Thermal radiation effect on an unsteady magnetohydrodynamic flow past inclined porous heated plate in the presence of chemical reaction and viscous dissipation

Applied Mathematics and Computation 226 (2014) 423–434

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Thermal radiation effect on an unsteady magnetohydrodynamic flow past inclined porous heated plate in the presence of chemical reaction and viscous dissipation R.N. Barik a,⇑, G.C. Dash b a b

Department of Mathematics, Trident Academy of Technology, Infocity, Bhubaneswar 751024, Odisha, India Department of Mathematics, S.O.A. University, Bhubaneswar 751030, Odisha, India

a r t i c l e

i n f o

Keywords: Thermal radiation MHD flow Heat transfer Mass transfer Porous medium Chemical reaction Viscous dissipation Finite difference method

a b s t r a c t An analysis is made for the unsteady magnetohydrodynamic (MHD) flow of a viscous, incompressible, electrically conducting fluid in a porous medium. Considering the viscous dissipative term in energy equation which is important in free convective flow. The coupled non-linear partial differential equations are solved by using an implicit finite difference method of Crank Nicolson type. The effects of chemical reaction, viscous dissipation and radiation on velocity, temperature and concentrations are discussed. More importantly, the results of numerical solution of the present study agree well with the analytical solution of the earlier study in a particular case (i.e. without viscous dissipation). It is interesting to note that the Hartmann number reduces the velocity at all points of the flow domain as expected. An increase in viscous dissipation contributes slightly but uniformly to the rise of temperature as well as velocity distribution. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction Several problems related to transfer of mass over inclined beds finds numerous applications in geophysical, petroleum, chemical, bio-mechanical and chemical technology. The viscous drainage over an inclined porous plane is a subject of considerable interest to both theoretical and experimental investigators, especially in the flow of oil through porous rock, the extraction of geo-thermal energy from the deep interior of the earth to the shallow layers, flow of liquids through ion exchange beds, chemical reactor for economical separation or purification of mixtures. Another area of bio-medical application is the drug permeation through human glands, flow of fluids in lungs, blood vessels, arteries and so on. Raptis and Kafousias [1] studied the magnetohydrodynamic free convective flow and mass transfer through a porous medium bounded by an infinite vertical porous plate with constant heat flux. Gebhart [2] and Gebhart and Mollendorf [3] showed that the viscous dissipative heat in natural convection is important when the free convective flow field is of extreme size or at extremely low temperature or in high gravity field. Reddy et al. [4] investigated mass transfer and radiation effects of unsteady MHD free convective fluid flow embedded in porous medium with heat generation/absorption. The effects of the chemical reaction and radiation absorption on free convective flow through porous medium with variable suction in the presence of uniform magnetic field was studied by Sudheer Babu and Satyanarayana [5]. Bhaskar et al. [6] studied radiation and mass transfer effects on MHD free convection flow past an impulsively started isothermal vertical plate with dissipation. Muthucumaraswamy and Vijayalakshmi [7] investigated MHD and chemical reaction on flow past an impulsively started

⇑ Corresponding author. E-mail address: [email protected] (R.N. Barik). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.09.077

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Nomenclature A Bo Cp Gr Gc g K0 K M Pr Sc N Q Ec kr D T0 T 0w T 01 C C0 C 0w C 01 t0 t u0 u Nu

a constant magnetic field component along y0 -axis specific heat at constant pressure Grashof number modified Grashof number acceleration of gravity the permeability of medium the permeability parameter Hartmann number Prandtl number Schmidt number thermal radiation parameter heat absorption parameter Eckert number chemical reaction parameter chemical molecular diffusivity temperature of fluid near the plate temperature of the fluid far away of the fluid from the plate temperature of the fluid at infinity concentration of the fluid concentration of fluid near the plate concentration of the fluid far away of the fluid from the plate concentration of the fluid at infinity time in x0 ; y0 coordinate system time in dimensionless co-ordinates velocity component in x0 -direction dimensionless velocity component in x0 -direction Nusselt number a Sherwood number qr radiative heat flux Rex Reynolds number k1 mean absorption coefficient  chemical reaction of first order with rate constant K x0 ; y0 co-ordinate system x; y dimensionless coordinates Uo reference velocity Greek symbolsb coefficient of volume expansion for heat transfer b coefficient of volume expansion for mass transfer j thermal conductivity of the fluid r electrical conductivity of the fluid m kinematic viscosity h non-dimensional temperature q density of the fluid a angle s skin-friction l viscosity, Ns/m2

