Effect of Buoyancy and Magnetic Field on Unsteady Convective Diffusion of Solute in a Boussinesq Stokes Suspension Bounded by Porous Beds

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1 Available at Appl. Appl. Math. ISSN: Vol. 12 Issue 1 June 2017 pp Applications and Applied Mathematics: An International Journal AAM Effect of Buoyancy and Magnetic Field on Unsteady Convective Diffusion of Solute in a Boussinesq Stokes Suspension Bounded by Porous Beds 1 Nirmala P. Ratchagar & 2 R. Vijayakumar 1 Department of Mathematics Annamalai University Annamalainagar India nirmalapasala@yahoo.co.in & 2 Mathematics Section Faculty of Engineering and Technology Annamalai University Annamalainagar India rathirath viji@yahoo.co.in Received: July ; Accepted: February Abstract Hydromagnetic free and forced convection in a parallel plate channel bounded by porous bed and transverse magnetic field has been considered. When there is a uniform axial temperature variation along the walls the primary flow shows incipient flow reversal at the upper plate for an increase in temperature along that plate. Similarly flow reversal at the lower plate occurs with a decrease in temperature along that plate. The magnetic field arising as a body couple in the governing equations is shown to increase the axis dispersion coefficient. The effect of various physical parameters such as Hartmann number Grashof number porous parameter and couple stress parameter on the velocity temperature and dispersion coefficient mean concentration skin friction coefficient and Nusselt numbers are computed and analyzed through graphs. Keywords: Hartmann number; Heat and mass transfer; Magnetohydrodynamic; couple stress fluid; generalized dispersion model 540

2 AAM: Intern. J. Vol. 12 Issue 1 June MSC 2010 No.: 76S05 76W05 76D05 1. Introduction The external regulation of dispersion in plane parallel flow is very important from the point of view of its applications in biomechanical chemical engineering and in many industrial problems. One way of regulating dispersion is by means of influencing the flow by appropriate thermal means at the boundaries. Miscible dispersion of passive solute also depends on the fluid solvent. Many applications involve solvents with micron sized suspended particles and these results in change of solvent viscosity. The particles also have relative spin with respect to the solvent. Stoke s couple stress fluid is one such fluid which models suspensions. The unsteady convective diffusions of passive solute in this fluid has been analyzed by Rudraiah et al The bounding walls and their problem are assumed to be permeable. Lighthill 1966 obtained the exact solution of the unsteady convection diffusion equation which is asymptotically valid for small time. Chatwin 1971 also studied the theory for large time by the Laplace transform technique. Barton 1986 has resolved certain technical difficulties in the Aris 1956 method of moments and obtained the solutions of the second and third moments equations of the distribution of the solute valid for all time. However Sankarasubramanian and Gill 1973 have developed an analytical method to analyze the transient dispersion of a non-uniform initial distribution called generalized dispersion in a channel. The effect of buoyancy forces caused by a density difference due to concentration difference of a solvent in a straight horizontal pipe studied by Erdogan and Chatwin Barton and Stokes 1986 computed shear dispersion in parallel flow numerically. Mazumder 1979 studied the dispersion of solute in the combined free and forced convective laminar flow between two parallel plates in presence of uniform axial temperature variation along the channel walls for asymptotically large time and found that effective Taylor diffusion coefficient increases with the Grashof number. Bestman 1983 studied the unsteady low Reynolds number flow in a heated tube of slowly varying section. In that analysis the effect of forced and free convection heat transfer on flow in an axisymmetric tube whose radius varied slowly in the axial direction was addressed. Siddheshwar and Thangaraj 2005 investigated the dispersion of solute in a fully developed flow of a Boussinesq Stokes suspension through a parallel channel with axial variation of temperature along the bounding walls. Sivasankaran 2007 examined the effect of variable thermal conductivity on buoyant convection in a cavity with internal heat generation. Kafoussias et al described the two dimensional steady and laminar free-forced convective boundary layer flow of a biomagnetic fluid over a semi infinite vertical hot plate under the action of a localized magnetic field. Tzirtzilakis et al studied the forced and free convective boundary layer flow of a magnetic fluid over a flat plate under the action of a localized magnetic field. Sibanda and Makinde 2010 investigated the MHD flow and heat transfer past a rotating disk in a porous medium with ohmic

