On MHD Free Convection Slip Flow of An Elastico-Viscous Fluid with Heat Source and Mass Transfer Through A Porous Medium

On MHD Free Convection Slip Flow of An Elastico-Viscous Fluid with Heat Source and Mass Transfer Through A Porous Medium

*M. M. Hamza, *B.Y Isah, *N. S Dauran and **I. D Yale

*Department of Mathematics, Usmanu Danfodiyo University, Sokoto, Nigeria

**Department of Mathematics, Kebbi State University of Science and Technology, Aleiro, Nigeria

ABSTRACT

MHD free convection elastic-viscous slip fluid flow with Heat source and chemical reaction through a porous medium is analyzed, by taking into account the effect of mass transfer and Slip parameter. A uniform magnetic field acts perpendicular to the porous surface, which absorbs the fluid with a constant suction velocity. The dimensionless governing equations for this investigation are solved analytically using two-term harmonic and non-harmonic functions. The effects of various controlling parameters associated with this flow problem are discussed with the help of graphs. The results show that the velocity increases with increasing Slip and permeability parameters, while decreases with increasing Prandtl number, Magnetic parameter, Schmidt number, Chemical reaction parameter, and Heat source parameter. The study also revealed that the visco-elastic parameter has no significant effect on velocity profile.

1. In many practical engineering applications the free convective heat and mass transfer flow of an electrically conducting fluid is of immense importance in the design of control systems for modern free convection heat exchange devices, magnetohydrodynamic (MHD) generator, oil exploration, plasma studies and geothermal energy extractions. Many common liquids such as oils, certain paints, polymer solution, some organic liquids and many new material of industrial importance exhibit both viscous and elastic properties. Therefore, the above fluid called elastico- viscous fluids is being studied extensively. Many researchers have shown their interest in the fluctuating and non fluctuating flow of a viscous incompressible fluid past an infinite or semi- infinite flat plate. For class of problems related to geothermal energy system there is a need for including constant heat sources or at times heat generating in porous media. Despite the importance of the unsteady heat and mass transfer of an elastico-viscous fluid with heat source in many engineering systems and applications, the attention it has received to date is not enough.

   Mention may be made of the research studies of Raptis and Tzivanidis (1981), Chowdhury and Islam (2000). Basant Kumar Jha and Ravindra Prasad (1991) have studies the effects of heat source on MHD free convection and mass transfer flow through a porous medium. Rahman and Sattar (1999) analyzed MHD free convection and mass transfer flow with oscillating plate velocity and constant heat source in a rotating frame of reference. Recently Rajesh (2011) investigated the effects of heat source and mass transfer on MHD flow of an elasto-viscous fluid through a porous medium.

   The phenomenon of slip flow regime has attracted the attention of a large numbers of researchers due to its wide ranging applications. The fluid slippage at the solid boundaries appeared in many applications such as micro channels or Nano channels etc. Several excellent studies have also considered slip condition and chemical reaction effects on unsteady heat and mass transfer flow of an electrically conducting fluid. Excellent articles include those by Watanebe et al (1998), Khalid and Vafai (2004), Makinde and Osalusi (2006), Mehmood and Ali (2007). More recently Seth et al (2010) investigated the unsteady MHD convective flow within a parallel plate rotating channel with thermal source/ Sink in a porous medium under slip conditions. Rakesh (2011) studied the effect of slip conditions and hall current on unsteady MHD flow of a visco-elastic fluid past an infinite vertical porous plate through porous medium.

   This paper investigate the combined effects of mass transfer, heat source, chemical reaction and slip conditions on MHD free convection flow of a visco-elastic fluid past an infinite vertical oscillating plate through a channel filled with porous medium.

