telangana, India.

2 Professor, Department of Mathematics, Matrusri Engineering College, Saidabad, Hyderabad, 500059, Telangana, India. *Corresponding author

Abstract

In this research work, the combined effects of thermal diffusion (Soret) and diffusion thermo (Dufour) effects on an unsteady Rivlin-Ericksen fluid flow towards a vertically inclined plate in presence of magnetic field, thermal radiation, heat and mass transfer effects studied using numerical solutions. The effects of diffusion thermo and thermal radiation are considered in energy equation and the effects of diffusion thermo and chemical reaction are studied in concentration equation. The numerical solutions for the linear parabolic governing equations in the boundary layer are discussed by applying finite difference technique. The effects of various fluid flow parameters are discussed through graphs and tables. We found an excellent agreement of the present results by comparing with the published results. It is found that Soret and Dufour parameters regulate the heat and mass transfer rate. Nonlinear thermal radiation effectively enhances the thermal boundary layer thickness. Keywords: Thermal Diffusion; Rivlin-Ericksen fluid; Diffusion Thermo; MHD; Natural Convection; Finite difference method;

NOMENCLATURE

List of variables:

yx , Co-ordinate system ( m )

pC Specific heat at constant pressure KKgJ 1

oB Uniform Applied Magnetic field ( 1mA )

A A constant

g Acceleration of gravity ( 2sm )

wC Characteristic concentration of fluid ( 3mmol )

wT Characteristic temperature K

D Chemical molecular diffusivity ( 12 sm )

C Concentration of fluid in free stream ( 3mmol )

C Concentration of fluid within boundary layer

( 3mmol )

sC Concentration susceptibility ( 1molem )

rK Dimensional chemical reaction parameter

yx, Dimensionless coordinates ( m )

u Dimensionless velocity component in x direction

( 1sm )

Du Dufour number

T Fluid temperature within free stream ( K )

Gr Grashof number for heat transfer Gc Grashof number for mass transfer

K Homogeneous Chemical reaction of first order M Magnetic parameter which is square of Hartman number

mD Mass diffusivity ( 12 sm )

mT Mean fluid temperature K

t Non-dimensional time ( s ) Pr Prandtl number

rq Radiative heat flux

Nu Rate of heat transfer coefficient Sh Rate of mass transfer coefficient

oU Reference velocity ( 1sm )

xRe Reynold’s number

Sc Schmidt number Cf Skin-friction Coefficient ( pascal )

Sr Soret number *a Stefan-Boltzmann Constant

T Temperature of fluid within boundary layer K

K The permeability of medium ( 2m ) K The permeability parameter ( 2m )

Tk Thermal diffusion ratio

R Thermal radiation parameter t Time ( s ) u Velocity component in x direction


704

Greek Symbols

Coefficient of volume expansion for heat transfer

( 1K ) * Coefficient of volume expansion for mass transfer

( 13 Kgm )

Density of the fluid ( 3mKg )

Electrical conductivity of the fluid ( 1mS )

Kinematic viscosity ( 12 sm )

Non-dimensional concentration of fluid ( 3mmol )

Non-dimensional temperature ( K )

w Shear stress ( pascal )

Thermal conductivity of the fluid ( 11 KmW ) Angle of inclination of plate ( reesdeg )

Kr Chemical reaction parameter Rivlin-Ericksen fluid parameter

1 Kinematic viscoelasticity

Superscripts:

Dimensionless properties

Subscripts:

Free stream conditions p Plate

w Conditions on the wall

1. INTRODUCTION:

The study of heat and mass transfer with chemical reaction is of considerable importance in chemical and hydrometallurgical industries. The study of heat generation or absorption in moving fluid is important in many problems which are related with chemical reactions and those concerned with dissociating fluids. The possible effects of non-uniform heat generation effects may change the temperature distribution and consequently, the rate of particle deposition in nuclear reactors, semi-conductor waters and electronic chips. Chemical reaction can be codified as either heterogeneous or homogeneous processes. Geetha and Muthucumaraswamy [1] was studied the effects of thermal radiation and chemical reaction on unsteady free convection and mass transfer of a viscous incompressible fluid past an accelerated infinite isothermal vertical plate in the presence of magnetic field. Chamkha et al. [2] studied MHD mixed convection radiation interaction along a permeable surface immersed in a porous medium in the presence of Soret and Dufour effects. Prabhu et al. [3] considered the effects of chemical reaction, heat and mass transfer on MHD flow over a vertical stretching surface with heat source and thermal stratification effects. Postelnicu [4] studied the influence of chemical reaction on heat and mass transfer by natural convection from vertical surfaces in porous media by considering Soret and Dufour effects.