3 542 N.P. Ratchagar & R. Vijayakumar heating and viscous dissipation. Rashidi et al applied MHD flow in medicine science. They studied the dual control mechanisms of transverse magnetic field and porous media filtration in a buoyancy-driven blood flow regime in a vertical pipe as a model of a blood separation configuration. Makinde et al described the buoyancy effects on MHD stagnation point flow and heat transfer of a nanofluid past a convectively heated stretching/ shrinking sheet. Bhuvaneswari and Sivasankaran 2014 investigated the free convection flow in an inclined plate with variable thermal conductivity by scaling group transformations. Rundora and Makinde 2015 investigate the combined effects of suction/injection and Navier slip at the channel walls on the heat transfer characteristics in such flows. Chinyoka and Makinde 2015 studied the unsteady flow of a reactive variable viscosity third-grade fluid between two parallel porous plates filled with a porous medium and acted upon by both nonconstant pressure and buoyancy effects. Both the left-hand side and right-hand side walls of the channel are subjected to asymmetric convective heat exchange with the ambient and allow for uniform suction/injection in the transverse direction. The objective of the present chapter is to study the dispersion of a solute in combined free and forced convective laminar flow through a horizontal channel bounded by porous beds with uniform axial temperature variation along the walls using generalized dispersion of a solute Gill and Sankarasubramanian Mathematical Formulation For a steady fully developed laminar flow the velocity u in the x direction is a function of y only. Consider the combined free and forced convective laminar flow of a viscous incompressible fluid bounded by porous layers separated by a distance 2h apart in the presence of a uniform linear axial temperature variation along the channel. y g B 0 Region 2 y = h Region 1 y=0 x y = - h Region 2 -Xs/2 Xs/2 B 0 Figure 1. Physical Configuration The equations governing the flow of a couple stress fluid are given by Srinivasacharya et al.

4 AAM: Intern. J. Vol. 12 Issue 1 June Region 1: Fluid Film Conservation of mass for an incompressible flow Conservation of momentum Conservation of energy ρc p D T Dt Conservation of species V = 0 1 ρ D V Dt = p + µ 2 V λ 4 V + J B + ρg 2 = K T 2 T + µ[ V : V T + V : V ] +4λ[ ω : ω T ] + 4λ [ ω : ω T ] + J.J σ 0 3 D C Dt = D 2 C. 4 Region 2: Porous Tissue Conservation of mass for an incompressible flow Conservation of momentum V p = 0 5 ρ D V p Dt = p + µ 2 Vp µ k 1 + β V p + ρg 6 where D = + V. is the material derivative Dt t V = u v 0 is the velocity vector ρ the blood density µ is the dynamic viscosity of the blood T is the temperature of the blood λ and λ are the couple stress parameter C p is the specific heat at the constant pressure K T is the thermal conductivity ω is the rotation vector C is the concentration D is the molecular diffusivity k is the permeability parameter of porous medium and g is gravity. In deriving the governing equation and the corresponding boundary conditions the following assumptions are made. Blood is treated as a couple stress fluid non-newtonian. Flow region may be classified into two sub-regions fluid film and porous tissuefigure 1. The biomagnetic fluid flow is laminar steady unidirectional and incompressible. The induced magnetic field and the electric field produced by the motion of blood are negligible since blood has low magnetic Reynolds number

5 544 N.P. Ratchagar & R. Vijayakumar A uniform magnetic field B 0 is applied in the y direction to the flow of blood. The effect of viscous dissipation and Joule heating are considered in the energy equation. The Boussinesq approximation is applied. Concentration C is introduced as a slug which is a function of time t and coordinates x and y. Under the above stated assumptions equations 1 and 6 reduces to Region 1: Fluid Film u x = 0 0 = p x + µ 2 u y 2 λ 4 u y 4 B2 0σ 0 u 7 0 = p y and energy equation 3 becomes 2 T 2 u 0 = K T y + µ + λ 2 y ρg 8 2 u y σ 0 B 2 0u 2. 9 The concentration C satisfying the convective diffusion equation 4 gives C 2 t + u C x = D C x + 2 C y 2 Region 2: Porous Tissue u p x = 0 0 = p x µ1 + β 1 u p 11 k 0 = p y The boundary conditions on the velocity and temperature are The couple stress conditions ρg 12 u y = α u u p T = T 1 at y = h 13 k u y = α k u u p T = T 0 at y = h 14 2 u = 0 at y = ±h 15 y2