2. We consider unsteady free convection and mass transfer flow of a viscous incompressible and electrically conducting visco-elastic fluid past an infinite vertical plate through a porous medium in the presence of first order chemical reaction and heat source. A magnetic field of uniform strength B0 is applied transversely to the plate. The flow is assumed to be in x-direction which is taken along the vertical plate in the upward direction. The y-axis is taken to be normal to the plate. The plate is subjected to a constant suction velocity. The effect of viscous dissipation is assumed to be negligible. Then by usual Boussinesq approximation, the flow governing equations are as follows:

   u'

   u' 2u'

   ' '

   * ' '

   3u'

   B2u' u'

   o g (T T ) g

   (C C ) ko

   o

   (1)

   t'

   y'

   y'2

   y't' K '

   T ' T '

   k 2T ' Q* (T ' T ' )

   o

   (2)

   t' y'

   C y'2 C

   p p

   C'

   C'

   2C'

   ' '

   t'

   o y'

   D y'2

   Kr (C C )

   (3)

   The boundary conditions are

   ' * u'

   ' ' ' '

   it

   ' ' ' '

   it '

   u

   y'

   0, T

   T (T T )e

   , C C (C C )e

   , at y 0

   (4)

   u' 0,

   T ' T ' ,

   C' C' ,

   as y'

   On introducing the following dimensionless quantities and parameters:

   u'

   2t'

   y'

   '

   Cp

   2t'

   Q *

   u , t o , t o , , Pr

   2

   , t o , F

   2

   0 o

   k o Cp

   T ' T '

   C' C'

   g (T ' T ' ) g * (C' C' )

   (5)

   , C

   T ' T '

   , Gr

   C' C'

   , Gc

   3 3

   o o

   B2

   K ' 2

   k 2

   K ' *

   M o ,

   K o ,

   S o o ,

   Kr r , Sc , o

   2

   2 2 2

   D

   o o

   Where u ' is the axial velocity, t '

   is the time,T '

   denotes the temperature of the fluid at the plate,

   T

   C

   ' denotes the temperature of the fluid far away from the plate, g the gravitational force, C '

   denotes the concentration of the fluid at the plate,

   ' ,denotes the concentration of the fluid far

   0

   away from the plate, D is mass diffusivity, K is the mean permeability of the porous medium ,

   is the frequency of the oscillation, is the coefficient of thermal expansion, *

   is the

   coefficient of concentration expansion, Bo the electromagnetic induction,e the conductivity of the fluid, the density of the fluid, v is the kinematics viscosity coefficient, vo is mean suction velocity, Gr is the Grashoff number, Gc is the mass Grashoff number, M is the Hartmann number, Pr is the Prandtl number, is the Slip parameter, Sc is the Schmidt number, is the

   dimensionless temperature, C is the dimensionless concentration, Kr is the chemical reaction parameter,S is the visco-elastic parameter and F is the heat source parameter.

   Equations (1) to (3) with the help of (5) reduce to

   u u 2u Gr u

   3u

   t y

   y2

   GcC Mu

   K S y2t

   (6)

   1 2

   F

   t y Pr y2

   (7)

   C C 1 2C

   t y Sc y2

   Kr C

   (8)

   The corresponding boundary conditions become

   u u 0, 1 eit , C 1 eit at y 0

   y u 0,

   0,

   C 0

   as y

   (9)

3. In order to solve equations (6) to (8) under the boundary conditions (9), we assumed the fluid velocity, temperature and concentration of the fluid as follows:

   U ( y, t) u ( y) u ( y)eit

   o 1

   it

   1

   ( y, t) o ( y) ( y)e

   (10)

   C( y, t) C ( y) C ( y)eit

   o 1

   Putting equation (10) into equations (6) to (8), we obtain

   u'' u' m u Gr GcC

   (11)

   0 0 1 o o o

   (1 iA)u'' u' m u Gr GcC

   (12)

   1 1 2 1 1 1

   0 0

   o

   '' Pr ' F Pr 0

   (13)

   1 1 1

   '' Pr ' Pr(i F ) 0

   (14)

   C'' ScC' ScK C 0 (15)

   0 0 r o

   1 1

   r

   C'' ScC' Sc(i K

   )C1 0

   (16)

   The corresponding boundary conditions become

   u u' 0,u

   u' 0,

   1, C C

   1, at y 0

   o o

   uo u1 0,

   1 1 o

   o 1 0,

   1 o 1

   Co C1 0

   as y

   (17)

   Where the primes denoted differentiation.