semi-infinite vertical plate with thermal radiation. The radiation and mass transfer effects on an unsteady MHD convection flow past a semi-infinite vertical permeable moving plate embedded in a porous medium with viscous dissipation was studied by Prasad and Reddy [8]. Kumar and Verma [9] studied the radiation effects on MHD flow past an impulsively started exponentially accelerated vertical plate with variable temperature in the presence of heat generation. They [10] also studied the thermal radiation and mass transfer effects on MHD flow past a vertical oscillating plate with variable temperature and variable mass diffusion. Pattanaik et al. [11] studied the radiation and mass transfer effects on MHD free convective flow through porous medium past an exponentially accelerated vertical plate with variable temperature. Rajesh and Verma [12] considered the radiation and mass transfer effects on MHD free convection flow past an exponentially accelerated vertical plate with variable temperature. The radiation effects on MHD free convective flow over a vertical plate with heat and mass flux was studied by Sivaiah et al. [13].

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425

Wafosoh and Murcithi [14] computed exact and numerical solutions of a fully developed flow of a generalized secondgrade fluid with power law temperature dependent viscosity in an inclined plate. The influence of viscous dissipation and radiation on unsteady MHD free convective flow past an infinite heated vertical plate in a porous medium with time dependent suction was studied by Israel-Cookey et al. [15]. Soundalgekar [16] investigated the viscous dissipative effects on unsteady free convective flow past a vertical plate with constant suction. Recently, Reddy et al. [17] have discussed the effect of thermal radiation on an unsteady magnetohydrodynamic flow past an inclined porous heated plate in the presence of chemical reaction with heat and mass transfer by using a closed analytical form method. They have not considered the viscous dissipation term in the energy equation. In the present study we have considered the problem of Reddy et al. retaining the viscous dissipation in the energy equation so that the present analysis may be useful under the above mentioned circumstance. The numerical solution for the velocity, temperature and concentrations are obtained. The effects of pertinent parameters are shown in the graphs. More importantly, the results of analytical solution obtained by Reddy et al. [17] agree with our numerical solution. The objective of the present work is to investigate the effect of thermal radiation and viscous dissipation on MHD free convective flow past an impulsively started semi-infinite inclined heated porous plate with variable temperature in the presence of chemical reaction and heat absorption of first order. The method of solution applied here is finite difference method which is more economical from computational point of view and the results so obtained are in good agreement with the results of Reddy et al. [17]. 2. Mathematical formulation Consider a two dimensional unsteady MHD free convection flow of a viscous, incompressible, electrically conducting fluid past a semi infinite tilted porous plate with chemical reaction and thermal radiation. The x0 -axis is taken to be along the plate and the y0 -axis normal to the plate. Since the plate is considered infinite in x0 -direction, hence all physical quantities will be independent of x0 -direction. The wall is maintained at constant temperature ðT 0w Þ and concentration ðC 0w Þ higher than the ambient temperature ðT 01 Þ and concentration ðC 01 Þ respectively. A uniform magnetic field of magnitude Bo is applied normal to the plate. The transverse applied magnetic field and magnetic Reynolds number are assumed to be very small, so that the induced magnetic field is negligible. 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 a  between the diffusing species and to the vertical. The homogeneous chemical reaction of first order with rate constant K the fluid is assumed. It is assumed that there is no applied voltage which implies the absence of an electric field. The fluid has constant kinematic viscosity and constant thermal conductivity, and the Boussinesq’s approximation have been adopted for the flow. 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. It is considered to be negligible in x0 -direction as compared in y0 -direction. At time t 0 > 0 the plate is given an impulsive motion in the direction of flow i.e. along x0 -axis against the gravity with constant velocity U o ; it is assumed that the plate temperature and concentration at the plate are varying linearly with time. The concentration of the diffusing species in the binary mixture is assumed to be very small in comparison with the other chemical species, which are present and hence Soret and Dufour effects are negligible. Under these assumptions the equations governing the flow are: Momentum equation:

@u0 @ 2 u0 0 0 0 0   0 ¼ gbðT  T 1 Þðcos aÞ þ gb ðC  C 1 Þðcos aÞ þ m @t @y02

!

rB20 m 0 þ u q K0

ð1Þ

Energy equation:

qC p

 0 2 @T 0 @ 2 T 0 @qr @u ¼ j  þ m  Q 0 ðT 0  T 01 Þ @t 0 @y02 @y0 @y0

ð2Þ

Species diffusion equation:

@C 0 @2C0  K r ðC 0  C 01 Þ 0 ¼ D @t @y02

ð3Þ

With the following initial and boundary conditions:

u0 ¼ 0; T 0 ¼ T 01 ; C 0 ¼ C 01 for all y0 8 0 0 0 0 0 0 > < u ¼ U o ; T ¼ T 1 þ ðT w  T 1 ÞAt ; 0 0 0 0 0 0 0 t > 0 : C ¼ C 1 þ ðC w  C 1 ÞAt ; at y ¼ 0 > : 0 u ! 0; T 0 ! T 01 ; C 0 ! C 01 as y0 ! 1 t0 6 0 :

2

where A ¼ Umo .

9 > > > = > > > ;

ð4Þ

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The radiative heat flux qr , under Rosseland approximation has the form

qr ¼ 

4r1 @T 04 3j1 @y0

ð5Þ

where r1 and j1 are the Stefan–Boltzmann constant and the absorption coefficient respectively. It is assumed that, the temperature differences within the flow are sufficiently small such that T 04 may be expressed as a linear function of the temperature T 01 . This is obtained by expanding T 04 in a Taylor series about T 01 and neglecting higher order terms. Thus, we get 0 04 T 04 ffi 4T 03 1 T  3T 1

ð6Þ

By using equations (5) and (6), equation (2) reduces to

@T 0 j @ 2 T 0 16r1 T 031 @ 2 T 0  0 ¼ @t qC p @y02 3k1 qC p @y02

ð7Þ

Introducing the following non-dimensional parameters in Eqs. 1, 2, 3, and (7) quantities: 0

0

y ¼ y mUo ;

t ¼ t Um o ;

0

0

w

1

0

0

Gr ¼ gbmðTUw3T 1 Þ ;

1 ; C ¼ CC0 C C 0

o

0



0

u ¼ Uuo ;

0

lC

Gc ¼ gb mðCUw3 C1 Þ ; Pr ¼ j p ;  2 o 2 M ¼ rqBo Um2 ; Ec ¼ C p ðTU0 oT 0 Þ ; p

o

0

ð8Þ

> Sc ¼ Dm ; > > > > > > 2 > jU o > N ¼ 4m2 r T 30 ; > > 1 1 > > > ;

w

o

0

Q ¼ qCmQU2 ;

9 0 0 1 h ¼ TT0 T 0 ;> > w T 1 > > > > > > > > > > > > =

2

K ¼ K mU2o ;

kr ¼ KUr2m

1

o

In Eqs. 1, 2, 3, and (7) reduces to

  @u @ 2 u 1 u þ ðGrÞhðcos aÞ þ ðGcÞCðcos aÞ ¼ 2 Mþ @t @y K ðPrÞ

   2 @h 4 @2h @u ¼ 1þ þ ðPrÞðEcÞ  ðPrÞðQ Þh @t 3N @y2 @y

@C 1 @2C ¼  ðkr ÞC @t Sc @y2

ð9Þ

ð10Þ

ð11Þ

The corresponding initial and boundary conditions in dimensionless form are:

t60: t>0:

u ¼ 0; h ¼ 0; C ¼ 0 for all y  u ¼ 1; h ¼ t; C ¼ t at y ¼ 0 u ! 0;

h ! 0;

C ! 0 as y ! 1

9 = ;

ð12Þ

All the physical parameters are defined in the nomenclature. Knowing the velocity, temperature and concentration distributions it is important to study the effects of viscous dissipation on the skin-friction, rate of heat transfer and the rate of mass transfer. Skin-friction in non-dimensional form is given by

s¼



sw @u ; sw ¼ l @y qu2w

 y0 ¼0

¼ qU 2o u0 ð0Þ ¼



 @u @y y¼0

ð13Þ

The dimensionless local surface heat flux (i.e. Nusselt number) and local mass flux (i.e. Sherwood number) are given by