6 AAM: Intern. J. Vol. 12 Issue 1 June The initial and boundary conditions on concentration C 0 x x s 2 C = 0 x > x at t = 0 16 s 2 C y = 0 at y = ±h 17 C = C x = 0 at x = 18 where u represents the axial velocity of the blood p is the pressure B 0 the magnetic field σ 0 is the electric conductivity of the blood K T is the the thermal conductivity T is the temperature of blood T 0 and T 1 are the temperatures at the lower and upper wall of blood. Equation 11 is the modified Darcy equation modified in the sense of incompressible couple stress parameter u p is the Darcy velocity α is the slip parameter and C 0 is the initial concentration of the initial slug input of length x s. Equations 13 and 14 are Beavers and Joseph 1967 slip condition at the lower and upper permeable surfaces. Equation 15 specifies the vanishing of couple stress at the boundaries. The term σ 0 B0u 2 in 7 represents the Lorentz force per unit volume and arises due to the electrical conductivity of the fluid whereas the term σ 0 B0u 2 2 in 9 represents the Joule heating. These two terms arise due to the MHD Cramer and Pai 1973 and Hughes and Young Assuming the uniform axial temperature variation along the walls the temperature of the blood can be written as T T 0 = N x + φy 19 where N is a constant temperature gradient in the x direction φy is certain function of temperature. The equation of state under the Boussinesq approximation Sekar and Raju 2013 is assumed to be ρ = ρ 0 1 β T T 0 20 where ρ 0 is the density of a reference state and β is the coefficient of volume expansion. Substituting equations 19 and 20 in 8 and integrating with respect to y we get p = ρ 0 gy + ρ 0 gβ Nxy ρ 0 gβ φydy + ψ 1 where ψ 1 = ψ 1 x is a y integration constant. Differentiating with respect to x we get p x = ρ 0gβ Ny + ψ 1 x. 21

7 546 N.P. Ratchagar & R. Vijayakumar Substituting 21 in 7 we obtain Introducing the non-dimensional variables 0 = ρ 0 gβ Ny dψ 1 dx + µ 2 u y 2 λ 4 u y 4 σ 0B 2 0u. 22 U = uh νp x U p = u ph νp x η = y h P x = h3 ρ 0 ν 2 dψ 1 dx T = T T 0 T 1 T 0. Equations 9 and 22 in non-dimensional form are Region 1: Fluid Film and 4 U η 2 U 4 a2 η + 2 a2 M 2 U = a 2 1 Gη 23 a 2 2 T EcP r η + 2 a2 U η U η a 2 M 2 U 2 = Region 2: Porous Tissue Integrating 12 with respect to y then substituting in 11 and using non-dimensional variables we get 1 U p = σ β 1 25 The boundary conditions of equations 13 to 15 in non-dimensional form are U η = ασu U p T = 1 at η = 1 26 U η = ασu U p T = 0 at η = U = 0 at η = ±1 28 η2 where a = h λ is the couple stress parameter l = µ l is the material constant characterizing the couple stress property of the fluid M 2 = B2 0σ 0 h 2 µ is the square of the Hartmann number G = β gnh 4 v is the Grashof number E ν 2 c = 0P 2 x 2 is the Eckert number P r = µc p P x h 2 C p T 1 T 0 K T is the Prandtl number σ = h k is the porous parameter ν is the kinematic viscosity νpx ρ h Re = h is the Reynolds number P x λ along the channel walls. is the positive values of G correspond to heating

8 AAM: Intern. J. Vol. 12 Issue 1 June Method of Solution Velocity and Temperature Distribution Region 1: Fluid Film Equation 22 is a fourth order differential equation with constant coefficient we get the complementary functioncf as and the particular integralpi as CF = C 1 e m 1η + C 2 e m 1η + C 3 e m 3η + C 4 e m 3η P I = 1 Gη M 2 We obtain Uη as the sum of CF and PI applying the boundary conditions 13 to 15 the velocity of blood are obtained as Uη = C 1 e m 1η + C 2 e m 1η + C 3 e m 3η + C 4 e m 3η + 1 Gη M Applying the boundary conditions 13 and 14 using equation 9 the temperature of blood is obtained as T = C 6 η + C 5 + I 4 I 5 e 2ηm 1 + I 4 I 6 e 2ηm 3 I 4 I 7 C2 C 3 e ηm 3 m 1 + C 1 C 4 e ηm 1 m 3 + I 4 I 9 e 2ηm 1 I 4 I 8 C1 C 3 e ηm 1+m 3 + C 2 C 4 e η m 1+m 3 + I 4 a 2 η 4 G 2 M 2 + 4a 2 η 3 GM 2 I 11 η 2 + I 4 I 10 e 2ηm 3 + I 12 I 4 C1 e ηm 1 GM 2 ηm Gm 2 1 m 1 M 2 + I 4 C 2 e η m 1 GM 2 ηm Gm 2 1 m 1 M 2 + I 13 I 4 C3 e ηm 3 GM 2 ηm Gm 2 3 m 3 M 2 + I 4 C 4 e η m 3 G GM 2 ηm m 2 3 m3 M 2 30 where C 1 C 2 C 3 C 4 C 5 C 6 I 4 I 5 I 6 I 7 I 8 I 9 and I 10 are constants given in the Appendix. The normalized axial components of velocity obtained from 29 is U = Ū U where Ū = Generalized dispersion model Uηdη = C 1 + C 2 sinh m 1 m 1 + C 3 + C 4 sinh m 3 m M 2.

