11 maths ijamss unsteady mhd mixed madhusudhana rao by IASET Journals

International Journal of Applied Mathematics & Statistical Sciences (IJAMSS) ISSN(P): 2319-3972; ISSN(E): 2319-3980 Vol. 2, Issue 5, Nov 2013, 93-116 ÂŠ IASET

UNSTEADY MHD MIXED CONVECTION OF A VISCOUS DOUBLE DIFFUSIVE FLUID OVER A VERTICAL PLATE IN POROUS MEDIUM WITH CHEMICAL REACTION, THERMAL RADIATION AND JOULE HEATING B. MADHUSUDHANA RAO1, G. VISWANATHA REDDY2 & M. C. RAJU3 1

Department of Mathematics, R.M.K. Engineering College, Chennai, Tamil Nadu, India 2

Department of Mathematics, S.V. University, Tirupati, Andhra Pradesh, India

Department of Mathematics, Annamacharya Institute of Technology and Sciences (Autonomous), Rajampet, Andhra Pradesh, India

ABSTRACT The problem of 2-dimensional unsteady boundary layer MHD mixed double diffusive flow of a viscous incompressible, electrically conducting fluid along a semi-infinite vertical permeable moving plate in presence of a transverse magnetic field, chemical reaction, heat absorption and thermal radiation is considered. The dimensionless governing partial differential equations for this study are solved analytically by using 2-term harmonic and non-harmonic functions. Furthermore, the plate is assumed to move with a constant velocity in fluid flow direction while the free stream velocity is assumed to follow the perturbation rule. Numerical evaluation of the analytical results are performed and few graphical results for the velocity, temperature and concentration distributions and tabulated results for Skin friction, Nusselt number and Sherwood numbers are discussed and presented.

KEYWORDS: MHD, Mixed Convection, Heat and Mass Transfer, Thermal Radiation, Heat Absorption, Thermal Diffusivity, Permeability and Chemical Reaction

1. INTRODUCTION Mixed convection flows with simultaneous heat and mass transfer in porous media under the influence of a magnetic field and chemical reaction are frequently encountered in many transport processes in nature. Its application is found in many industries viz. in the chemical industry, power and cooling industry for drying, chemical vapour deposition on surfaces, cooling of nuclear reactors and magneto hydrodynamic power generators. Simultaneous heat transfer and evaporation of crude oil in different stages of refining process are physical examples of heat and mass transfer. Many transport processes exist in nature and in industrial applications in which the simultaneous heat and mass transfer occurs as a result of combined buoyancy effects of diffusion of chemical species. We are particularly interested in cases in which diffusion and chemical reaction occur at roughly the same speed. When diffusion is much faster than chemical reaction, then only chemical factors influence the chemical reaction rate. When diffusion is not much faster than reaction, the diffusion and kinetics interact to produce very different effects. The study of heat generation or absorption effects in moving fluids is important in view of several physical problems, such as fluids undergoing exothermic or endothermic chemical reaction. Thermal diffusion effect has been utilized for isotopes separation in the mixture between gases with very light molecular weight (hydrogen and helium) and medium molecular weight. Chambre and Young [1] have presented a first order chemical reaction in the neighborhood of a horizontal plate. Dekha et al. [2] investigated the effect of the first order homogeneous chemical reaction on the process of an unsteady flow past a vertical plate with a constant heat and mass transfer. Muthucumaraswamy [3] presented heat and mass transfer effects on a continuously moving isothermal

B. Madhusudhana Rao, G. Viswanatha Reddy & M. C. Raju

vertical surface with uniform suction by taking into account the homogeneous chemical reaction of first order. A comprehensive description of the theoretical work for both laminar and turbulent mixed convection boundary layer flows has been given in a review paper by Chen and Armaly [4] and in the book by Pop and Ingham [5]. The problem of mixed convection under the influence of magnetic field has attracted numerous researchers viz. Soundalgekar et al. [6], Elbasheshy [7], Abel et al. [8, 9] in view of its applications in geophysics and astrophysics. The effect of radiation on MHD flow and heat transfer problems has become industrially more important. At high operating temperatures, radiation effect can be quite significant. The effect of variable viscosity on hydro magnetic flow and heat transfer past a continuously moving porous boundary with radiation has been studied by Seddeek [10]. The same author investigated [11] thermal radiation and buoyancy effects on MHD free convective heat generating flow over an accelerating permeable surface with temperature-dependent. The radiation effects on MHD free -convection flow of a gas past a semi-infinite vertical plate is studied by Takhar et al. [12]. The fundamental problem of flow through and past porous media has been discussed by Cheng [13] and Rudraiah [14] on thermal radiation as a mode of energy transfer and emphasize the need for inclusion of radiactive transfer in these processes. In most chemical reactions, the reaction rate depends on the concentration of the species itself. A chemical reaction is said to be of first order, if the rate of reaction is directly proportional to concentration itself. During chemical reaction between two species concentration heat is also generated. Chemical reaction effects on heat and mass transfer laminar boundary layer flow have been discussed by various authors [15, 16, 17, 18, and 19] in various situations. Ramachandra Prasad and Bhaskar Reddy [20] investigated radiation and mass transfer effects on unsteady MHD free convection flow past a heated vertical plate in a porous medium with viscous dissipation. Seddeek et al. [21] analyzed the effects of chemical reaction, radiation and variable viscosity on hydromagnetic mixed convection heat and mass transfer for Hiemenz flow through porous media. Ibrahim et al. [22] analyzed the effects of the 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 Chaudhary et al. [23] have analyzed the effect of radiation on heat transfer in MHD mixed convection flow with simultaneous thermal and mass diffusion from an infinite vertical plate with viscous dissipation. Recently, Pal and Talukdar [24] studied the combined effect of MHD and Ohmic heating in unsteady two-dimensional boundary layer slip flow, heat and mass transfer of a viscous incompressible fluid past a vertical permeable plate with the diffusion of species in the presence of thermal radiation incorporating the first-order chemical reaction. Madhusudhana Rao B, Viswanatha reddy G and Raju M.C [25] studied MHD transient free convection and chemically reactive flow past a porous vertical plate with radiation and temperature gradient dependent heat source in slip flow regime.

2. MATHEMATICAL ANALYSIS We consider a two-dimensional unsteady free convection flow of an incompressible viscous fluid past an infinite vertical porous plate. In rectangular Cartesian coordinate system, we take x-axis along the plate in the direction of flow and y-axis normal to it. It is assumed that the plate moves with a constant velocity in the flow direction in the presence of a transverse applied magnetic field. It is also assumed that the temperature and the concentration at the wall as well as the suction velocity are exponentially varying with time. The equations of continuity, linear momentum, angular momentum, energy and diffusion, which govern the flow field, are solved by using a regular perturbation method. The behavior of the velocity, temperature, concentration, skin-friction, Nusselt number and Sherwood number has been discussed for variations in the physical parameters. Further the flow is considered in presence of temperature gradient dependent heat source and effect of radiation and chemical reaction.