Ibrahim et al. [5] analysed the effects of chemical reaction and radiation absorption on the unsteady MHD free convection flow past a semi-infinite vertical permeable moving plate with heat source and suction. Unsteady natural convective power-law fluid flow past a vertical plate embedded in a non-Darcian porous medium in the presence of a homogeneous chemical reaction was studied by Chamkha [6]. Recently Pal and Mondal [7] studied the influence of chemical reaction and thermal radiation on mixed convection heat and mass transfer over a stretching sheet in Darcy porous medium with Soret and Dufour effects. Devi et al. [8] investigated chemical reaction effects on heat and mass transfer MHD boundary layer laminar type flow over a wedge considering suction/injection. Kandasamy et al. [9] studied chemical reactive MHD flow through a stretching surface with the effect of thermal stratification and heat source. The study of MHD temperature dependent viscosity and thermal diffusive flow with Hall and ion-slip currents with the influence of chemical reaction was carried out by Elgazery [10].The analysis of a first-order chemical reaction on MHD thermo-solutal Marangoni convection flow was examined by Zhang and Zheng [11]. Ibrahim et al. [12] studied the effect of chemical reaction and thermal radiation absorption on the unsteady MHD free convection flow past a semi-infinite vertical permeable moving plate with heat source and suction. They found that the velocity profiles and concentration profile increased due to a decrease in the chemical reaction parameter. Al-Odat and Al-Azab [13] studied the influence of chemical reaction on transient MHD free convection over a moving vertical plate. They found that the velocity as well as concentration decrease with increasing the chemical reaction parameter. Seddeek et al. [14] examined the effects of chemical reaction and variable viscosity on hydro magnetic mixed convection heat and mass transfer for Hiemenz flow through porous media with radiation, and they found that the local Sherwood number significantly increases with the chemical reaction parameter. Pal and Talukdar [15] used perturbation analysis to study unsteady magneto hydrodynamic convective heat and mass transfer in a boundary layer slip flow past a vertical permeable plate with thermal radiation and chemical reaction. They solved the non-linear coupled partial differential equations by perturbation technique and found that the velocity as well as concentration decrease with increasing the chemical reaction parameter. Simultaneous influence of heat and mass transfer have significant role in many physical, biological and chemical processes like distillation extraction, absorption, melting, crystallization, condensation and evaporation. Simultaneous effects of heat and mass transfer for peristaltic flow can be observed in oxygenation, haemodialysis and chemical impurities etc. Recently many investigators described the heat and mass transfer effects on flow involving different type of fluids ([16], [17], [18], [19], [20] and [21]). It is perceived that energy flux can be generated not only by temperature gradients but also due to concentration gradients. The mass flux induced by the temperature gradient is termed as Soret (thermal-diffusion) effect. If the energy fluxes are generated by concentration gradient then it is known as Dufour (diffusion-thermo) effect. In many studies these effects are ignored because of a small order of magnitude

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when compared with the effects characterized by the Fourier’s and Fick’s laws. The significant role of Soret effect can be seen in biological systems, solar ponds and the micro-structure of the world oceans. Soret effect is also used for isotope separation and in a mixture of gases with small molecular weight H2, He and of medium molecular weight N2, Air. Having such view point the peristaltic flow of third grade fluid through Soret and Dufour effects in curved channel is investigated by Hayat et al. [22]. MHD peristaltic transport of fluid saturating porous medium and thermal-diffusion and diffusion-thermo effects in a rotating system is studied by Hayat et al. [23]. Reddy et al. [24] examined the Soret and Dufour effects in MHD flow of Nano fluids with heat generation. They remarked that enhancing values of Soret number and reducing Dufour number for both (Al2O3-H2O) and (TiO2-H2O), the heat transfer rate increases and mass transfer rate decreases. Soret and Dufour effects on the flow of power-law fluid with thermophoretic effect is explored by Hsiao et al. [25]. They also considered the constant magnetic field and porous effects. Hayat et al. [26] analysed the peristaltic flow of Casson fluid in a channel with chemical reaction and thermal-diffusion and diffusion-thermo effects. Recently some attempts examined chemical reaction concept in heat and mass transfer analysis ([27] and [28]). Weaver et al. [29] have pointed out that when the differences of the temperature and the concentration are large or when the difference of the molecular mass of the two elements in a binary mixture is great, the coupled interaction is significant. Anghel et al. [30] concluded that thermal-diffusion (Soret) and diffusion-thermo (Dufour) effects appreciably influence the flow field in free convection. Mahdy [31] examined the combined effect of spatially stationary surface waves and the presence of fluid inertia on the free convection along a heated vertical wavy surface embedded in an electrically conducting fluid saturated porous medium, subject to the diffusion-thermo (Dufour), thermo-diffusion (Soret) and magnetic field effects. Mahdy [32] investigated the effect of Soret and Dufour effects on non-Newtonian mixed convection in porous media. The effect of melting and/or thermo-diffusion on convective transport in a non-Newtonian fluid saturated non-Darcy porous medium is presented by Kairi and Murthy [33] and Srinivasacharya and RamReddy [34]. Kundu et al. [35] discussed the problem of combined effects of thermophoresis and chemical reaction magneto hydrodynamics mixed convection flow. The influence of thermophoresis and chemical reaction on MHD micro polar fluid flow with variable fluid properties was investigated by Das [36]. Different authors studied heat and mass transfer with or without thermal diffusion, diffusion thermo, radiation and chemical reaction effects on the flow past accelerated vertically inclined plate by considering different surface conditions, but the study on the effects of magnetic field on natural convection heat and mass transfer with thermal radiation, porous medium chemical reaction, thermal diffusion and diffusion thermo in Rivlin-Ericksen fluid flow through an accelerated vertically inclined plate has not been found in literature and hence the motivation to undertake the present study. Thus the present investigation is concerned with the study of combined effects of thermal diffusion, diffusion thermo on magnetic field and first-order