9 548 N.P. Ratchagar & R. Vijayakumar Introducing the non-dimensional quantities θ = C C 0 ; equation 10 becomes ξ = Dx h 2Ū ; ξ s = Dx s h 2 ū ; η = y h ; τ = Dt h 2 ; U = ū u ; P e = hū D. θ τ + U θ = 1 2 θ ξ P e 2 ξ + 2 θ 31 2 η 2 We define the axial coordinate moving with the average velocity of flow as x 1 = x τū which is in dimensionless form ξ = ξ τ where ξ = x 1. Then equation 31 becomes hp e θ τ + U θ ξ = 1 2 θ P e 2 ξ + 2 θ 32 2 η 2 where P e = Ūh D is the Peclet number and U = Ū U coordinate system. non-dimensional velocity in a moving The initial and boundary conditions 16 to 18 in dimensionless form are 1 ξ ξ s θ = 2 0 ξ > ξ at τ = 0 33 s 2 θ η = 0 at η = ±1 34 θ = θ ξ = 0 at ξ =. 35 The solution of equation 32 is obtained using generalized dispersion model of Gill and Sankarasubramanian 1970 is that is θτ ξ η = θ m τ ξ + f 1 τ η θ m ξ + f 2τ η 2 θ m ξ θτ ξ η = θ m τ ξ + k=1 f k τ η k θ m ξ k 36 where θ m is the dimensionless cross sectional average concentration given by θ m τ ξ = θτ ξ ηdη. 37 Integrating equation 32 with respect to η in [ 1 1] and using the equation 37 we get θ m τ = 1 P e 2 2 θ m ξ θ η dη ξ 1 1 U θ dη 38

10 AAM: Intern. J. Vol. 12 Issue 1 June Substituting equation 36 in 38 we obtain θ m τ = 1 P 2 e 2 θ m ξ ξ 1 1 U θ m τ ξ + f 1 τ η θ m τ ξ +... dη. 39 ξ It is assumed that the process of distributing θ m is diffusive in nature from the time ero then following Gill and Sankarasubramanian 1970 the generalized dispersion model for θ m can be written as = θ m τ i=1 Substituting the equation 40 in 39 we obtain K i τ i θ m ξ i. 40 θ m K 1 ξ + K 2 θ m 2 ξ +K 3 θ m 2 3 ξ +... = 1 3 Pe 2 2 θ m ξ ξ 1 1 U θ m τ ξ 41 + f 1 τ η θ m ξ + f 2τ η 2 θ m τ ξ +...dη ξ2 Equating the coefficients of θ m ξ 2 θ m... we get ξ2 where δ ij K i τ = δ ij P 2 e is the Kroneckar delta defined by U f i 1 τ ηdη i = and j = δ ij = Substituting equation 36 in 32 we get θ m τ ξ + f 1 τ η θ m τ +U ξ θ m τ ξ + f 1 τ η θ m = 1 P 2 e 2 ξ 2 Substituting equation 40 in 43 using { 1 if i = j 0 if i j. ξ τ ξ + f 2τ η 2 θ m ξ ξ τ ξ + f 2τ η 2 θ m ξ θ m τ ξ + f 1 τ η θ m k+1 θ m τ ξ k τ ξ τ ξ ξ τ ξ + f 2τ η θ η 2 m τ ξ + f 1 τ η θ m ξ +... = i=1 K i τ k+i θ m ξ k+i