Unsteady MHD Mixed Convection of a Viscous Double Diffusive Fluid over a Vertical Plate in Porous Medium with Chemical Reaction, Thermal Radiation and Joule Heating

Introduce the boundary layer and Boussineq’s approximations. Under the above assumptions, the equations governing the conservation of mass (continuity), momentum, energy and concentration can be taken as follows.

v1 0 y 1

(1)

2 1 u 1 1 p1  2u 1   B0  1 1 u 1 1 1 1 v   g T (T  T )  g c (C  C )   12   1  u  x1   t 1 y1 y k

(2)

1 B0 2  12  T 1  2T 1   1 T  v     u  2 1 C p  Cp t 1 y 1 y 1  C p k

(3)

 u 1  Q 1 q r  1    0 (T 1  T1 ) 1 C p  y  C p y

1 C 1  2C 1  2T 1 1 C 1 1 v  D 12  R(C  C )  D1 12 t 1 y1 y y

(4)

The boundary conditions relevant to the problem are;

u1  U 1p , T 1  Tw1   (Tw1  Tw1 )e n t , C1  Cw1   (Cw1  Cw1 )e n t 11

y 1  0 u1  U 1  U 0 (1  e n t ) , 11

T 1  T1 , C 1  C 1 as y 1   Where

(5)

u 1 and v 1 are the components of velocity along x-axis and y-axis directions, t is the time, g is the

acceleration due to gravity,

 T and  c are the coefficients of thermal expansion and concentration expansion respectively,

 is the kinematic viscosity, k 1 is the permeability of the porous medium,  is the density of the fluid,  is the magnetic permeability of the fluid ,

B0 is the uniform magnetic field, T 1 is dimensional temperature, α is the fluid thermal

diffusivity, C p is the specific heat at constant pressure,

q r is the radioactive heat flux, Q0 is the dimensional heat source,

Tw1 is the temperature of the wall as well as the temperature of the fluid at the plate, T1 is the temperature of the fluid far 1

away from the plate, C is dimensional concentration, D is the molecular diffusivity, as well as the concentration of the fluid at the plate. dimensional velocity,

C 1 is the concentration of the wall

U 1p is the wall dimensional velocity, U 1 is the free stream

U 0 and n 1 are constants, R is chemical reaction parameter and D1 is thermal diffusivity

The equation of continuity (1) yields that the suction velocity v1 at the plate is either a constant or function of time; hence it takes exponential form;

v1  V0 (1  Aen t ) 11

Where

(6)

V0 >0 is a scale of suction velocity at the plate and n 1 is a positive constant, here the negative sign

indicates that the suction velocity acts towards the plate, A is a real positive constant, ε and εA are very small quantities less than unity. Outside of the boundary layer, the equation (2) gives that

B. Madhusudhana Rao, G. Viswanatha Reddy & M. C. Raju

1 p1 dU 1  1 B0 1   1  1 U  U  x1  dt k 2

(7)

The fourth and fifth terms on RHS of energy equation (3) denote the thermal radiation effect and heat absorption effects respectively, The radio active heat flux is given by A.C. Cogley [25] as

q r  4(T 1  T1 ) I 1 y 1 



Where I 1  K  0

(8)

e d , Kλ is the absorption coefficient at the wall and e λ is plank’s function. T 1

3. METHOD OF SOLUTION The following non-dimensional variables are employed,

u

n

2 2 U 1p y 1V0 V K1 V t1 U1 u1 v1 y , v , U   , U p  , K 0 2 t 0    U0 U0 U0 V0

n1 V0

T

C 1  C1 T 1  T1 , C  , Tw1  T1 C w1  C1

(9)

In presence of the above discussed equations (6)--(9), the general equations (2)--(4) take the form;

u u dU   2u  (1  Aent )   2  N (U   u)  GT T  Gc C t y dt y  u  T T 1  2T  (1  Aent )   S  Q T  NEc u 2  Ec   2 t y Pr y  y 

(10)

(11)

C C 1  2C  2T  (1  Aent )   S  C 0 t y S c y 2 y 2 Where,

GT 

gT (Tw1  T1 ) U 0V0

Gc 

(12)

g c (C w1  C1 ) U 0V0

Q

Q0 C pV0 2

4I 1 S , C pV0 2

2 B0 2 U0 D1  Tw1  T1  1   C p    , S  , E  , S  , Pr   , N M  , M  c c 0  K K D   Cw1  C1  C p Tw1  T1  V0 2



R V0

Here GT is thermal Grashof number, Gc is the solutal Grashof number, Q is dimensionless heat absorption coefficient, S is the radiation parameter, Pr is Prandtl number, M is magnetic field parameter, K is a permeability parameter of the porous medium, Ec is Eckert number, Sc is the Schmidit number, S0 is Soret effect parameter and reaction parameter. Moreover, the dimensionless grammar of the boundary conditions (5) takes the form;

 is Chemical

Unsteady MHD Mixed Convection of a Viscous Double Diffusive Fluid over a Vertical Plate in Porous Medium with Chemical Reaction, Thermal Radiation and Joule Heating

u  U p , T  1  e nt , C  1  e nt , at y 1  0 u  U   1  e nt , T  0, C  0, as y 1  

(13)

4. SOLUTION OF THE PROBLEM The partial differential equations (10), (11), (12) cannot be solved in closed form. So we solve analytically by converting them into ordinary differential equations in dimension less grammar. Therefore the expressions for velocity, temperature and concentration can be represented in the following form.

u ( y, t )  f 0 ( y )  e nt f1 ( y )  O( 2 )

(14)

T ( y, t )  g 0 ( y )  e nt g1 ( y )  O( 2 )

(15)

C ( y, t )  h0 ( y )  e nt h1 ( y )  O( 2 )

(16)

Substituting above equations (14)--(16) into (10)--(12), equating the harmonic and non-harmonic terms that is coefficients of  ,  0

and neglecting the higher order terms of

O( 2 ) , one obtains the following pairs of ordinary

differential equations for (f0,g0,h0) and (f1,g1,h1).

f 011  f 01  Nf 0   N  GT g 0  Gc h0

(17)

f111  f11  ( N  n) f1   Af 01  GT g1  Gc h1  ( N  n)

(18)



g 011  Pr g 01  Pr (S  Q) g 0  Pr Ec Nf 0  f 01 2



(19)



g111  Pr g11  Pr ( S  Q  n) g1   APr g 01  2 Pr Ec Nf 0 f1  f 01 f11

 (20)

1 11 h0  h01  h0   S 0 g 011 Sc

(21)

1 11 h1  h11  (n   )h1   Ah01  S 0 g111 Sc

(22)

With the boundary conditions,

f 0  U p , f1  0, g 0  1, g1  1, h0  1, h1  1 at y  0 (23)

f 0  1, f1  1, g 0  0, g1  0, h0  0, h1  0 as y   But the above equations (17)--(22) are still coupled. Therefore, we expand the flow variables as an asymptotic series solution about the Eckert number Ec (Ec<<1 for an incompressible flow)

 

f 0 ( y)  f 00 ( y)  Ec f 01 ( y)  O Ec

B. Madhusudhana Rao, G. Viswanatha Reddy & M. C. Raju

  ( y)  OE 

f1 ( y)  f10 ( y)  Ec f11 ( y)  O Ec g0 ( y)  g00 ( y)  Ec g01

(24)

  h ( y)  h ( y)  E h ( y)  OE  h ( y)  h ( y)  E h ( y)  OE  g1 ( y)  g10 ( y)  Ec g11 ( y)  O Ec

c 01

c 11

On substituting above equations (24) in to (17)--(22), equating the coefficients of E c ,

and neglecting higher

order terms of Ec, we obtain the following sequence of ordinary differential equations. Zero Order

f 00  f 00  Nf 00  N  GT g00  Gc h00

(25)

f10  f10  N  n f10  ( N  n)  Af001  GT g10  Gc h10

(26)

g00  Pr g00  Pr (S  Q) g00  0

(27)

g10  Pr g10  Pr S  Q  ng10   APr g 00 11

(28)

1 11 1 11 h00  h00  h00   S 0 g 00 Sc

(29)

1 11 1 11 1 h10  h10  (n   )h10   Ah00  S 0 g 10 Sc

(30)

First Order

f 01  f 01  Nf 01  GT g 01  Gc h01 11

(31)

f11  f11  N  n f11   Af01  GT g11  Gc h11 11

(32) 2

g 01  Pr g 01  Pr (S  Q) g 01   Pr ( Nf 00  f 001 ) 11



1 g11  Pr g11  Pr S  Q  ng11   APr g01  2Pr Nf 00 f10  f 00 f10 11



(33)

(34)

1 11 1 11 h01  h01  h01  S 0 g 01 Sc 1 11 1 11 1 h11  h11  (n   )h11   Ah01  S 0 g11 Sc

(35)

(36)