chemical reaction in two-dimensional MHD flow, heat and mass transfer of a viscous incompressible fluid past a permeable vertically inclined plate embedded in a porous medium in the presence of Rivlin-Ericksen fluid flow using finite difference technique. The effects of various physical parameters on the velocity, temperature and concentration profiles as well as on local skin-friction coefficient, Sherwood number and local Nusselt number are shown graphically. Validation of the analysis has been performed by comparing the present results with those of Geetha and Muthucumaraswamy [1]. 2. MATHEMATICAL FORMULATION:

Consider an unsteady MHD natural convection flow of a viscous, incompressible, electrically conducting non-Newtonian Rivlin-Ericksen fluid flow past a vertically inclined porous plate with chemical reaction, thermal radiation, thermal diffusion, diffusion thermo, heat and mass transfer. The physical model and the coordinate system are shown in Fig. 1.

For this research work, we made the following assumptions.

i. In Cartesian coordinate system, let x axis is taken to be along the plate and the y axis normal to the plate.

ii. Since the plate is considered infinite in x direction, hence all physical quantities will be independent of x direction.

iii. A uniform magnetic field of magnitude oB is applied normal to the plate.

iv. The transverse applied magnetic field and magnetic Reynold’s number are assumed to be very small, so that the induced magnetic field is negligible.

v. Consider an unsteady magneto hydrodynamic free convection flow of a viscous, incompressible, electrically conducting fluid past a semi-infinite tilted porous plate with an angle to the vertical.

vi. The homogeneous chemical reaction of first order with rate constant K between the diffusing species and the fluid is assumed.

vii. It is assumed that there is no applied voltage which implies the absence of an electric field.

viii. The fluid has constant kinematic viscosity and constant thermal conductivity, and the Boussinesq’s approximation have been adopted for the flow.

ix. The fluid is considered to be gray absorbing-emitting radiation but non-scattering medium and the Roseland’s approximation is used to describe the radiative heat flux.

x. At time 0t the plate is given an impulsive motion in the direction of flow i.e. along x axis against the

gravity with constant velocity oU , it is assumed that the plate temperature and concentration at the plate are varying linearly with time.

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Fig. 1. Physical configuration and coordinates system a --- Momentum boundary layer, b --- Thermal boundary layer, c --- Concentration boundary layer

Under these assumptions the equations governing the flow are (Goethe and Muthucumaraswamy [1]):

Momentum Equation:

2

2

2 3

1 2

*

o

u g T T (cos )t

ug C C (cos )y

B uu uK t y

(1)

Energy Equation:

2

2

2

2 1yC

CCkD

yq

CyT

CtT

Sp

Tmr

pp

(2)

Species Diffusion Equation:

2

2

2

2

yT

TkDCCK

yCD

tC

m

Tmr (3)

The corresponding initial and boundary conditions of the flow field are

yasCCTTu

yattACCCCTTtUut

yallforCCTTut

wwo

,,0

0,,:0

,,0:03

(4)

The local radiant rq for the case of an optically thin gray gas is expressed (Geetha and Muthucumaraswamy [1]) by

44*4 TTayqr

(5)

It is assumed that, the temperature differences within the flow are sufficiently small such that 4T may be expressed as a linear function of the temperature

T . This is obtained by

expanding 4T in a Taylor series about T and neglecting

higher order terms.

