Effect of Mass Transfer and Mixed Convection on A Steady MHD Flow over A Porous Flat Plate

Effect of Mass Transfer and Mixed Convection on A Steady MHD Flow over A Porous Flat Plate

Vibhor Tomer1

Department of Mathematics, Statistics and Computer Science

G. B. Pant University of Agriculture & Technology Pantnagar – 263145, Uttarakhand, India

Manoj Kumar2

Department of Mathematics, Statistics and Computer Science

1. B. Pant University of Agriculture & Technology Pantnagar – 263145, Uttarakhand, India

   Abstract A steady mixed convection flow over a porous plate has been considered to investigate the combined effects of suction parameter, radiation parameter, Schmidt number, Prandtl number. The governing boundary layer equations are transformed into a non-dimensional form by group transformation and finally solved by using Runga-Kutta method with shooting technique. The numerical results have been depicted graphically to illustrate the influence of the mixed convection parameter and other various parameters along with Prandtl number on velocity, temperature and concentration profiles. The results for the skin-friction coefficient, Nusselt number and Sherwood number have also been analyzed. Good agreement is found between the numerical results of the present paper with published result for special case.

   Key Words: Mixed convection, variable viscosity, MHD flow, radiation, thermal conductivity, flat plate

   1. INTRODUCTION

      The problem of laminar hydrodynamic and thermal boundary layers over the flat plate in a uniform stream of fluid is a thoroughly researched problem in fluid mechanics. Hamad et al. [1], studied magnetic field effects of a nano-fluid past a vertical semi-infinite flat plate using group transformation. Reviews for the applications of group theory to differential equations can be found in the various researches done by [2-7]. The radiative flow of an electrically conducting fluid and heat and mass transfer situation arises in many practical applications, such as, in electrical power generation, solar power technology, space vehicle re-entry, nuclear reactors. It also occurs in many geophysical and engineering applications such as nuclear reactors, migration of moisture through air contained in fibrous insulations, nuclear waste disposal, dispersion of chemical pollutants through water-saturated soil and others as studied by Arasu et al. [7] and Chamakha et al. [9]. Radiation effect on boundary layer flow with and without applying a magnetic field has been investigated researchers [10-13]. Similarity representation of MHD flow with heat transfer taking into consideration variable viscosity and thermal conductivity by Seddeek et al. [14]. Mahanti et al.

      [15] investigated the effects of variable viscosity and thermal conductivity, which vary linearly on steady free convective flow of a viscous incompressible fluid along an isothermal vertical plate in the presence of heat sink. Recently, thermal convective surface boundary conditions were used by Aziz [16] and Makinde et al. [17]. They studied to solve different types of boundary layer

      equations. Recently, Hamad et al. [18] studied a steady laminar 2-D MHD viscous incompressible flow over a permeable flat plate with thermal convective boundary condition and radiation effects. The viscosity and thermal conductivity of fluid are assumed to vary linearly with temperature.

      The objective of present investigation is to study mixed convection flow over a permeable porous plate. To find the solution, authors are using similarity and group method of transformation. The attempt has also been made to study the effects of radiation, suction and thermal convective parameters on the fluid flow and the rate of heat and mass transfer.

   2. MATHEMATICAL FORMULATION OF THE PROBLEM

      Consider the steady mixed convective flow of a viscous incompressible electrically conducting fluid past an infinite vertical porous plate in a porous medium of time independent permeability in presence of a transverse magnetic field B0 as shown in the figure of physical model. Let x -axis be along the plate in the direction of flow and y- axis is normal to it. The velocity components along x and y axes are u and v , T and C be the fluid temperature and concentration. Further , , , k, R and M are the coefficient of viscosity, density, electric conductivity, thermal conductivity, radiation parameter and magnetic parameter of the fluid.

      Alam et al. [19] considered and it has been assumed that the magnetic Reynolds number is much less than unity so that the induced magnetic field is neglected in comparison to the applied magnetic field B0. The suction or injection are imposed on the permeable plate. The temperature of the plate surface is held uniform at Tw which is higher than the ambient temperature T. The physical model has been given below:

      u u du 2u

      1 T u

      u v u e

      x y e dx y2

      T y y

      (7)

      B 2

      0 (u

      u e )

      • u

      K

      gC

      C

      Assumed the viscosity and thermal conductivity as linearly temperature dependent [19]:

      (t) [1 b0 (Tf T)],

      k(t) k[1 c(T T )]

      Fig. 1. Physical model

      Where,

      and

      k are the constant undisturbed

      The species concentration at the plate surface is maintained uniform at Cw while the ambient fluid concentration is assumed to be C. Further, all the fluid properties are assumed to be constant except that of the dynamic viscosity and thermal conductivity. The bottom surface of the plate is heated by convection from a hot fluid

      viscosity and thermal conductivity, b0 > 0, c are constants depend on fluid.