Unsteady MHD Mixed Convection of a Viscous Double Diffusive Fluid over a Vertical Plate in Porous Medium with Chemical Reaction, Thermal Radiation and Joule Heating

And the boundary conditions (23) take the shape,

f 00  U p , f10  f 01  f11  0 g 00  g10  1, g 01  g11  0 h00  h10  1, h01  h11  0

At y=0

f 00  f10  1, f 01  f11  0 g 00  g10  g 01  g11  0

y 

h00  h10  h01  h11  0

(37)

On solving the 2nd order differential equations with constant coefficients (25)--(36) under the boundary conditions (37), we get

f 00  1  A33 e  m4 y  A3 e  m1 y  A4 e  m3 y

(38)

f10  1  A39 e  m8 y  A34 e  m4 y  A35 e  m1 y  A36 e  m3 y  A37 e  m2 y  A38 e  m7 y

(39)

g 00  e  m1 y

(40)

g10  A27 e  m2 y  A1e  m1 y

(41)

h00  A28 e  m3 y  A2 e  m1 y

(42)

h10  A32 e  m7 y  A29 e  m3 y  A30 e  m2 y  A31e  m1 y

(43)

f 01  A51e  m9 y  A40  A41e  m5 y  A42 e 2 m4 y  A43e 2 m1 y  A44 e 2 m3 y  A45 e m1 y  A46 e ( m1  m3 ) y  A47 e m3 y  A48 e m4 y  A49 e ( m1  m4 ) y  A50 e ( m3  m4 ) y

(44)

f11  A126 e  m12 y  A100 e  m9 y  A101e  m5 y  A102 e 2 m4 y  A103e 2 m1 y  A104 e 2 m3 y  A105 e  m1 y  A106 e ( m3  m1 ) y  A107 e m3 y  A108 e m4 y  A109 e ( m1  m4 ) y  A110 e ( m3  m4 ) y  A111e m10 y  A112 e ( m4  m8 ) y  A113e ( m2  m4 ) y  A114 e ( m7  m4 ) y  A115 e m8 y  A116 e m2 y  A117 e m7 y  A118 e ( m1  m8 ) y  A119 e ( m1  m2 ) y (45)  A120 e ( m1  m7 ) y  A121e ( m3  m8 ) y  A122 e ( m2  m3 ) y  A123e ( m3  m7 ) y  A124 e m11 y  A125 e m6 y

g 01  A15 e  m5 y  A5  A6 e 2 m4 y  A7 e 2 m1 y  A8 e 2 m3 y  A9 e  m1 y  A10 e ( m1  m3 ) y  A11e m3 y  A12 e m4 y  A13e ( m1  m4 ) y  A14 e ( m3  m4 ) y

(46)

g11  A74 e  m10 y  A52 e  m5 y  A53e 2 m4 y  A54 e 2 m1 y  A55 e 2 m3 y  A56 e  m1 y  A57 e  ( m3  m1 ) y  A58 e m3 y  A59 e m4 y  A60 e ( m1  m4 ) y  A61e ( m3  m4 ) y  A62 e ( m4  m8 ) y  A63e ( m2  m4 ) y  A64 e ( m7  m4 ) y  A65 e m8 y  A66 e m2 y  A67 e m7 y  A68 e ( m1  m8 ) y  A69 e ( m1  m2 ) y  A70 e ( m1  m7 ) y  A71e ( m3  m8 ) y  A72 e ( m2  m3 ) y  A73 e ( m3  m7 ) y

(47)

100

B. Madhusudhana Rao, G. Viswanatha Reddy & M. C. Raju

h01  A26 e  m6 y  A16 e  m5 y  A17 e 2 m4 y  A18 e 2 m1 y  A19 e 2 m3 y  A20 e m1 y  A21e ( m1  m3 ) y  A22 e m3 y  A23e m4 y  A24 e ( m1  m4 ) y  A25 e ( m3  m4 ) y

(48)

h11  A99 e  m11 y  A75 e  m6 y  A76 e  m5 y  A77 e 2 m4 y  A78 e 2 m1 y  A79 e 2 m3 y  A80 e  m1 y  A81e  ( m3  m1 ) y  A82 e m3 y  A83e m4 y  A84 e ( m1  m4 ) y  A85 e ( m3  m4 ) y  A86 e m10 y  A87 e ( m4  m8 ) y  A88 e ( m2  m4 ) y  A89 e ( m7  m4 ) y  A90 e m8 y  A91e m2 y  A92 e m7 y  A93e ( m1  m8 ) y  A94 e ( m1  m2 ) y  A95 e ( m1  m7 ) y

(49)

 A96 e ( m3  m8 ) y  A97 e ( m2  m3 ) y  A98 e ( m3  m7 ) y Where the constants are given in the appendix. In view of the above solutions (38)--(49) and the equations (24) & (14)--(16), the velocity distribution u (y, t), the temperature distribution T(y, t) and the concentration distribution C(y, t) in the boundary layer are given as,

u ( y, t )  1  A e 33

 m4 y

 A3 e

 m1 y

 A4 e



 m3 y



 A40  A41e  m5 y  A42 e 2 m4 y  A43 e 2 m1 y  A44 e 2 m3 y  A45 e  m1 y   c ( m1  m3 ) y  m3 y ( m3  m4 ) y  m4 y ( m1  m4 ) y   A e  A e  A e  A e  A e 47 48 49 50  46 

  E  A e

 m9 y



 1  A39 e m8 y  A34 e m4 y  A35 e m1 y  A36 e m3 y  A37 e m2 y  A38 e m7 y     A126 e m12 y  A100 e m9 y  A101e m5 y  A102 e 2 m4 y  A103e 2 m1 y  A104 e 2 m3 y  A105 e m1 y        A106 e ( m3  m1 ) y  A107 e m3 y  A108 e m4 y  A109 e ( m1  m4 ) y  A110 e ( m3  m4 ) y  A111e m10 y  A112 e ( m4  m8 ) y   e nt  E  c   A e ( m2  m4 ) y  A e ( m7  m4 ) y  A e m8 y  A e m2 y  A e m7 y  A e ( m1  m8 ) y  A e ( m1  m2 ) y  113 114 115 116 117 118 119    ( m1  m7 ) y ( m3  m8 ) y ( m2  m3 ) y ( m3  m7 ) y  m6 y  m11 y   A e  A e  A e  A e  A e  A e 121 122 123 124 125  120 

(50)

 A15 e  m5 y  A5  A6 e 2 m4 y  A7 e 2 m1 y  A8 e 2 m3 y  A9 e  m1 y   T ( y, t )  e m1 y  Ec    A e ( m1  m3 ) y  A e m3 y  A e m4 y  A e ( m1  m4 ) y  A e ( m3  m4 ) y  11 12 13 14  10 

 

  A74 e m10 y  A52 e m5 y  A53e 2 m4 y  A54 e 2 m1 y  A55 e 2 m3 y  A56 e m1 y  A57 e ( m3  m1 ) y    (51)   m3 y ( m3  m4 ) y ( m4  m8 ) y  m4 y ( m1  m4 ) y ( m2  m4 ) y    A e  A e  A e  A e  A e  A e  58 59 60 61 62 63  e nt  A27 e m2 y  A1e m1 y  Ec   ( m  m ) y m y m y ( m  m ) y m y ( m  m ) y   A64 e 7 4  A65 e 8  A66 e 2  A67 e 7  A68 e 1 8  A69 e 1 2     ( m1  m7 ) y   A71e ( m3  m8 ) y  A72 e ( m2  m3 ) y  A73e ( m3  m7 ) y   A70 e  





 A26 e  m6 y  A16 e  m5 y  A17 e 2 m4 y  A18 e 2 m1 y  A19 e 2 m3 y  A20 e  m1 y   C ( y, t )  A28 e m3 y  A2 e m1 y  Ec    A e ( m1  m3 ) y  A e m3 y  A e m4 y  A e ( m1  m4 ) y  A e ( m3  m4 ) y  22 23 24 25  21 