 Nu ðx0 Þ ¼ 

x0 @T 0 0 0 ðT w  T 1 Þ @y0

 then Nu ¼ y0 ¼0

  Nu ðx0 Þ @h ¼ Rex @y y¼0

ð14Þ

R.N. Barik, G.C. Dash / Applied Mathematics and Computation 226 (2014) 423–434

 and Sh ðx0 Þ ¼ 

x0 @C 0 0 0 ðC w  C 1 Þ @y0

 y0 ¼0

then Sh ¼

  Sh ðx0 Þ @C ¼ Rex @y y¼0

427

ð15Þ

0

where Rex ¼  Uomx is the Reynolds number. 3. Method of solution Equations (9)–(11) are coupled non-linear partial differential equations and are solved by using initial and boundary conditions (12). However, exact or approximate solutions are not possible for this set of equations and hence we solve these equations by an implicit finite difference method of Crank–Nicolson type for a numerical solution. The equivalent finite difference scheme of Eqs. (9)–(11) is as follows:

    u  1 u ui1; j  2ui; j þ uiþ1; j 1 1 ui; jþ1 þ ui; j i; jþ1  ui; j i1; jþ1  2ui; jþ1 þ uiþ1; jþ1  þ M þ ¼ 2 2 K Dt Dy 2 Dy 2 Dy     1 hi; jþ1 þ hi; j 1 C i; jþ1 þ C i; j þ ðGcÞðcos aÞ þ ðGrÞðcos aÞ 2 2 Dy Dy        hi; jþ1  hi; j hi1; jþ1  2hi; jþ1 þ hiþ1; jþ1 hi1; j  2hi; j þ hiþ1; j hi; jþ1 þ hi; j 1 4 1 ¼  1þ ðPrÞ þ QðPrÞ 2 3N 2 Dt Dy2 Dy2 Dy  2 1 ui; jþ1 þ ui; j þ ðEcÞðPrÞ 2 Dy !     C i;jþ1  C i;j 1 ui1;jþ1  2ui;jþ1 þ uiþ1;jþ1 ui1;j  2ui;j þ uiþ1;j 1 C i;jþ1 þ C i;j  ¼ ðScÞ þ ðScÞðk Þ r 2 2 Dt Dy ðDyÞ2 ðDyÞ2

ð16Þ

ð17Þ

ð18Þ

Here the suffix i corresponds to y and j corresponds to t. Also Dt ¼ tjþ1  t j and Dy ¼ yiþ1  yi The complete solution of the discrete Eqs. (16)–(18) proceeds as follows: Step–(1) Knowing the values of C, h and u at a time t ¼ j, calculate C and h at a time t ¼ j þ 1 using Eqs. (17) and (18) and solving tri-diagonal linear system of equations. Step–(2) Knowing the values of h and C at time t ¼ j, solve the Eq. (16) (via tri-diagonal matrix inversion) to obtain u at a time t ¼ j þ 1. We can repeat steps (1) and (2) to proceed from t ¼ 0 to the desired time value. The implicit Crank–Nicolson method is a second order method OðDt2 Þ in time and has no restrictions on space and timesteps, Dy and Dt, i.e. the method is unconditionally stable. The finite difference scheme used, involves the values of the function at the six grid points. A linear combination of the ‘‘future’’ points is equal to another linear combination of the ‘‘present’’ points. To find the future values of the function, one must solve a system of linear equations, whose matrix has a tri-diagonal form. The computations were carried out for Gr = 1.0, Gc = 1.0, Pr = 0.71 (Air), Sc = 0.22 (Hydrogen), M = 1.0, K ¼ 1.0, N = 1.0, Q = 1.0, Ec = 0.001, a ¼ p4 ; kr = 1.0 and Dy = 0.1, Dt = 0.001 and the procedure is repeated till y = 4. In order to check the accuracy of numerical results, the results of the present study is compared with the available theoretical solution of Reddy et al. [17] and is found to be in good agreement. 4. Results and discussion For the purpose of discussion, the effects of the permeability of the medium, viscous dissipation and other pertinent parameters on free convection and mass transfer flow which arise from a combination of temperature and concentration differences on the flow field are to be considered. For the numerical calculations, here we have considered Pr ¼ 0:71 (air), Sc ¼ 0:60 (representing water vapor at approximately 250 C and one atmosphere) and Eckert number (Ec) which may be interpreted as the addition of heat due to viscous dissipation, is equal to 0.01 which is more appropriate for incompressible fluids. The values of Grashof number ðGrÞ; modified Grashof number ðGcÞ, Hartmann number ðMÞ, permeability parameter ðKÞ, thermal radiation parameter ðNÞ, heat absorption parameter ðQ Þ and chemical reaction parameter ðkr Þ, time (t) are taken arbitrarily positive in order to investigate their effects on the flow field. The positive values of Gr and Ec correspond to the cooling of the surface. From Figs. 3–5, 11 and 13 it is observed that velocity gets retarding effect due to increase in Prandtl number ðPrÞ, Schmidt number ðScÞ, Hartmann number ðMÞ, heat absorption parameter ðQ Þ and chemical reaction parameter ðkr Þ but velocity profiles exhibited in Figs. 1, 2, 6, 8 and 9 increase with increasing the values of Grashof number ðGrÞ; modified Grashof number ðGcÞ, Eckert number ðEcÞ, permeability parameter ðKÞ, thermal radiation parameter ðNÞ respectively. The same observation was also made by Reddy et al. [17] in the absence of viscous dissipation. Therefore, it may be remarked that both the