Thus, we get 434 34 TTTT (6)


707

By using Eqs. (5) and (6), Eq. (2) reduces to

2

23*

2

2 16yC

CCkD

CTTTa

yT

CtT

Sp

Tm

pp

(7)

Introducing the following non-dimensional quantities in Eqs. (1), (3) and (7),

xUTTCC

CCkDDuUTaR

CCCTTTkDSr

UKKrUKK

DScUA

UBM

UCCgGc

UU

TTgGrCCCC

TTTT

UuuUttUyy

ox

wpS

wTm

o

p

wm

wTm

o

roo

o

o

o

w

o

o

w

wwo

oo

Re,)(

)(,16,Pr,)()(

,,,,,,

,,,,,,

2

23*

22

22

2

2

3

*

2

21

3

2

(8)

then we get

2

2

3

2

1u ( M )u Gr (cos )Ky

u uGc (cos )tt y

(9)

ty

DuRy

2

2

2

2

PrPr1

(10)

t

Kry

SrySc

2

2

2

21

(11)

The corresponding initial and boundary conditions in dimensionless form are:

yasuyatttu

t

yallforut

0,0,00,1,

:0

0,0,0:0

(12)

All the physical parameters are defined in the nomenclature. It is now important to calculate the physical quantities of primary interest, which are the local wall shear stress, the local surface heat and mass flux. Given the velocity field in the boundary layer, we can now calculate the local wall shear stress (i.e., skin-friction) is given by and in dimensionless form, we obtain knowing the temperature field, it is interesting to study the effect of the free convection and radiation on the rate of heat transfer. This is given by which is written in dimensionless form as

0

22 )0(

yo

w

w

yuuU

uCf

(13)

Where 0

yw y

u

The dimensionless local surface heat flux (i.e., Nusselt number) is obtained as

0)(

)(

ywu y

TTT

xxN

then

0Re)(

yx

u

yxNNu

(14)

The definition of the local mass flux and the local Sherwood number are respectively given by with the help of these equations, one can write

0)(

)(

ywh y

CCC

xxS

then 0Re

)(

yx

h

yxSSh

(15)

The mathematical formulation of the problem is now completed. Eqs. (9)- (11) present a coupled non-linear system of partial differential equations and are to be solved by using initial and boundary conditions (12). However, exact solutions are difficult, whenever possible. Hence, these equations are solved by finite difference method. 3. NUMERICAL SOLUTIONS BY FINITE

DIFFERENCE METHOD:

Fig. 2. Finite difference space grid


708

The non-linear momentum, energy and concentration equations given in equations (9), (10) and (11) are solved under the appropriate initial and boundary conditions (12) by the implicit finite difference method. The transport equations (9), (10) and (11) at the grid point (i, j) are expressed in difference form using Taylor’s expansion.

11 1

2

1 1 11 1 1 1

2

2

1

2 2

j j j j ji i i i i

j j ji i i

j j j j j ji i i i i i

u u u u ut y

M u Gr cos Gc cosK

u u u u u ut y

(16)

11 1

2

1 12

21

2

j j j j ji i i i i

j j jj i i i

i

t Pr y

R DuPr y

(17)

11 1

2

1 12

21

2

j j j j ji i i i i

j j jj i i i

i

t Sc y

Kr Sry

(18)

Where the indices i and j refer to y and t respectively. Thus the values of u, θ and ϕ at grid point t = 0 are known; hence the temperature field has been solved at time

ttt ii 1 using the known values of the previous time

itt for all 1........,,2,1 Ni . Then the velocity field is evaluated using the already known values of temperature and concentration fields obtained at ttt ii 1 . These processes are repeated till the required solution of u, θ and ϕ is gained at convergence criteria:

310,,,, numericalexact uuabs (19)

4. VALIDATION OF CODE:

Table-1.: Comparison between present Skin-friction Cf results with the Skin-friction results of Geetha and Muthucumaraswamy [1] for different values of Gr, Gc, M, Sc, Pr, Kr (K) and t

Gr Gc M Sc Pr Kr (K) t

Present

Skin-friction Cf results

Skin-friction results of Geetha and

Muthucumaraswamy (2013)