 A32 e m7 y  A29 e m3 y  A30 e m2 y  A31e m1 y     A99 e m11 y  A75 e m6 y  A76 e m5 y  A77 e 2 m4 y  A78 e 2 m1 y  A79 e 2 m3 y  A80 e m1 y  A81e ( m3  m1 ) y  (52)       A82 e m3 y  A83e m4 y  A84 e ( m1  m4 ) y  A85 e ( m3  m4 ) y  A86 e m10 y  A87 e ( m4  m8 ) y  A88 e ( m2  m4 ) y   e nt  E  c   A e ( m7  m4 ) y  A e m8 y  A e m2 y  A e m7 y  A e ( m1  m8 ) y  A e ( m1  m2 ) y  A e ( m1  m7 ) y  89 90 91 92 93 94 95    ( m3  m8 ) y ( m2  m3 ) y ( m3  m7 ) y   A e  A e  A e 97 98  96  SKIN FRICTION The expression for the rate of mass transfer in terms of skin-friction (  ) at the plate is,

 du 

 df 

      0   e nt  1   dy  y 0  dy  y 0  dy  y 0

101

Unsteady MHD Mixed Convection of a Viscous Double Diffusive Fluid over a Vertical Plate in Porous Medium with Chemical Reaction, Thermal Radiation and Joule Heating

= A127  e A128 nt

(53)

NUSSELT NUMBER The expression for the rate of heat transfer in terms of Nusselt number (N u) is,

 dT   dg   dg     0   e nt  1  N u    dy  y 0  dy  y 0  dy  y 0 = A129  e A130 nt

(54)

SHERWOOD NUMBER The expression for the stretching sheet (Sh) is,

 dC   dh   dh     0   e nt  1  S h    dy  y 0  dy  y 0  dy  y 0 = A131  e nt A132

(55)

5. RESULTS AND DISCUSSIONS The non-linear equations (14) to (22) subject to the boundary conditions (23) describing heat and mass transfer double diffusive flow past a semi infinite vertical plate immersed in a porous medium in the presence of thermal radiation, chemical reaction and joul heating under the influence of magnetic field are solved analytically by perturbation method. In order to find physical insight of the problem, the effects of various parameters (Magnetic parameter M, Heat absorption parameter Q, Chemical reaction parameter γ, Schimidt number Sc, Radiation parameter S, Prandtl number Pr, Thermal Grashof number GT, Solutal Grashof number Gc, Permeability number K and Soret number S0) are analyzed on Velocity, Temperature and Concentration distributions with the help of following figures and tables. 1.3 1.2

M=0.5, 1.0, 1.5, 2.0

1.1 1

0.9 0.8 0.7 Sc=0.2,Pr=0.7,n=0.1,E=0.005, GT=2.0,Gc=2.0,Q=0.3,K=0.5, A=0.5,Up=0.5,v=0.1,S=0.59, t=1.0,Ec=0.01,So=3.7.

0.6 0.5 0.4 0

0.5

1.5

2.5 Y

3.5

4.5

Figure 1: Effect of Magnetic Field M on Velocity u From Figure 1, it is observed that the velocity decreases as the existence of magnetic field (M) increases, because the application of transverse magnetic field results in Lorentz force. In Figure 2 the velocity decreases as the heat

102

B. Madhusudhana Rao, G. Viswanatha Reddy & M. C. Raju

absorption (Q) increases and this causes the thermal buoyancy effects to decrease, which results in a total reduction in the fluid velocity. In the figures 3&4, it is seen that the velocity decreases as the chemical reaction (γ) and Schimidt number (Sc) increase respectively. The effects of radiation parameter (S) and Prandtl number (Pr) on velocity distribution are presented in the figures 5&6. From these figures it is noticed that as S and Pr increase, the velocity decreases due to decrease in boundary layer thickness for t=1.0. The velocity distribution for distinct values of Thermal Grashof number (GT) and Solutal Grashof number (Gc) are shown in the figures 7&8 respectively for t=1.0, and it seen that an increase in GT and Gc lead to increase the velocity profile. From these figures it is concluded that the peak values of velocity increases rapidly near the porous plate’s wall as the Grashof number is increased which ultimately decays to the relevant free stream velocity. From the Figure 9, it is observed that due to increase in the plate velocity (Up) there is an increase in the velocity near the porous plate and its effects diminish away from the plate. The resistance offered by the porous medium decreases as the permeability of porous medium increase, and so it is observed that as Permeability parameter (K) increases, the velocity increases across the boundary layer from the Figure 10. 1.3 1.2

Q = 0.3, 0.5, 0.7, 0.9

1.1

1 0.9 0.8

Sc=0.22,Pr=0.7,n=0.1,E=0.005, GT=2.0,Gc=2.0,M=0.5, K=0.5,A=0.5,Up=0.5,v=0.1, S=0.59,t=1.0,Ec=0.01,So=3.7.

0.7 0.6 0.5 0

0.5

1.5

2.5 y

3.5

4.5

Figure 2: Effect of Heat Absorption Parameter Q on Velocity u 1.3 1.2

v = 0.1, 0.3, 0.5, 0.7

1.1 1

0.9 0.8 Sc=0.22,Pr=0.7,n=0.1,E=0.005, GT=2.0,Gc=2.0,M=0.5,Q=0.3 K=0.5,A=0.5,Up=0.5, S=0.59,t=1.0,Ec=0.01,So=3.7.

0.7 0.6 0.5 0.4 0

0.5

1.5

2.5 y

3.5

4.5

Figure 3: Effect of Chemical Reaction Parameter γ and on Velocity u

103

Unsteady MHD Mixed Convection of a Viscous Double Diffusive Fluid over a Vertical Plate in Porous Medium with Chemical Reaction, Thermal Radiation and Joule Heating 1.4 1.3

Sc = 0.16, 0.22, 0.60, 1.0

1.2 1.1

1 0.9 0.8 0.7 Pr=0.7,n=0.1,E=0.005, GT=2.0,Gc=2.0,M=0.5,Q=0.3 K=0.5,A=0.5,Up=0.5,v=0.1, S=0.59,t=1.0,Ec=0.01,So=3.7.

0.6 0.5 0.4 0

0.5

1.5

2.5 y

3.5

4.5

Figure 4: Effect of Schimidt Number Sc on Velocity u 1.4 1.3 S=0.1, 0.3, 0.5, 0.7 1.2 1.1

1 0.9 0.8 Sc=0.22,Pr=0.7,n=0.1,E=0.005, GT=2.0,Gc=2.0,M=0.5,Q=0.3, K=0.5,A=0.5,Up=0.5,v=0.1, t=1.0,Ec=0.01,So=3.7.

0.7 0.6 0.5 0.4 0

0.5

1.5

2.5 Y

3.5

4.5

Figure 5: Effect of Radiation Parameter S on Velocity u 1.4 1.3 Pr = 0.7, 1.0, 3.0, 7.0

1.2 1.1

1 0.9 0.8 Sc=0.22,E=0.005,n=0.1, GT=2.0,Gc=2.0,M=0.5, Q=0.3,K=0.5,A=0.5,Up=0.5, v=0.1,S=0.3,t=1.0, Ec=0.01,So=3.7.

0.7 0.6 0.5 0.4 0

0.5

1.5

2.5 Y

3.5

4.5

Figure 6: Effect of and Prandtl Number Pr on Velocity u

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B. Madhusudhana Rao, G. Viswanatha Reddy & M. C. Raju

1.3 1.2 1.1 1 GT=1.0, 2.0, 3.0, 4.0

0.9 0.8 0.7

Sc=0.22,E=0.005,n=0.1, Gc=2.0,M=0.5,Pr=0.7, Q=0.3,K=0.5,A=0.5,Up=0.5, v=0.1,S=0.3,t=1.0, Ec=0.01,So=3.7.