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R.N. Barik, G.C. Dash / Applied Mathematics and Computation 226 (2014) 423–434

1

0.5

= 1.0, 2.0, 3.0, 4.0

0 0

1

2

Fig. 1. Velocity profiles for different values of Gr.

1

0.5

= 1.0, 2.0, 3.0, 4.0

0 0

1

2

Fig. 2. Velocity profiles for different values of Gc.

1

0.5

= 0.025, 0.71, 7.0, 11.62

0 0

1

2

Fig. 3. Velocity profiles for different values of Pr.

1

0.5

= 0.22, 0.30, 0.60, 0.66

0 0

1

2

Fig. 4. Velocity profiles for different values of Sc.

methods of solution i.e. analytical method (Reddy et al.) and numerical method (the present study) yield the same result. This aids to the conformity of the present solution. Figs. 1 and 2 exhibit the effect of Grashof number and modified Grashof number on the velocity profile when other parameters are fixed. The Grashof number ðGrÞ signifies the relative effect of the thermal buoyancy force to the viscous

R.N. Barik, G.C. Dash / Applied Mathematics and Computation 226 (2014) 423–434

429

hydrodynamic force in the boundary layer. It is observed that an increase in thermal buoyancy force leads to increase the velocity of the flow. It is also noticed that the velocity increases with increasing values of the modified Grashof number ðGcÞ. Fig. 3 depicts the effect of Prandtl number on velocity profiles in presence of foreign species such as mercury (Pr = 0.025), air (Pr = 0.71), water (Pr = 7.00) and water at 4o C (Pr = 11.62). It is observed that the velocity decreases with increasing of Prandtl number ðPrÞ. The effect of Schmidt number on velocity profiles in presence of foreign species such as hydrogen (Sc = 0.22), helium (Sc = 0.30), oxygen (Sc = 0.60) and water–vapor (Sc = 0.66) are shown in fig. 4. The flow field suffers a decrease in velocity at all points in presence of heavier diffusing species. The effect of the Hartmann number (M) is shown in Fig. 5. It has a significant reducing effect on the velocity profile in the presence of viscous dissipative heat. The effect of viscous dissipative heat is shown in Figs. 6 and 7. It is observed that for constant Gr; Gc; K; N; Pr; M; Q an increase in viscous dissipative heat ðEcÞ leads to decrease both in velocity and temperature profiles. Fig. 8 displays the variation of velocity with the permeability of the porous medium. The variation is insignificant both near and farther from the embedded plate. The variation of K also includes the effect of the variation in free stream velocity U 0 since K ¼

K 0 U 20

m2

; for

K0

m2

= constant.

1

0.5

= 1.0, 2.0, 3.0, 4.0

0 0

1

2

Fig. 5. Velocity profiles for different values of M.

1

0.5

= 0.001, 0.01, 0.1, 1.0

0 0

1

2

Fig. 6. Velocity profiles for different values of Ec.

1

= 0.001, 0.01, 0.1, 1.0

0.5

0 0

1

2

3

Fig. 7. Temperature profiles for different values of Ec.