5.0 5.0 2.0 0.16 0.71 2.0 0.2 - 0.519952185488 - 0.5215

5.0 5.0 2.0 0.16 0.71 2.0 0.4 - 2.511172649837 - 2.5110

5.0 5.0 2.0 0.16 0.71 2.0 0.6 - 6.932214856218 - 6.9488

5.0 5.0 2.0 0.16 0.71 2.0 0.8 - 15.17001485621 - 15.1779

5.0 5.0 2.0 0.16 0.71 2.0 1.0 - 29.11985124632 - 29.1185

5.0 5.0 2.0 0.16 0.71 0.2 0.2 - 0.519551201486 - 0.5208

5.0 5.0 2.0 0.16 0.71 0.2 0.4 - 2.512488951649 - 2.5266

5.0 5.0 2.0 0.16 0.71 0.2 0.6 - 6.999854147852 - 7.0550

5.0 5.0 2.0 0.16 0.71 0.2 0.8 - 15.55269411879 - 15.5548

5.0 5.0 2.0 0.16 0.71 0.2 1.0 - 30.11786655249 - 30.1247

5.0 5.0 2.0 0.16 0.71 5.0 0.2 - 0.511895032168 - 0.5216

5.0 5.0 2.0 0.16 0.71 5.0 0.4 - 2.479665234812 - 2.4800

5.0 5.0 2.0 0.16 0.71 5.0 0.6 - 6.762193048621 - 6.7721

5.0 5.0 2.0 0.16 0.71 5.0 0.8 - 14.59123266945 - 14.6014

5.0 5.0 2.0 0.16 0.71 5.0 1.0 - 27.65123059402 - 27.6612


709

To verify the accuracy of the numerical methods used to solve the problems considered in this study, a numerical test of the effect of thermal radiation and chemical reactions on convective and unsteady mass transfer was carried out. The viscosity of an infinite isothermal vertical plate accelerated in the presence of a magnetic field, as reported by Geetha and Muthucumaraswamy [1], is incompressible. The values calculated for the skin-friction coefficient are shown in tables 2, 3 and 4 for the current heating solution and mass transfer rate α = Sr = Du = λ = 0. The permeability parameter → ∞ and the results were published by Geetha and Muthucumaraswamy [1]. Tables 2, 3 and 4 show a good agreement between the results, which gives confidence in this numerical programme code.

Table-2.: Comparison between present rate of heat transfer coefficient (Nusselt number) results with the rate of heat

transfer coefficient (Nusselt number) results of Geetha and Muthucumaraswamy (2013) for different values of t and Pr

t Pr Present Nusselt number results

Nusselt number results of

Geetha and Muthucumaraswamy

(2013) 0.2 0.17 0.105154896213 0.1082 0.4 0.17 0.154662136548 0.1587 0.6 0.17 0.198543015648 0.2014 0.2 7.0 0.684216594532 0.6941 0.4 7.0 1.021564548231 1.0186 0.6 7.0 1.294121564893 1.2923 0.2 4.0 0.527845126998 0.5247 0.4 4.0 0.775126648953 0.7700 0.6 4.0 0.980156248863 0.9769

Table-3.: Comparison between present rate of mass transfer coefficient (Sherwood number) results with the rate of mass

transfer coefficient (Sherwood number) results of Geetha and Muthucumaraswamy (2013) for different values of t, Sc and

Kr (K)

t Sc Kr (K)

Present Sherwood number results

Sherwood number results of Geetha and Muthucumaraswamy

(2013) 0.2 0.16 0.2 0.021445231659 0.0274 0.4 0.16 0.2 0.079551324896 0.0861 0.6 0.16 0.2 0.172215492458 0.1737 0.2 6.0 0.2 0.048812477893 0.0493 0.4 6.0 0.2 0.148552169532 0.1458 0.6 6.0 0.2 0.264854778921 0.2797 0.2 2.5 0.2 0.095477893165 0.0964 0.4 2.5 0.2 0.268849512487 0.2745 0.6 2.5 0.2 0.495521467872 0.5079

(Here K, Gr and Gc are Chemical reaction parameter, thermal and mass Grashof numbers of Geetha and

Muthucumaraswamy (2013))

5. RESULTS AND DISCUSSIONS:

The solution of the equations of nonlinear connection of partial derivatives (9) with (11) under the limiting condition (12) cannot be obtained in a closed form, therefore, these equations are solved numerically using the finite difference method. Results of properties describing the effect of various parameters on velocity, temperature and concentration profiles, including skin-friction, Nusselt number and Sherwood number coefficients for different values of material parameters at t = 1.0. In this case, the y → ∞ boundary condition is replaced by ymax = 10 is sufficiently large value in which the velocity, temperature, and concentration distributions can be approximated at relatively free flow. To get a physical idea of the problem, the effect of various control parameters on physical quantities is calculated and shown and discussed in detail in Figs. 3-21.