11 550 N.P. Ratchagar & R. Vijayakumar we obtain [ ] f1 τ 2 f 1 η 2 + U θm + K 1 τ ξ + [ fk+2 + k=1 τ [ f2 τ 2 f 2 η 2 + U f 1 + K 1 τf 1 + K 2 τ 1 2 f k+2 η 2 + U f k+1 + K 1 τf k+1 + ] k+2 + K i f k+2 i i=3 P 2 e K 2 τ 1 P 2 e ] 2 θ m ξ 2 f k k+2 θ m ξ k+2 = 0 44 k θ m with f 0 = 1. Equating the coefficients of k = in equation 44 to zero we ξ k obtain the following set of partial differential equations. f 1 τ f 2 τ f k+2 τ = 2 f 1 η 2 U K 1 τ 45 = 2 f 2 η 2 U f 1 K 1 τf 1 K 2 τ + 1 = 2 f k+2 η 2 U f k+1 K 1 τf K+1 P 2 e 46 K 2 τ 1 P 2 e k+2 f k i=3 K i f k+2 i. 47 Since θ m is chosen in such a way to satisfy the initial and boundary conditions on θ 33 and 34 on f k function becomes for k = f k 0 η = 0 48 f k τ 1 η = 0 49 f k τ 1 η = 0 50 Also from equation 35 we have 1 1 f k τ ηdη = 0 51 for k = To find K i s we know the f k s and its corresponding initial and boundary conditions. From equation 42 for i = 1 using f 0 = 1 we get K 1 as K 1 τ = 0. 52

12 AAM: Intern. J. Vol. 12 Issue 1 June From equation 42 for i = 2 we get K 2 as To evaluate K 2 τ K 2 τ = 1 P 2 e U f 1 dη. 53 put f 1 = f 10 η + f 11 τ η 54 where f 10 η corresponds to an infinitely wide slug which is independent of τ and f 11 is τ dependent satisfying 1 df 10 dη = 0 at η = ±1 55 f 10 dη = Using the 54 in 45 gives 1 d 2 f 10 dη 2 U = 0 57 f 11 τ = 2 f 11 η Solving the equation 57 with conditions 55 and 56 we get f 10 = 1 C 1 e ηm 1 + C 2 e η m 1 + C 3e ηm 3 + C 4 e η m 3 1 um 2 + u m 2 1 m 2 3 M 2 C 9 η C 10. η 2 2 η3 G 6M 2 Equation 58 is heat conduction type and its solution satisfying condition f 11 τ η = f 10 η of the form f 11 = B n e λ2nτ cosλ n η 60 n=1 1 where B n = 2 and λ n = nπ. Substituting 59 in 61 we get B n = 1 2C1 + C 2 m 1 cosnπ sinhm 1 ū m 2 1m n 2 π 2 ūm cosnπ +. M 2 ūn 2 π f 10 η cosλ n ηdη C 3 + C 4 m 3 cosnπ sinhm 3 m 2 3m n 2 π 2

13 552 N.P. Ratchagar & R. Vijayakumar Substituting 59 and 60 in equation 54 we get f 1 = 1 C 1 e ηm 1 + C 2 e η m 1 + C 3e ηm 3 + C 4 e η m 3 + u m 2 1 m 2 3 C 10 + e π2τ cosπτ π 3 u 2π 1 + M 2 u M 2 1 um 2 M 2 η 2 2 η3 G C 6M 2 9 η 2π 3 C 3 + C 4 m π 2 m 1 sin hm 3 +C 1 + C 2 m π 2 m 3 sin hm 1 m 1 m 3 m π 2 m π 2 e 4π2τ cos2πτ 8π 3 u 4π 1 + M 2 u M 2 16π 3 C 3 + C 4 m π 2 m 1 sin hm 3 +C 1 + C 2 m π 2 m 3 sin hm 1 m 1 m 3 m π 2 m π 2 + e 9π2τ cos3πτ 27π 3 u 6π 1 + M 2 u M 2 54π 3 C 3 + C 4 m π 2 m 1 sin hm 3 +C 1 + C 2 m π 2 m 3 sin hm 1 m 1 m 3 m π 2 m π 2. Substituting f 1 into equation 42 and integrating we get the solution of dispersion coefficient with help of MATHEMATICA 8.0 where C 9 and C 10 are constants given in the Appendix. Similarly K 3 τ K 4 τ and so on are obtained and we found that K i τ i > 2 are negligibly small compared to K 2 τ. Hence the dispersion model 40 now leads to θ m τ = K 2 θ m 2 ξ The exact solution of 62 satisfying the conditions 33 to 36 is obtained using Fourier TransformSankara 1995 as where T = τ 0 [ θ m ξ τ = 1 ξs erf + ξ ξs erf ξ ] 2 T 2 63 T K 2 ηdη and erfx = 2 π x e z2 dz. 0 The flow and heat transfer characteristics are the local skin friction coefficient C f and the local rate of heat transfer coefficient. Define C f = 2τ 1 ρ νpx h 2 Nu = qh K T T 1 T 0 64