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R.N. Barik, G.C. Dash / Applied Mathematics and Computation 226 (2014) 423–434

The effect of thermal radiation parameter ðNÞ is exhibited through the Figs. 9 and 10. It is seen that both velocity and temperature increase with an increase in thermal radiation parameter. Further, it is observed that an increase in heat absorption parameter ðQ Þ in Figs. 11 and 12 leads to decrease both velocity and temperature in the flow domain but the close observation reveals that effect on temperature distribution is significant than the velocity distribution. It is also observed that the both the hydrodynamic (velocity) and the thermal (temperature) boundary layers decrease as the heat absorption effects increase. It is seen from Figs. 13 and 14 that the destructive chemical reaction parameter ðkr > 0Þ has a reducing effect on both velocity and concentration distribution. Figs. 15 and 16 exhibit the effect of Prandtl number ðPrÞ and Schmidt number ðScÞ on the temperature and concentration field respectively. Prandtl number ðPrÞ and Schmidt number ðScÞ measure the relative importance of viscosity with thermal diffusivity and viscosity with mass diffusivity respectively. It is observed that the temperature and concentration distribution experience significant reduction for increasing values of Pr and Sc. This implies that heavier species with low thermal diffusivity causes significant fall in concentration and temperature distribution. Figs. 17 and 18 display the effect of the time ðtÞ on temperature and concentration profiles respectively. From these two figures we observe that, both temperature and concentration are increasing with increasing values of time ðtÞ. The numerical values of skin-friction ðsÞ; Nusselt number ðNuÞ and Sherwood number ðShÞ are entered in Table 1–3 respectively. The effects of pertinent parameters on the shearing stress are shown in Table 1. The skin-friction ðsÞ increases

1

0.5

= 1.0, 2.0, 3.0, 4.0

0 0

1

2

Fig. 8. Velocity profiles for different values of K.

1

0.5

= 1.0, 2.0, 3.0, 4.0

0 0

1

2

Fig. 9. Velocity profiles for different values of N.

1

= 1.0, 2.0, 3.0, 4.0

0.5

0 0

1

2

3

Fig. 10. Temperature profiles for different values of N.

R.N. Barik, G.C. Dash / Applied Mathematics and Computation 226 (2014) 423–434

431

1

0.5

= 1.0, 2.0, 3.0, 4.0

0 0

1

2

Fig. 11. Velocity profiles for different values of Q.

1

0.5

= 1.0, 2.0, 3.0, 4.0

0 0

1

2

Fig. 12. Temperature profiles for different values of Q.

1

0.5

= 1.0, 2.0, 3.0, 4.0

0 0

1

2

Fig. 13. Velocity profiles for different values of kr.

under the effects of Grashof number ðGrÞ, modified Grashof number ðGcÞ; Eckert number ðEcÞ, permeability parameter ðKÞ and thermal radiation parameter ðNÞ but decreases with an increase in the values of Prandtl number ðPrÞ, Schmidt number ðScÞ, Hartmann number ðMÞ; heat absorption parameter ðQ Þ and chemical reaction parameter ðkr Þ .Therefore, it is concluded that heavier species with low diffusivity under the influence of magnetic field reduces the shearing stress at the plate whereas other parameters are not favorable for reduction. From Table 2 it is evident that rate of heat transfer i.e. Nusselt number ðNuÞ at the plate gets increased due to Eckert number ðEcÞ;thermal radiation parameter ðNÞ and time ðtÞ but the reverse effect is observed in case of Prandtl number ðPrÞ; and heat absorption parameter ðQ Þ. Table 3 shows that the rate of mass transfer i.e. Sherwood number ðShÞ decreases with heavier species under the influence of destructive chemical reaction but finally as time elapses the rate of mass transfer gets increased. Table 4 presents a comparison of our result with that of Reddy et al. [17] which establishes the stability of the numerical method used here and the conformity of the result predicted in the present study under the action of forcing forces giving rise to various pertinent parameters for Gr = Gc = 1.0, Pr = 0.71, Sc = 0.22, M = 1.0, K = 1.0, N = 1.0, Q = 1.0 and kr = 1.0.

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R.N. Barik, G.C. Dash / Applied Mathematics and Computation 226 (2014) 423–434

1

0.5

= 1.0, 2.0, 3.0, 4.0

0 0

1

2

Fig. 14. Concentration profiles for different values of kr.

1

= 0.025, 0.71,1.0, 7.0 0.5

0 0

1

2

3

Fig. 15. Temperature profiles for different values of Pr.

1

= 0.22, 0.30, 0.60, 0.66,

0.5

0 0

1

2

3

Fig. 16. Concentration profiles for different values of Sc.

1

0.5

= 0.5, 1.0, 1.5, 2.0

0 0

1

2

Fig. 17. Temperature profiles for different values of t.

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1

0.5

= 0.5, 1.0, 1.5, 2.0

0 0

1

2

Fig. 18. Concentration profiles for different values of t.