Fig. 3. Gr effect on velocity profiles

Fig. 4. Gc effect on velocity profiles

0

1.25

2.5

0 2 4 6 8 10

u

y

Gr = 5.0, 10.0, 15.0, 20.0

0

1.25

2.5

0 2 4 6 8 10

u

y

Gc = 5.0, 10.0, 15.0, 20.0


710

In Fig. 3 shows a typical velocity profiles in the boundary layer for different values of the heat transfer of the Grashof number (Gr). It is worth noting that an increase in Gr leads to an increase in the value of velocity due to an increase in thrust. Here, the positive value of Gr corresponds to the cooling of the plate. In addition, it should be noted that the velocity increases sharply near the wall of the perforated plate, while Gr increases and, consequently, decreases to the value of free flow. For the case of different mass exchange values of the Grashof number (Gc), the distribution of the velocities of the boundary layer is shown in Fig. 4. The velocity of the flow reaches a maximum unique value next to the board, and then decreases accordingly to a value that approaches the free flow. As expected, as buoyancy increases , represented by Gc, the velocity of the fluid increases, and the peak becomes more pronounced.

Fig. 5. M effect on velocity profiles

Fig. 6. K effect on velocity profiles

The effect of magnetic field parameters is shown in Fig. 5. It is observed that the velocity of the fluid decreases with increasing values of the magnetic field parameter. As the magnetic field parameter increases, the decrease in velocity occurs due to the presence of a magnetic field in a conductive fluid, creating a force called the Lorentz force, which flows when the magnetic field is applied in that direction. Normal, as shown in this study. The component of resistance to the rate of slowing fluid, as shown in Fig. 5, the velocity

distribution and concentration, along with several layers, reduce buoyancy. The effect of the Permeability parameter is shown in Fig. 6. As can be seen from the figure, the fluid velocity increases with an increase in the permeability parameter of the boundary layer region. The physics underlying this phenomenon is that an increase in the permeability parameter means that the resistance of a porous medium decreases, leading to an increase in fluid flow. As can be seen from Fig. 7, for a fluid with Pr < 1, the thickness of the moment boundary layer increases, and Pr > 1.The Prandtl number effectively describes the relationship between the pulse diffusion coefficient and the thermal diffusion coefficient, thereby controlling the relative thickness of the pulse and the thermal confinement layers. When Pr is low, that is, Pr = 0.025, it is felt that heat spreads very quickly in velocity (impulse).This means that for fluid metals the thickness of the reservoir or heat limit is much greater than the velocity of the boundary layer.

Fig. 7. Pr effect on velocity profiles

Fig. 8. Pr effect on temperature profiles

Fig. 8 shows the temperature characteristics of different Prandtl values. It was noted that an increase in the Prandtl number leads to a decrease in the thickness of the thermal boundary layer and, as a rule , leads to a lower average temperature inside the boundary layer. The reason is that a lower Pr value corresponds to an increase in the thermal conductivity of the fluid, so the heat can heat up from the heated surface to a higher Pr value. Therefore, in the case where the Prandtl number is the smallest, the thermal

0

1.25

2.5

0 2 4 6 8 10

u

y

M = 1.0, 2.0, 3.0, 4.0

0

1.25

2.5

0 2 4 6 8 10

u

y

K = 1.0, 2.0, 3.0, 4.0

0

1.25

2.5

0 2 4 6 8 10

u

y

Pr = 0.025, 0.71, 7.0, 11.62

0

0.5

1

0 2 4 6 8 10

θ

y

Pr = 0.025, 0.71, 7.0, 11.62


711

boundary layer is thick and the heat transfer rate decreases. The influence of the Schmidt number Sc on the velocity distribution is shown in Fig. 9. It was noted that at that time all other parameters remained constant and the participation of Sc increased, but the result, as a rule, decreases velocity. In addition, we believe that when we leave the board, the velocity of the fluid will decrease.