   Solving equations (11) to (16) with the help of (17), and using (10), we have

   ( y, t) ep y eit p y

   C( y, t) er1 y eit r2 y

   (18)

   (19)

   1 1 2 2 3

   U ( y, t) B e2 y Dep y D er1 y (B e4 y D ep y D er2 y )eit

   (20)

   The shear stress at the lower wall of the channel is given by

   U B h D r D ( B

   • h D

   • r D )eit

   (21)

   y y0

   2 1 1 1 1 4 2 2 2 2 3

   Where

   1

   1 Pr

   Pr2 4F Pr

   Pr

   Pr2 4 Pr(i F )

   m1 M K , m2 M K i , p 2 , p 2 ,

   Sc Sc2 4ScK

   Sc Sc2 4Sc(i K )

   1 1 4m

   r r , r

   r ,

   1 ,

   1

   1

   2

   1 4(1 iA)m2

   2

   , D

   Gr

   2

   , D

   Gc

   2 2

   , D

   Gr ,

   2 2 p h m

   1 r 2 r m

   2 (1 iA)p h m

   1 1 1

   1 1 1

   2 2 2

   D Gc , B

   [1 p ]D [1 r1 ]D1 , B

   [1 p ]D2 [1 r2 ]D3

   3 (1 iA)r 2 r m 1 1 2 1

   2 2 2 2 4

4. In order to investigate the physical situation of the problem, we plotted velocity profiles for different parameters such as Slip parameter, Magnetic parameter, chemical reaction parameter, permeability parameter, Heat source parameter, visco-elastic parameter, Prandtl number and Schmidt number. Figure (1) displays the influences of Slip parameter ( = 0, 0.5, 1.0, 1.5) on the velocity field. From this figure we observed that, as the values of Slip parameter increases the velocity increase. Figure (2) represent the velocity profiles for different values of the Heat source parameter (F = 0, 0.5, 1.0, 1.5), it is found that the velocity decreases with increasing Heat source parameter. Figure (3) illustrate the velocity profiles for different values of the Prandtl number (Pr

   = 0.71, 0.85, 1.0, 2.0) from this figure it is observed that the velocity decreases with increasing the Prandtl number. Figure (4) represent the velocity profiles for different values of the Magnetic parameter (M = 1, 2, 3, 4), it is found that the velocity decreases with increasing magnetic parameter. It is because that the application of transverse magnetic field will result a resistivity type force (Lorentz force) similar to drag force, which tends to resist the fluid flow and thus reducing its velocity. Figure (5) display the velocity profiles for different values of the Schmidt number (Sc = 0.22, 0.60, 0.78, 2.00). From this figure the velocity is found to decrease with increasing Schmidt number. Figure (6) illustrate the velocity profile for different values of the visco-elastic parameter (S = 0, 5, 10, 100). It is observed that the velocity does not change with increasing visco-elastic parameter. Figure (7) display the velocity profiles for different values of the permeability parameter (K = 0.1, 0.3, 0.5, 0.7). It is observed that the velocity increases with increasing permeability parameter. This is due to the fact that the presence of a porous medium

   increases the resistance to flow. The influences of chemical reaction parameter (Kr = 1, 2, 3, 4) on the velocity profiles are presented in figure (8). It is observed that the velocity decreases with increasing chemical reaction parameter.

   Figure1: Velocity profile for different values of Slip parameter.

   Figure2: Velocity profiles for different values of the Heat source parameter.

   Figure3: Velocity profiles for different values of the Prandtl number.

   Figure4: Velocity profiles for different values of the Magnetic parameter.

   Figure5: Velocity profiles for different values of the Schmidt number.

   Figure6: Velocity profiles for different values of the Visco-elastic parameter.

   Figure7: Velocity profiles for different values of the Permeability parameter.

   Figure8: Velocity profiles for different values of the Chemical reaction parameter.

We investigate the combined effects of mass transfer, heat source, chemical reaction and slip conditions on MHD free convection flow of a visco-elastic fluid past an infinite vertical oscillating plate through a channel filled with porous medium. The governing equations are solved using perturbation method. The result shows that the velocity rises with increasing the slip parameter and permeability parameter, and decreases with increasing the values of the Prandtl number, heat source, magnetic parameter, Schmidt number and chemical reaction parameter. While there is a negligible effect of visco-elastic parameter on velocity profiles.

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