0.6 0.5 0.4 0

0.5

1.5

2.5 Y

3.5

4.5

Figure 7: Effect of Thermal Grashof Number GT on Velocity u 1.3 1.2 1.1 1

Gc=1.0, 2.0, 3.0, 4.0 0.9 0.8 Sc=0.22,E=0.005,n=0.1, GT=2.0,M=0.5,Pr=0.7, Q=0.3,K=0.5,A=0.5,Up=0.5, v=0.1,S=0.3,t=1.0, Ec=0.01,So=3.7.

0.7 0.6 0.5 0

0.5

1.5

2.5 Y

3.5

4.5

Figure 8: Effect of Solutal Grashof Number Gc on Velocity u 1.6 1.4 1.2

1 Up = 0.0, 0.5, 1.0, 1.5

0.8 0.6

Sc=0.22,E=0.005,n=0.1, GT=2.0,M=0.5,Pr=0.7, Q=0.3,K=0.5,A=0.5,Gc=2.0, v=0.1,S=0.3,t=1.0, Ec=0.01,So=3.7.

0.4 0.2 0 0

0.5

1.5

2.5 Y

3.5

Figure 9: Effect of Plate Velocity Up on Velocity u

4.5

105

Unsteady MHD Mixed Convection of a Viscous Double Diffusive Fluid over a Vertical Plate in Porous Medium with Chemical Reaction, Thermal Radiation and Joule Heating 1.3 1.2 1.1 1

K = 0.5, 1.0, 1.5, 2.0

0.9 0.8 0.7

Sc=0.22,E=0.005,n=0.1, GT=2.0,M=0.5,Pr=0.7, Q=0.3,Up=0.5,A=0.5,Gc=2.0, v=0.1,S=0.3,t=1.0, Ec=0.01,So=3.7.

0.6 0.5 0.4 0

0.5

1.5

2.5 Y

3.5

4.5

Figure 10: Effect of Permeability Parameter K on Velocity u From Figure 11, it is observed that temperature profile decreases as heat absorption (Q) increases. This is due to the fact that heat of the fluid is absorbed by the porous plate and hence higher the heat absorption parameter, lower the temperature profile in the boundary layer. In Figure 12, it is seen that the increase in the radiation parameter (S) results in decrease in temperature due to the fact that the divergence of radiation heat flux

q r decreases as the absorption y 1

coefficient Kλ increases at the wall which causes the fluid temperature to decrease. It is noticed from Figure 13 that temperature profile decreases as Prandtl number(Pr) increases, and this is due to fact that thermal boundary layer decreases with an increase in Pr. In figure 14, it is shown that temperature distribution decreases with an increase in an increase in Soret effect number (So). From Figure 15, it is observed that the concentration decreases as the chemical reaction parameter (γ) increases, and this is due to the fact that solutal boundary layer decreases with chemical reaction parameter. In the Figures 16&17, it is seen that concentration profile decreases as Schimidt number (Sc) and Soret number (So) increase respectively. The variations in Skin friction (  ), Nusselt number (Nu) and Sherwood number (Sh) against respective parameters are discussed and presented in the Tables 1, 2 & 3. 1.4 Sc=0.22,Pr=0.7,n=0.1, E=0.005,GT=2.0,Gc=2.0, M=0.5,K=0.5,A=0.5,Up=0.5, v=0.1,S=2,t=0.5,Ec=0.01,So=2.

1.2

0.8

0.6

0.4 Q=0.5,1.0,1.5,2.0 0.2

0 0

0.5

1.5

2.5 Y

3.5

4.5

Figure 11: Effect of Heat Absorption Parameter Q on Temperature T

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B. Madhusudhana Rao, G. Viswanatha Reddy & M. C. Raju

1.4 Sc=0.22,Pr=0.7,n=0.1,E=0.005, GT=2.0,Gc=2.0,M=0.5, K=0.5,A=0.5,Up=0.5,v=0.1, Q=0.3,t=0.5,Ec=0.01,So=2.

1.2

0.8

0.6 S=0.1,0.3,0.5,1.0

0.4

0.2

0 0

0.5

1.5

2.5 Y

3.5

4.5

Figure 12: Effect of Radiation Parameter S on Temperature T 1.4 Sc=0.22,S=1.0,n=0.1,E=0.005, GT=2.0,Gc=2.0,M=0.5, K=0.5,A=0.5,Up=0.5,v=0.1, Q=0.3,t=0.5,Ec=0.01,So=2.

1.2

0.8

0.6

0.4

Pr=0.7,1.0, 3.0, 7.0

0.2

0 0

0.5

1.5

2.5 Y

3.5

4.5

Figure 13: Effect of Prandtl Number Pr on Temperature T 1.2 Sc=0.22,Pr=0.7,n=0.1,E=0.005, GT=2.0,Gc=2.0,M=0.5,Q=0.3, K=0.5,A=0.5,Up=0.5,v=0.1, S=0.59,t=1.0,Ec=0.01.

0.8

0.6

0.4 So=1, 2, 3, 4 0.2

-0.2 0

0.5

1.5

2.5 Y

3.5

4.5

Figure 14: Effect of Soret Number So on Temperature T

107

Unsteady MHD Mixed Convection of a Viscous Double Diffusive Fluid over a Vertical Plate in Porous Medium with Chemical Reaction, Thermal Radiation and Joule Heating 1.2 Sc=0.22,Pr=0.7,n=0.1,E=0.005, GT=2.0,Gc=2.0,M=0.5,Q=0.3, K=0.5,A=0.5,Up=0.5,So=3 S=0.59,t=1.0,Ec=0.01.

0.8

0.6

0.4 v=0.1,0.3,0.5,0.7

0.2

-0.2 0

0.5

1.5

2.5 Y

3.5

4.5

Figure 15: Effect of Chemical Reaction Parameter Îł on Concentration C 1 v=0.1,Pr=0.7,n=0.1,E=0.005, GT=2.0,Gc=2.0,M=0.5,Q=0.3, K=0.5,A=0.5,Up=0.5,So=3 S=0.59,t=1.0,Ec=0.01.

0.8 0.6 0.4

Sc=0.16,0.22,0.60,1.0

0.2 0 -0.2 -0.4 -0.6 -0.8 0

0.5

1.5

2.5 Y

3.5

4.5

Figure 16: Effect of Schimidt Number Sc on Concentration C 1.2 v=0.1,Pr=0.7,n=0.1,E=0.005, GT=2.0,Gc=2.0,M=0.5,Q=0.3, K=0.5,A=0.5,Up=0.5,Sc=0.22 S=0.59,t=1.0,Ec=0.01.

0.8

0.6 So=1,2,3,4 0.4

0.2

-0.2 0

0.5

1.5

2.5 Y

3.5

4.5

Figure 17: Effect of Soret Number So on Concentration C

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B. Madhusudhana Rao, G. Viswanatha Reddy & M. C. Raju

Table 1 k 0.5 1.0 1.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

S 0.5 0.5 0.5 1.0 1.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

γ 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.3 0.1 0.1 0.1 0.1

Q 0.5 0.5 0.5 0.5 0.5 1.0 1.5 0.5 0.5 0.5 0.5 0.5 0.5

Sc 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.22 0.60 0.16 0.16

Pr 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 1.0 3.0

τ 1.9110 2.7631 3.0389 -1.0379 -2.0677 -1.0687 -2.1685 -2.1231 -2.3945 -3.3399 -3.7276 -6.9262 -8.0790

Table 2 Q 0.5 1.0 1.5 0.5 0.5 0.5 0.5 0.5 0.5

S 0.5 0.5 0.5 1.0 1.5 0.5 0.5 0.5 0.5

Pr 0.7 0.7 0.7 0.7 0.7 1.0 3.0 0.7 0.7

So 2 2 2 2 2 2 2 3 4

Nu -1.1971 -1.5138 -1.5917 -1.5258 -1.5827 -1.2789 -1.4195 -1.7849 -1.9039

Table 3 γ 0.1 0.2 0.3 0.1 0.1 0.1 0.1

Sc 0.16 0.16 0.16 0.22 0.60 0.16 0.16

So 2 2 2 2 2 3 4

Sh -0.7849 -0.9977 -1.1238 -0.8241 -0.9148 -0.9271 -1.0039

6. CONCLUSIONS The governing partial differential equations were transformed to ordinary differential equations by suitable transformation and then equations were solved analytically by perturbation method. The computed values obtained from analytical solutions of Velocity, Temperature and Concentration fields and are discussed graphically. The Skin friction, Nusselt number and Sherwood number are presented in tabular form. Thus, we conclude the following. 