Fig. 9. Sc effect on velocity profiles

Fig. 10. Sc effect on concentration profiles

Fig. 11. R effect on velocity profiles

Fig. 12. R effect on temperature profiles

Fig. 13. Du effect on velocity profiles

Fig. 14. Du effect on temperature profiles

Fig. 15. Sr effect on velocity profiles

0

1.25

2.5

0 2 4 6 8 10

u

y

Sc = 0.22, 0.30, 0.60, 0.78

0

0.5

1

0 2 4 6 8 10

ϕ

y

Sc = 0.22, 0.30, 0.60, 0.78

0

1.25

2.5

0 2 4 6 8 10

u

y

R = 1.0, 2.0, 3.0, 4.0

0

0.5

1

0 2 4 6 8 10

θ

y

R = 1.0, 2.0, 3.0, 4.0

0

1.25

2.5

0 2 4 6 8 10

u

y

Du = 1.0, 2.0, 3.0, 4.0

0

0.5

1

0 2 4 6 8 10

θ

y

Du = 1.0, 2.0, 3.0, 4.0

0

1.25

2.5

0 2 4 6 8 10

u

y

Sr = 1.0, 2.0, 3.0, 4.0


712

Fig. 16. Sr effect on concentration profiles

Fig. 17. Kr effect on velocity profiles

Fig. 18. Kr effect on concentration profiles

Fig. 19. α effect on velocity profiles

Fig. 20. t effect on velocity profiles

Fig. 21. t effect on concentration profiles

Fig. 22. λ effect on velocity profiles

The effect of the Schmidt number on the concentration is shown in Fig. 10. An increase in Sc was observed to help reduce fluid concentration. In addition, we see that the Sc does not make a large contribution to the field of concentration when we are away from the boundary surface. The influence of the parameter R of thermal radiation on the distribution of velocity and temperature in the boundary layer is shown in Figs. 11 and 12. An increase in the thermal radiation

0

0.5

1

0 2 4 6 8 10

ϕ

y

Sr = 1.0, 2.0, 3.0, 4.0

0

1.25

2.5

0 2 4 6 8 10

u

y

Kr = 1.0, 2.0, 3.0, 4.0

0

0.5

1

0 2 4 6 8 10

ϕ

y

Kr = 1.0, 2.0, 3.0, 4.0

0

1.25

2.5

0 2 4 6 8 10

u

y

α = π/6, π/4, π/3, π/2

0

1.25

2.5

0 2 4 6 8 10

u

y

t = 1.0, 2.0, 3.0, 4.0

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10

ϕ

y

t = 1.0, 2.0, 3.0, 4.0

0

1.25

2.5

0 2 4 6 8 10

u

y

λ = 1.0, 2.0, 3.0, 4.0


713

parameter R significantly increases the thermal state of the fluid and its thermal boundary layer. This increase in fluid temperature causes an increase in flow in the boundary layer, which leads to an increase in fluid velocity. For different values of Du (Dufour number), the velocity and temperature profiles are shown in Figs. 13 and 14. The Dufour Du number represents the contribution of the concentration gradient to the flow of thermal energy in the flow. It was found that increasing the amount of Dufour leads to an increase in velocity and temperature through the boundary layer. However, for Du > 1, a significant deformation of the velocity near the plate is observed, and from there the profile drops to zero at the boundary layer boundary. Figs. 15 and 16 show Soret number on the velocity and concentration of the distribution of different values of the number. Soret number Sr certain temperature gradient, diffusion causes a significant effect of mass. It is seen that increasing the amount of Soret Sr leads to an increase in velocity and concentration in the boundary layer. As it was, the presence of a chemical reaction

significantly influenced the concentration distribution and the velocity profile in Figs. 17 and 18. It must say that in this case it was about chemical reactions and reaction reactions. In fact, when it increases, the expected velocity profile decreases significantly, and the presence of a peak indicates that the maximum velocity appears in the fluid near the surface, not on the surface. In addition, the parameters of the chemical reaction increase, and the concentration decreases. The influence of the angle of inclination of the plate (α) on the velocity range is shown in Fig. 19. It is seen that with an increase in the angle of inclination of the panel, the velocity field decreases. Figs. 20 and 21 show the effect of time t on both the velocity and concentration distributions. It is clear that the profiles of the velocity and concentration of the fluid increase with time. The influence of Rivlin-Ericksen fluid parameter (λ) on velocity profiles is shown in Fig. 22. From this figure, the velocity profiles are decreasing with increasing values of Rivlin-Ericksen fluid parameter.