      Using Rosselands approximation for radiation from [21], we obtained

      4 T4

      of temperature Tf it generates a heat transfer coefficient hf as taken by Aziz [16].

      qr 1

      3k y

      (8)

      1

      Under the above assumptions, the governing equations for the problem can be written as Kays et al. [20].

      Where 1 is the StefanBoltzman constant and k1 is

      u v 0

      x y

      (1)

      the absorption coefficient. It is assumed that the

      temperature variation within the flow is such that T4 may be expanded in a Taylor series about T and neglecting higher order terms, we get

      2 T4 4TT3 3T4

      (9)

      u u v u 1 P u 1 T u

      x y

      x

      y2

      T y y

      (2)

      Equations (8) and (9) give

      B2 u

      0 u

      gC

      C

      q 16 T

      3 2T

      K

      r 1 (10)

      y 3k1 y

      2

      u T v T

      x y

      k(T) T 1 qr

      y y c y

      p

      (3)

      Using equations (8) and (10), the energy equation (3) becomes

      C C

      2C

      16 3

      2

      u v D

      (4)

      u v 1 S 1

      (11)

      m 2

      x y c y y 3 c y2

      x y y p 1 p

      The boundary conditions are given by

      Here S c(Tf T ) is thermal conductivity parameter.

      u 0, v v

      ,C C ,k T h T T

      at y 0

      Now, the following dimensionless variables have been

      w w y

      f f w

      introduced as considered by Hamad et al. [18]:

      u ue (x),T T ,C C

      as y

      (5)

      x y Re

      u v Re u

      In the free stream flow, u = ue(x ) and hence momentum equation (2) becomes

      x , y

      l

      T T ,

      , u , v

      l u

      C C

      , ue e ,

      u u

      dx

      1 p

      x

      B2

      0 ue (x)

      (6)

      Tf T

      u ,

      Cw C

      v

      (12)

      Using equations (2) and (6), equation of momentum becomes

      y x

      Where Re = u l/ is the Reynolds number, is the stream function, l being the characteristic length and u is reference velocity.

      Hence, equations (7), (11) and (3) reduce in the following form:

      3

      y xy2

      3 4

      x y3 y4

      2 2 3 A y2 y2 y y3

      (19)

      2 2 3 2 du

      2

      A ue e

      y xy x y2 y3 y y2 dx

      (13)

      y2

      (M c ) y 0

      e

      M u C 0

      y y

      By using above group transformation in equation (19), we get the following relation

      1

      4R 2

      1 32 23 42 3 22 3 2 3

      (20)

      y x x y

      Pr 1 S

      3 y2

      (14)

      On solving the equation, we get 1 3 , 2 0

      1 2

      S 0

      Pr y

      Similarly equations (14), (15) and (16) are also giving

      1

      2

      0

      (15)

      1 3 , 2 0 , so these equations show invariant under the group transformation (18).

      y x

      x y

      Sc y2

      Now the characteristic equations are

      Subject to boundary conditions,

      dx dy d d d

      (21)

      0, vw , 1,

      x 0 0 0

      y x u Re

      (16)

      Which give the following similarity transformations:

      lh f 1 0

      y

      Re 1 S0

      at y 0

      y, xf ,

      ,

      and

      (22)

      Using these transformations, the momentum, energy

      y ue , 0, 0

      as y

      and mass equations become

      f (A f)f M(f 1) f 2 cf (23)

      Where

      B2l

      gl 3C

      • C

      cp

      1

      1 S 4R 3

      S2 Prf

      (24)

      M 0 , Gr w , Pr ,

      u

      4 T3

      k2 k

      Scf

      (25)

      Sc

      , R 1

      Dm k1k

      Subject to the boundary conditions

      A b(T T ),

      Gr

      , k

      , c

      l

      (17)

      f fw

      , f 0, 1 b , 1

      1 Sb

      at 0

      f Re5/2

      U KU

      f 1,

      0,

      0

      as

      (26)

      The application of group transformations has been considered to find similarity reduction of equations (13),

      (14) and (15). Consider the following group transformations

      The physical quantities of interest are the Skin friction coefficient Cf, Nusselt number Nu and Sherwood number Sh, which are defined as

      u x T

      x# x1 ,

      y# y2 ,

      # 3 ,

      # ,

      # (18)

      Cf u 2 y

      , Nu

      T T

      y ,

      (27)

      Where 1, 2, 3 are constants and is the parameter of

      e

      Sh x

      C C

      y0

      C

      y

      f

      y0

      point transformation. Now finding the relation among s

      such that

      w

      y0

      # # # # #

      3 #

      3

   3. METHOD OF SOLUTION

      j (x , y , , , ,….., 3 ) H j (x, y,,, ,….., 3 )

      y# y

      3

      j (x, y,,, ,….., y3 )(j 1,2,3)

      1, 2 and 3 are conformally invariant under the group transformation (18), [2].