The Velocity profile decreases as the values of Magnetic parameter, Heat absorption parameter, Chemical reaction parameter, Schimidt number, Radiation parameter, Prandtl number increase where as it increases with an increase in Permeability number of porous medium, Thermal and Solutal Grashof numbers.



The Temperature distribution decreases as the values of Heat absorption parameter, Radiation parameter, Prandtl number and Soret effect number increase in the boundary layer.



The Concentration distribution decreases as the values of Chemical reaction parameter, Schimidt number and Soret effect number increase.

Unsteady MHD Mixed Convection of a Viscous Double Diffusive Fluid over a Vertical Plate in Porous Medium with Chemical Reaction, Thermal Radiation and Joule Heating



109

The value of Skin friction decreases as Radiation parameter, Heat absorption parameter, Chemical reaction parameter, Schimidit number, prandtl number increase and it increases with an increase in permeability number.



The value of Nusselt number decreases as Radiation parameter, Heat absorption parameter, prandtl number and Soret effect number increases.



The value of Sherwood number decreases as Chemical reaction parameter, Radiation parameter, Soret effect number increase.

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Chambre PL and Young JD (). On the diffusion of a chemically reactive species in a laminar boundary layer flow.

Dekha R Das UN and Soundalgekar VM (1994). Effects on mass transfer on Flow past an impulsively started infinite ertical plate with constant heat flux and chemical reaction. Forschungim Ingenieurwesen 60, pp. 284-309.

Muthecumaraswamy R, (2002). Effects of a chemical reaction on a moving isothermal surface with suction. Acta Mechanica 155, pp.65-72.

Chen T.S and B.F. Armaly, Mixed convection in external flow, In: S KaKac, R. K. Shah, W. Aunng, editors, Handbook of single-phase convective heat transfer, Wiley, New York, 1987.

Pop I and D.B. Ingham, Convective heat transfer: mathematical and computational modelling of viscous fluids and porous media, Pergamon Press, Oxford, 2001.

Soundalgekar V.M, S.K. Gupta and N.S. Birajdar, Effects of mass transfer and free effects on MHD Stokes problem for a vertical plate, Nuclear Engineering Research 53; 309-346, 1979.

Elbasheshy E.M.A, Heat and mass transfer along a vertical plate surface tension and concentration in the presence of magnetic field, International journal of Sciences 34(5); 515-522, 1997.

Abel M.S, P.G. Siddheshwar and N. Mahesha, Effects of thermal buoyancy and variable thermal conductivityon the MHD flow and heat transfer in a power law fluid past a vertical stretching sheet in the presence of a non-uniform heat source, International Journal of Non-Linear Mechanics, 44; 1-12, 2009.

Abel M. S, P. S. Datti and N. Mahesha, Flow and heat transfer in a power-law fluid over a stretching sheet with variable thermal conductivity and nonuniform heat source. International Journal of Heat Mass Transfer, 52; 2902-2913, 2009.

10. Seddeek M.A. (2000), the effect of variable viscosity on hydromagnetic flow and heat transfer past a continuously moving porous boundary with radiation, Int. Commun. Heat Mass Transfer,Vol 27, 1037–1046. 11. Seddeek M.A. (2001), Thermal radiation and buoyancy effects on MHD free convective heat generating flow over an accelerating permeable Surface with temperature-dependent viscosity, Can. J. Phys. Vol.79, 725–732. 12. Takhar H.S., Gorla R.S.R and Soundalgekar V.M (1996), Radiation effects on MHD free convection flow of a gas past a semi-infinite vertical plate, Int. J. Numer. Meth. Heat Fluid Flow Vol. 6, 77–83. 13. Cheng P, Heat transfer in geothermal system, Advances in Heat Transfer, 14; 1-105, 1978. 14. Rudraiah N, Flow through and past porous media, Encyclopedia of fluid mechanics, Gulf Publications, 5; 567-647, 1986

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15. Das U.N, R.K. Deka and V.M. Soundalgekar, Effect of mass transfer on flow past an impulsivel started vertical plate with constant heat flux and chemical reaction, Forschung im Ingenieurwesen, 60(10); 284-287, 1994. 16. Muthucumarswamy R and P. Ganesan, Diffusion and first-order chemical reaction on impulsively started infinte vertical plate with variable temperature, International Journal of Thermal Sciences, 41(5); 475-479, 2002. 17. Prasad K.V, S. Abel and P.S. Datti, Diffusion of chemically reactive species of a non-Newtonian fluid immersed in a porous medium over a stretching sheet, International Journal of Non-Linear Mechanics, 38; 651-657, 2003. 18. Ghaly A.Y and M.A. Seddeek, Chebyshev finite difference method for the effect of chemical reaction, heat and mass transfer on laminar flow along a semiinfinite horizontal plate with temperature dependent viscosity, Choas Solitons Fractals, 19; 61-70, 2004. 19. Akyildiz F. T, H. Bellout and K. Vajravelu, Diffusion of chemically reactive species in a porous medium over a stretching sheet, Journal of Mathematical Analysis and Application, 320; 322-339, 2006. 20. Ramachandra Prasad V and Bhaskar reddy N(2007); Radiation and mass transfer effects on unsteady MHD free convection flow past a heated vertical plate in a porous medium with viscous dissipation, Theoret. Appl. mech., Vol.34, No.2, 135-160. 21. Seddeek M. A., Darwish A. A and Abdelmeguid M. S. (2007), Effects of chemical reaction and variable viscosity on hydromagnetic mixed convection heat and mass transfer for Hiemenz flow through porous media with radiation, Communications in Non-linear Science and Numerical Simulation Vol.12, 195 – 213. 22. Ibrahim F. S., Elaiw A.M. and Bakr A.A (2008), Effect of the 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, communications in Non-linear Science and Numerical Simulation Vol. 13, 1056-1066 23. Chaudhary R.C, B.K. Sharma and A.K. Jha, Radiation effect with simultaneous thermal and mass diffusion in MHD mixed convection flow, Romanian Journal of Physics, 51(7-8); 715-727, 2006. 24. Pal D and B. Talukdar, Perturbation analysis of 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, Communication in Nonlinear Science and Numerical Simulation, 15 (7); 1813-1830, 2010. 25. Madhusudhana Rao B, Viswanatha Reddy G and Raju M.C MHD transient free convection and chemically reactive flow past a porous vertical plate with radiation and temperature gradient dependent heat source in slip flow regime, IOSR Journal of Applied physics. 6(2013) PP 22-32.