Table-4.: Numerical values of Skin-friction coefficient for variations of

Gr, Gc, M, K, Pr, Sc, Sr, Du, R, Kr, t, α and λ

Gr Gc M K Pr Sc Sr Du R Kr t α λ Cf

5.0 5.0 1.0 1.0 0.71 0.22 1.0 1.0 1.0 1.0 1.0 0.785 1.0 0.101124568153

10.0 5.0 1.0 1.0 0.71 0.22 1.0 1.0 1.0 1.0 1.0 0.785 1.0 0.314051844421

5.0 10.0 1.0 1.0 0.71 0.22 1.0 1.0 1.0 1.0 1.0 0.785 1.0 0.452130185264

5.0 5.0 2.0 1.0 0.71 0.22 1.0 1.0 1.0 1.0 1.0 0.785 1.0 - 0.810145721564

5.0 5.0 1.0 2.0 0.71 0.22 1.0 1.0 1.0 1.0 1.0 0.785 1.0 0.154852140617

5.0 5.0 1.0 1.0 7.00 0.22 1.0 1.0 1.0 1.0 1.0 0.785 1.0 - 0.751421620158

5.0 5.0 1.0 1.0 0.71 0.30 1.0 1.0 1.0 1.0 1.0 0.785 1.0 - 0.775015488621

5.0 5.0 1.0 1.0 0.71 0.22 2.0 1.0 1.0 1.0 1.0 0.785 1.0 0.184169548215

5.0 5.0 1.0 1.0 0.71 0.22 1.0 2.0 1.0 1.0 1.0 0.785 1.0 0.210543215489

5.0 5.0 1.0 1.0 0.71 0.22 1.0 1.0 2.0 1.0 1.0 0.785 1.0 0.051324876012

5.0 5.0 1.0 1.0 0.71 0.22 1.0 1.0 1.0 2.0 1.0 0.785 1.0 0.021421633214

5.0 5.0 1.0 1.0 0.71 0.22 1.0 1.0 1.0 1.0 2.0 0.785 1.0 0.211548215648

5.0 5.0 1.0 1.0 0.71 0.22 1.0 1.0 1.0 1.0 1.0 1.570 1.0 0.011321561482

5.0 5.0 1.0 1.0 0.71 0.22 1.0 1.0 1.0 1.0 1.0 0.785 2.0 0.001625851249

Table-5.: Numerical values of Nusselt number coefficient for variations of Pr, Du and R

Pr Du R Nu

0.71 0.5 0.5 0.102132611854

7.00 0.5 0.5 0.041542188722

0.71 1.0 0.5 0.184201348564

0.71 0.5 1.0 0.024318774215

Table-6.: Numerical values of Sherwood number coefficient for variations of Sc, Sr, Kr and t

Sc Sr Kr t Sh 0.22 1.0 1.0 1.0 0.041548236158

0.30 1.0 1.0 1.0 0.000951248632 0.22 2.0 1.0 1.0 0.101523146895 0.22 1.0 2.0 1.0 0.001425462546 0.22 1.0 1.0 2.0 0.120548752144


714

The effects of thermal Grashof number (Gr), solutal Grashof number (Gc), Prandtl number (Pr), Schmidt number (Sc), Hartmann number (M), Permeability parameter (K), Soret number (Sr), Dufour number (Du), Thermal Radiation parameter (R), Rivlin-Ericksen fluid flow parameter (λ), Angle of inclination parameter (α), time (t) and Chemical reaction parameter (Kr) on skin-friction coefficient is presented in table-4 with the help of numerical values. From this table, it is observed that, the numerical values of skin-friction coefficient is increasing under the increasing of thermal Grashof number (Gr), solutal Grashof number (Gc), Permeability parameter (K), Soret number (Sr), Dufour number (Du) and time (t), while it decreasing under the increasing of Prandtl number (Pr), Schmidt number (Sc), Hartmann number (M), Thermal Radiation parameter (R), Rivlin-Ericksen fluid flow parameter (λ), Angle of inclination parameter (α) and Chemical reaction parameter (Kr). The effects Prandtl number (Pr), Dufour number (Du), Thermal Radiation absorption parameter (R) and time (t) on rate of heat transfer coefficient in terms of Nusselt number is discussed in table-5. From this table, it is observed that, the numerical values of Nusselt number coefficient is increasing with the increasing of Dufour number (Du) and time (t) and the reverse effect is observed with the increasing values of Prandtl number (Pr) and Thermal Radiation absorption parameter (R). Table-6, shows the numerical values of rate of mass transfer coefficient in terms of Sherwood number coefficient for different values of Schmidt number (Sc), Soret number (Sr), time (t) and Chemical reaction parameter (Kr). From this table, it is observed that Sherwood number coefficient is increasing with increasing values of Soret number (Sr) and time (t) and decreasing with increasing of Schmidt number (Sc) and Chemical reaction parameter (Kr).

6. CONCLUSIONS

Below we summarize the results on physical velocity, temperature and concentration distributions, as well as skin-friction, heat and mass transfer rates by variations of engineering parameters.

i. More and more Hartmann number or Angle of inclination or Rivlin-Ericksen fluid slow down the velocity of the flow field at all points.

ii. The effect of increasing the permeability parameter or the thermal radiation parameter or time or Dufour is to accelerate the flow field velocity at all points.

iii. More and more Prandtl or time reduces the temperature of the flow field at all points and increases with increasing parameters of thermal radiation.

iv. Dufour effects deeply influence the temperature profiles in the thermal boundary layer i.e. temperature profiles increases with the increase in the Dufour number.

v. Concentration decreases with increasing time and decreases with increasing Schmidt number.

vi. The Soret effect is an increase in the concentration distribution during the formation of a peak concentration with a higher value of the Soret parameter in the layer that limits the concentration.

vii. The velocity and concentration decrease with increasing of chemical reaction parameter.

viii. The obtained numerical results are compared with previously registered cases and are available in the open literature and are very consistent.

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