Table 1 Skin-friction results (s) for the values of Gr, Gc, Pr, Sc, M, K, Ec, N, Q, and kr. Gr

Gc

Pr

Sc

M

K

Ec

N

Q

kr

s

1.0 2.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

1.0 1.0 2.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

0.71 0.71 0.71 7.00 0.71 0.71 0.71 0.71 0.71 0.71 0.71

0.22 0.22 0.22 0.22 0.30 0.22 0.22 0.22 0.22 0.22 0.22

1.0 1.0 1.0 1.0 1.0 2.0 1.0 1.0 1.0 1.0 1.0

1.0 1.0 1.0 1.0 1.0 1.0 2.0 1.0 1.0 1.0 1.0

0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.100 0.001 0.001 0.001

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 1.0 1.0

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 1.0

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0

1.3583 2.5584 3.2069 1.2124 1.2236 1.1588 1.4170 1.3651 1.3844 1.3126 1.3115

Table 2 Rate of heat transfer (Nu) values for different values of Pr, Ec, N and Q. Pr

Ec

N

Q

t

Nu

0.71 7.00 0.71 0.71 0.71 0.71

0.001 0.001 0.100 0.001 0.001 0.001

1.0 1.0 1.0 2.0 1.0 1.0

1.0 1.0 1.0 1.0 2.0 1.0

1.0 1.0 1.0 1.0 1.0 2.0

0.6874 0.5541 0.7259 0.7143 0.6057 0.7015

Table 3 Rate of mass transfer (Sh) values for different values of Sc and jr. Sc

kr

t

Sh

0.22 0.30 0.22 0.22

1.0 1.0 2.0 1.0

1.0 1.0 1.0 2.0

1.4291 1.3248 1.3682 1.4457

Table 4 Comparison of present skin-friction results (s) with the skin-friction results (s⁄) obtained by Reddy et al. [17] for different values of Gr, Gc, Pr, Sc, M, K, N, Q, and kr. Gr

Gc

Pr

Sc

M

K

N

Q

kr

s

s⁄

1.0 2.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

1.0 1.0 2.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

0.71 0.71 0.71 7.00 0.71 0.71 0.71 0.71 0.71 0.71

0.22 0.22 0.22 0.22 0.30 0.22 0.22 0.22 0.22 0.22

1.0 1.0 1.0 1.0 1.0 2.0 1.0 1.0 1.0 1.0

1.0 1.0 1.0 1.0 1.0 1.0 2.0 1.0 1.0 1.0

1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 1.0 1.0

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 1.0

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0

1.3486 1.3774 1.4157 1.3210 1.3140 1.3115 1.3544 1.3612 1.3265 1.3301

1.3467 1.3768 1.4149 1.3201 1.3137 1.3109 1.3535 1.3606 1.3257 1.3297

434

R.N. Barik, G.C. Dash / Applied Mathematics and Computation 226 (2014) 423–434

5. Conclusion We summarize below the following results of physical interest on the velocity, temperature and concentration distributions of the flow field and also on the skin-friction, rate of heat and mass transfer at the wall. 1. Both the methods of solution i.e. analytical method (Reddy et al.) and numerical method (the present study) yield the same result. This aids to the conformity of the present solution. 2. Retarding effect of magnetic field on velocity distribution is magnified due to the inclusion of viscous dissipative heat. 3. An increase in viscous dissipative heat ðEcÞ leads to decrease both in velocity and temperature profiles. 4. Heavier species with low thermal diffusivity causes significant fall in concentration and temperature distribution. 5. Heavier species with low diffusivity under the influence of magnetic field reduces the shearing stress at the plate whereas other parameters do not favor for reduction. 6. Comparison of our result with that of Reddy et al. [17] which establishes the stability of the numerical method used here and the conformity of the result predicted in the present study under the action of forcing forces. 7. An increase in Hartmann number or Prandtl number or Schmidt number or heat absorption parameter or chemical reaction parameter retard the velocity of the flow field at all points. 8. The concentration decreases with increasing values of Schmidt number and chemical reaction parameter but the reverse effect is observed in case of time of the flow field at all points. 9. The rate of heat transfer decreases with increasing of Prandtl number and heat absorption parameter and increases with increasing of Eckert number and thermal radiation parameter. 10. The rate of mass transfer decreases with increasing of Schmidt number and chemical reaction parameter. 11. On comparing the skin-friction ðsÞ results of present study with the skin-friction (s ) results of Reddy et al. [17], it can be seen that they agree very well.

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