APPENDICES

Pr  Pr  4Pr ( S  Q) 2

m1  m5 

S c  S c  4S c

m2  m10 

1  1  4N 2

S c  S c  4(n   ) S c

m8  m12 

1  1  4(n  N ) 2

m7  m11 

m4  m9 

m3  m6 

Pr  Pr  4Pr ( S  Q  n)

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Unsteady MHD Mixed Convection of a Viscous Double Diffusive Fluid over a Vertical Plate in Porous Medium with Chemical Reaction, Thermal Radiation and Joule Heating

A1 

APr m1

A27  1  A1

m1  Pr m1  Pr ( S  Q  n)

A3 

 GT  A2 Gc

A4 

m1  m1  N 2

A9 

A11 

A13 

( N  m1 ) Pr A3

4m4  2 Pr m4  Pr ( S  Q) 2

 2 Pr NA3 m1  Pr m1  Pr ( S  Q ) 2

A7 

A10 

 2 Pr NA4 m3  Pr m3  Pr ( S  Q) 2

 1   Sc

A33  U p  1  A3  A4

m3  m3  N

( N  m4 ) Pr A33 2

A6 

A2 Gc  Gc

A2 

S o m1

4m1  2 Pr m1  Pr ( S  Q) 2

A5 

N S Q

( N  m3 ) Pr A4 2

A8 

4m3  2 Pr m3  Pr ( S  Q) 2

( N  m1m3 )2 Pr A3 A4 (m1  m3 ) 2  Pr (m1  m3 )  Pr ( S  Q)

A12 

 2 Pr NA33 m4  Pr m4  Pr ( S  Q ) 2

( N  m1m4 )2 Pr A3 A33 ( N  m3 m4 )2 Pr A4 A33 A14  2 (m1  m4 )  Pr (m1  m4 )  Pr ( S  Q) (m3  m4 ) 2  Pr (m3  m4 )  Pr ( S  Q)

S 0 m5 A15  1  2  m5  m5    Sc 

4S 0 m4 A6 4 S 0 m1 A7 4S 0 m3 A8 A19  A18   4  2  4 2  4  2  m4  2m4    m3  2m3    m1  2m1   S S  c  Sc   c

S 0 (m1  m3 ) 2 A10 S 0 m1 A9 S 0 m3 A11 A21  A22   1   1  2  1  2  (m1  m3 ) 2  (m1  m3 )    m1  m1    m3  m3   S S  c  Sc   c 2

A20 

A28  1  A2

 2 m1  m1   

A15   A5  A6  A7  A8  A9  A10  A11  A12  A13  A14 A16 

A17 

S 0 m4 A12 S 0 (m1  m4 ) 2 A13 A23  A24   1  2  1   m4  m4    (m1  m4 ) 2  (m1  m4 )    Sc   Sc  2

A25 

S 0 (m3  m4 ) 2 A14  1   Sc

 (m3  m4 ) 2  (m3  m4 )   

A26   A16  A17  A18  A19  A20  A21  A22  A23  A24  A25 A29 

AA28 m3  1   Sc

 2 m3  m3  (n   ) 

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A30 

A34 

A37 

A40 

A44 

A47 

A50 

 S 0 A27 m2  1   Sc

AA2 m1  S 0 A1m1 A32  1  A29  A30  A31  1  2  m1  m1  (n   )  Sc  2

 2 m2  m2  (n   ) 

AA33 m4 m4  m4  ( n  N ) 2

 GT A27  Gc A30 m2  m2  ( n  N ) 2

A31 

A35 

A38 

AA3 m1  GT A1  Gc A31 m1  m1  ( n  N ) 2

 Gc A32 m7  m7  ( n  N ) 2

A36 

AA4 m3  Gc A29 m3  m3  ( n  N ) 2

A39  1  A34  A35  A36  A37  A38

 GT A15  Gc A16  GT A6  Gc A17  GT A7  Gc A18 GT A5 A41  A42  A43  2 2 2 N m5  m5  N 4m4  2m4  N 4m1  2m1  N

 GT A8  Gc A19 4m3  2m3  N 2

 GT A11  Gc A22 m3  m3  N 2

A45 

A48 

 GT A9  Gc A20

A46 

m1  m1  N 2

 GT A12  Gc A23 m4  m4  N 2

A49 

 GT A10  Gc A21 (m1  m3 ) 2  (m1  m3 )  N  GT A13  Gc A24 (m1  m4 ) 2  (m1  m4 )  N

 GT A14  Gc A25 (m3  m4 ) 2  (m3  m4 )  N

A51   A40  A41  A42  A43  A44  A45  A46  A47  A48  A49  A50 A52 

APr A15 m5 m5  Pr m5  Pr ( S  Q  n) 2

2 Pr ( AA6 m4  NA33 A34  A33 A34 m4 ) 2

A53 

2 Pr ( AA7 m1  NA3 A35  A3 A35 m1 )

4m4  2 Pr m4  Pr ( S  Q  n) 2

A54 

A56 

4m1  2 Pr m1  Pr ( S  Q  n) 2

2 Pr ( AA8 m3  NA4 A36  A4 A36 m3 ) 2

A55 

4m3  2 Pr m3  Pr ( S  Q  n) 2

Pr ( AA9 m1  2 NA35  2 NA3 )

m1  Pr m1  Pr ( S  Q  n) P AA10 (m1  m3 )  2 NA3 A36  2 NA4 A35  2 A3 A36 m1m3  2 A4 A35 m1m3  A57  r (m1  m3 ) 2  Pr (m1  m3 )  Pr ( S  Q  n)

A58 

Pr ( AA11m3  2 NA36  2 NA4 ) m3  Pr m3  Pr ( S  Q  n) 2

A59 

Pr ( AA12 m4  2 NA33  2 NA34 ) m4  Pr m4  Pr ( S  Q  n) 2

A60 

Pr AA13 (m1  m4 )  2 NA33 A35  2 NA3 A34  2 A33 A35 m1m4  2 A3 A34 m1m4  (m1  m4 ) 2  Pr (m1  m4 )  Pr ( S  Q  n)

A61 

Pr AA14 (m3  m4 )  2 NA33 A36  2 NA4 A34  2 A33 A36 m3 m4  2 A4 A34 m3 m4  (m3  m4 ) 2  Pr (m3  m4 )  Pr ( S  Q  n)

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Unsteady MHD Mixed Convection of a Viscous Double Diffusive Fluid over a Vertical Plate in Porous Medium with Chemical Reaction, Thermal Radiation and Joule Heating

A62 

 2 Pr A33 A39 ( N  m8 m4 ) (m8  m4 ) 2  Pr (m8  m4 )  Pr ( S  Q  n)

A63 

 2 Pr A33 A37 ( N  m2 m4 ) (m2  m4 ) 2  Pr (m2  m4 )  Pr ( S  Q  n)

A64 

 2 Pr A33 A39 ( N  m7 m4 )  2 Pr NA39 A65  2 2 (m7  m4 )  Pr (m7  m4 )  Pr ( S  Q  n) m8  Pr m8  Pr ( S  Q  n)

A66 

 2 Pr NA37 m2  Pr m2  Pr ( S  Q  n) 2

A67 

 2 Pr NA38 m7  Pr m7  Pr ( S  Q  n) 2

A68 

 2 Pr A3 A39 ( N  m8 m1 ) (m8  m1 ) 2  Pr (m8  m1 )  Pr ( S  Q  n)

A69 

 2 Pr A3 A37 ( N  m2 m1 ) (m2  m1 ) 2  Pr (m2  m1 )  Pr ( S  Q  n)

A70 

 2 Pr A3 A38 ( N  m7 m1 ) (m7  m1 ) 2  Pr (m7  m1 )  Pr ( S  Q  n)

A71 

 2 Pr A4 A39 ( N  m8 m3 )  2 Pr A4 A37 ( N  m2 m3 ) A72  2 (m8  m3 )  Pr (m8  m3 )  Pr ( S  Q  n) (m2  m3 ) 2  Pr (m2  m3 )  Pr ( S  Q  n)

A73 

 2 Pr A4 A38 ( N  m7 m3 ) (m7  m3 ) 2  Pr (m7  m3 )  Pr ( S  Q  n)

A74   A52  A53  A54  A55  A56  A57  A58  A59  A60  A61  A62  A63  A64  A65  A66  A67  A68  A69  A70  A71  A72  A73 AA16 m5  S 0 A52 m5 2 AA17 m4  4S 0 A53 m4 A75  A76  A77   1  2  4  2  1  2  m5  m5  (n   )  m4  2m4  (n   )  m6  m6  (n   )  Sc   Sc   Sc  2