      By equation (13), we have

      The system of ordinary differential equations (23), (24) and (25) subject to the boundary conditions (26) have been solved numerically using Runga-Kutta method with shooting technique. The computations were carried out using step size of = 0.01 selected to be satisfactory for a convergence criterion of 10-6 in all cases.

      The physical quantities skin friction coefficient Cf, Nusselt number Nu and Sherwood number Sh indicate the wall shear stress, rate of heat transfer and rate of mass transfer respectively and these are proportional to the numerical values of f(0), (0) and (0) respectively.

   4. RESULTS AND DISCUSSION

The numerical results have been computed and represented in the form of the dimensionless velocity, temperature, concentration, wall heat transfer, the rate of heat and mass transfer. Prandtl number Pr = 0.7 for air at 1 atmospheric pressure, Schmidt number Sc = 0.22 for Hydrogen, Sc = 0.67 for water vapour, Sc = 0.78 for Ammonia were taken. The values for the skin friction

1.0

0.9

0.8

0.7

f, ,

0.6

0.5

0.4

0.3

0.2

0.1

0.0

f w = 0.1 , 0.5 , 1

f w = 0.1 , 0.5 , 1

f'

f w = 0.1 , 0.5 , 1

coefficient, Nusselt number and Sherwood number have been tabulated below:

Table 1. Effect on Skin friction coefficient Cf, Nusselt number Nu and Sherwood number Sh for fw = 0.5, Pr = 0.7, M

= 0.5, R = 1, Sc = 0.1, A = 0.1, S = 1, = 0.6, a = 1, b = 0.3

and c =0.2.

0 1 2 3 4 5 6 7 8 9

Fig. 2. Effect of suction parameter fw on velocity f, temperature , concentration , for Pr = 0.7, M = 0.1, R = 1 Sc = 0.1, A = 0.1, S = 0.5,

= 0.6, a = 1, b = 0.1 and c =0.2.

1.0

0.9

0.8

0.7

f' ,

0.6

0.5

0.4

R = 1, 5, 10

f'

0.3

0.2

0.1

0.0

R = 1, 5, 10

0 1 2 3 4

5 6 7 8 9

Fig. 3. Effect of radiation parameter R on velocity f, temperature , for fw

= 0.5, Pr = 0.7, M = 0.1, Sc = 0.1, A= 1, S = 1, = 0.6, a = 1, b = 0.5, and

c =0.2.

1.0

Figure 2 exhibits the effect of physical parameters on velocity f, temperature and concentration . It is seen that the suction has a significant effect on the boundary layer thicknesses. It can be observed that the velocity f rises with suction parameter whereas temperature and concentration fall with rising fw. It is also noticed that the thickness of momentum, thermal and concentration boundary layer reduce with an increase in fw. The variation of velocity f and temperature for different values of the radiation parameter R have been depicted in Figure 3. It reveals that the velocity f and temperature increase with

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Sc = 0.22, 0.67, 0.78

an increase in radiation parameter R. This is because rises in R have the tendency to increase the conduction effects

0 1 2 3 4

5 6 7 8 9

and to increase temperature at each point away from the surface. Therefore, higher value of radiation parameter implies higher surface heat flux. It is also observed that momentum boundary layer thickness decreases while the thermal boundary layer thickness increases with the increasing values of R.

Fig. 4. Effect of Schmidt number Sc on concentration for fw = 0.5, Pr = 0.7, M = 0.1, R = 1, A = 0.1, S = 1, = 0.7, a = 1, b = 0.3, and c =0.2.

The effect of Schmidt number on concentration is represented through figure 4. It has been observed that as Schmidt number increases, the mass transfer rate increases and concentration decreases. There is a little change in

temperature and concentration in case of moderate changes in Schmidt number Sc.

1.0

0.9

0.8

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    0.9

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    0.5

    0.4

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    Fig. 6. Effect of convective heat transfer parameter b on velocity f, temperature , concentration , for fw = 0.5 Pr = 0.7, M = 0.6, R = 1, Sc = 0.78, = 0.6, A =0.5, S = 0.3, a = 1 and c =0.2.

    To show the variations of thermal convective parameter b on the field variables velocity f, temperature and concentration respective we have drawn figure 6. This figure shows that velocity f concentration and temperature reduce with increasing value of b.

    The authors also attempted the case study the effect of injection parameter. The results were also seen with the good agreement as done by Hamad et al. [18].

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