AA26 m6

2 AA18 m1  4S 0 A54 m1 2 AA19 m3  4S 0 A55 m3 AA20 m1  S 0 A56 m1 A79  A80   4  2  4  2  1  2  m1  2m1  (n   )  m3  2m3  (n   )  m1  m1  (n   )  Sc   Sc   Sc  2

A78 

2 AA21 (m1  m3 )  S 0 A57 (m1  m3 ) 2 AA22 m3  S 0 A58 m3 A82   1   1  2  (m1  m3 ) 2  (m1  m3 )  (n   )  m3  m3  (n   )  Sc   Sc  2 AA24 (m1  m4 )  S 0 A60 (m1  m4 ) 2 AA23 m4  S 0 A59 m4 A84  A83   1   1  2  (m1  m4 ) 2  (m1  m4 )  (n   )  m4  m4  (n   ) S  c  Sc 

A81 

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B. Madhusudhana Rao, G. Viswanatha Reddy & M. C. Raju

A85 

A87 

A89 

A91 

2 AA25 (m3  m4 )  S 0 A61 (m3  m4 ) 2  S 0 A74 m10 A86   1   1  2  (m3  m4 ) 2  (m3  m4 )  (n   )  m10  m10  (n   )  Sc   Sc 

 S 0 A62 (m8  m4 ) 2  1   Sc

 (m8  m4 ) 2  (m8  m4 )  (n   )   S 0 A64 (m7  m4 ) 2

 1   Sc

 (m7  m4 ) 2  (m7  m4 )  (n   )   S 0 A66 m2

A90 

 S 0 A63 (m2  m4 ) 2  1   Sc

 (m2  m4 ) 2  (m2  m4 )  (n   )   S 0 A65 m8

 1   Sc

 S 0 A67 m7

 2 m8  m8  (n   ) 

 2  1  2 m2  m2  (n   )  m7  m7  (n   )   Sc   S 0 A68 (m8  m1 ) 2  S 0 A69 (m2  m1 ) 2 A93  A94   1   1   (m8  m1 ) 2  (m8  m1 )  (n   )  (m2  m1 ) 2  (m2  m1 )  (n   )  Sc   Sc 

A95 

A97 

 1   Sc

A92 

A88 

 S 0 A70 (m7  m1 ) 2  1   Sc

 (m7  m1 ) 2  (m7  m1 )  (n   )   S 0 A72 (m2  m3 ) 2

 1   Sc

 (m2  m3 ) 2  (m2  m3 )  (n   ) 

A96 

A98 

 S 0 A71 (m8  m3 ) 2  1   Sc

 (m8  m3 ) 2  (m8  m3 )  (n   )   S 0 A73 (m7  m3 ) 2

 1   Sc

 (m7  m3 ) 2  (m7  m3 )  (n   ) 

A99   A75  A76  A77  A78  A79  A80  A81  A82  A83  A84  A85  A86  A87  A88  A89  A90  A91  A92  A93  A94  A95  A96  A97  A98 AA51m9 AA41m5  GT A52  Gc A76 2 AA42 m4  GT A53  Gc A77 A100  2 A101  A102  2 2 m9  m9  (n  N ) m5  m5  ( n  N ) 4m4  2m4  ( n  N ) A103  A105 

A106 

A108 

2 AA43 m1  GT A54  Gc A78 4m1  2m1  (n  N ) AA45 m1  GT A56  Gc A80 2

A104 

2 AA44 m3  GT A55  Gc A79 4m3  2m3  (n  N ) 2

m1  m1  (n  N ) 2

AA46 (m1  m3 )  GT A57  Gc A81 AA47 m3  GT A58  Gc A82 A107  2 2 (m1  m3 )  (m1  m3 )  (n  N ) m3  m3  ( n  N ) AA48 m4  GT A59  Gc A83 m4  m4  ( n  N ) 2

A109 

AA49 (m1  m4 )  GT A60  Gc A84 (m1  m4 ) 2  (m1  m4 )  (n  N )

Unsteady MHD Mixed Convection of a Viscous Double Diffusive Fluid over a Vertical Plate in Porous Medium with Chemical Reaction, Thermal Radiation and Joule Heating

A110 

AA50 (m3  m4 )  GT A61  Gc A85  GT A74  Gc A86 A111  2 2 (m3  m4 )  (m3  m4 )  (n  N ) m10  m10  (n  N )

A112 

 GT A62  Gc A87  GT A63  Gc A88 A113  2 (m4  m8 )  (m4  m8 )  (n  N ) ( m4  m2 ) 2  ( m4  m2 )  ( n  N )

A114 

 GT A64  Gc A89  G A  Gc A90  G A  Gc A91 A115  2 T 65 A116  2 T 66 2 ( m 4  m 7 )  ( m 4  m7 )  ( n  N ) m8  m8  (n  N ) m2  m2  ( n  N )

 GT A68  Gc A93 (m1  m8 ) 2  (m1  m8 )  (n  N ) m7  m7  ( n  N )  GT A69  Gc A94  GT A70  Gc A95  A120  2 (m1  m2 )  (m1  m2 )  (n  N ) (m1  m7 ) 2  (m1  m7 )  (n  N )

A117 

A119

115

 GT A67  Gc A92 2

A118 

A121 

 GT A71  Gc A96  GT A72  Gc A97 A122  2 (m3  m8 )  (m3  m8 )  (n  N ) (m3  m2 ) 2  (m3  m2 )  (n  N )

A123 

 GT A73  Gc A98  Gc A99  Gc A75 A124  A125  2 2 2 (m3  m7 )  (m3  m7 )  (n  N ) m11  m11  (n  N ) m 6  m6  ( n  N )

A126   A100  A101  A102  A103  A104  A105  A106  A107  A108  A109  A110  A111  A112  A113  A114  A115  A116  A117  A118  A119  A120  A121  A122  A123  A124  A125  m9 A51  m5 A41  2m4 A42  2m1 A43  2m3 A44  m1 A45   A127  m4 A33  m1 A3  m3 A4  E c   ( m  m ) A  m A  m A  ( m  m ) A  ( m  m ) A 1 3 46 3 47 4 48 1 4 49 3 4 50  

A128  m8 A39  m4 A34 e  m4 y  m1 A35  m3 A36  m2 A37  m7 A38  m12 A126  m9 A100  m5 A101  2m4 A102  2m1 A103  2m3 A104  m1 A105  (m3  m1 ) A106  m3 A107  m4 A108  (m1  m4 ) A109     Ec   (m3  m4 ) A110  m10 A111  (m4  m8 ) A112  (m2  m4 ) A113  (m7  m4 ) A114  m8 A115  m2 A116  m7 A117  (m1  m8 ) A118    (m  m ) A  (m  m ) A  (m  m ) A  (m  m ) A  (m  m ) A  m A  m A  1 7 120 3 8 121 2 3 122 3 7 123 11 124 6 125  1 2 119 

 m5 A15  2m4 A6  2m1 A7  2m3 A8  m1 A9  (m1  m3 ) A10   A129  m1  Ec    m3 A11  m4 A12  (m1  m4 ) A13  (m3  m4 ) A14 

A130  m2 A27  m1 A1  m10 A74  m5 A52  2m4 A53  2m1 A54  2m3 A55  m1 A56  (m3  m1 ) A57  m3 A58  m4 A59  (m1  m4 ) A60     Ec   (m3  m4 ) A61  (m4  m8 ) A62  (m2  m4 ) A63  (m7  m4 ) A64  m8 A65  m2 A66  m7 A67    (m  m ) A  (m  m ) A  (m  m ) A  (m  m ) A  (m  m ) A  (m  m ) A  1 8 68 1 2 69 1 7 70 3 8 71 2 3 72 3 7 73  

 m6 A26  m5 A16  2m4 A17  2m1 A18  2m3 A19  m1 A20   A131  m3 A28  m1 A2  Ec    (m1  m3 ) A21  m3 A22  m4 A23  (m1  m4 ) A24  (m3  m4 ) A25 