14 AAM: Intern. J. Vol. 12 Issue 1 June where τ 1 = µ u y y=±h between the fluid and the wall. is the wall shear stress and q = K T T y y=±h is the heat flux By use of dimensionless quantities equation 64 can be written as: C f = 2 U Nu = T 65 Re η η η=±1 η=±1 where Nu is the Nusselt number. 4. Results and Discussion Dispersion of solute in combined free and forced convective fully developed flow of a couple stress fluid bounded by porous beds under the influence of magnetic field is studied using generalized dispersion model. The results are obtained to illustrate the influence of the Hartmann number M = Grashof number Gr = couple stress parameter a = 1 20 dimensionless time τ = and porous parameter σ = on the velocity temperature skin friction coefficient Nusselt numbers dispersion coefficient and the concentration profiles while the values of some of the physical parameters are taken as constant such as P r = 100 Ec = 0.2 α = 0.1 and β 1 = 0.1 in all the figures. We have extracted interesting insights regarding the influence of all the parameters that govern this problem. The influence of the parameters M Gr a τ and σ on horizontal velocity temperature skin friction coefficient Nusselt numbers dispersion coefficient and concentration profiles are analyzed from Figures 2 to 21. The expression for velocity profiles U are evaluated using equation 29 and are shown in Figures 3 and 4 for different values of the Grashof number Gr and couple stress parameter a with η. It is seen that the effect of increasing Grashof number and couple stress parameter decreases the velocity profile of the blood flow and are parabolic in nature. Figures 2 and 5 for different values of M and σ with η reveal that the velocity profile decreases with the increase of the Hartmann number M and porous parameter σ. The effect of M on the velocity profile is displayed. The presence of magnetic field normal to the flow in an electrical conducting fluid introduces a Lorentz force which acts against the flow. This resistive force tends to slow down the blood flow and hence the boundary layer decreases with the increase of the magnetic field. The expression for temperature distribution T are evaluated using equation 30 and are shown in Figures 6 and 9 for different values of M and σ with η. It is observed that temperature of the blood increases with decrease of the value of M and σ. Figures 7 and 8 for different values of Gr and a with η show that temperature profile increases with increase of Grashof number and couple stress parameter. It is also seen that the temperature is parabolic in nature increasing from its value at η = 1 to a maximum temperature around 3.2 and then decreasing steadily to its value at η = 1. The expression for dispersion coefficient K 2 P e 2 are numerically evaluated using equation 42

15 554 N.P. Ratchagar & R. Vijayakumar and are shown in Figures 10 to 13 for different values of M Gr a and σ with dimensionless time τ. Figure 10 shows that K 2 τ P e 2 decreases with increase in M. From Figures 11 to 13 reveal that the axial dispersion coefficient increases K 2 τ P e 2 with the increase of Grashof number couple stress and porous parameter which reflects the existence of larger velocity variation across the channel. The effects of above parameters on K 2 τ P e 2 is very significant when τ is very small and its effect is not so significant for large value of τ. For the values of Gr 2 it has been observed that K 2 τ P e 2 reaches a fixed value. That is where Gr 2 K 2 τ P e 2 becomes τ independent. The expression for mean concentration θ m are numerically evaluated using equation 63 and are shown in Figures 14 to 18 for different values of M Gr a σ and τ with axial distance ξ. Figures 15 to 17 reveal that there is marked variation of concentration with axial distance. It is apparent from these figures that the effect of increasing Gr a and σ is to decrease the peak value of the mean concentration. This implies that the concentration is more distributed in ξ direction for larger and larger values of Gr. The curves are bell shaped and symmetrical about the origin. θ m increases with the decrease of the values of M and τ as shown in Figures 14 and 18. These results are useful to understand the transport of solute at different times. Figure 19 depicts that the effect of skin friction coefficient against M for different values of Grashof number on the lower η = 1 and upper η = 1 wall. It shows that the shear stress grows rapidly for increasing the Grashof number at the lower and upper wall. C f decreases at the lower wall and increases at the upper wall. Negative values show that flow reversal arises within the boundary layer. The variation of Nusselt number with M is depicted in Figure 20 and in Figure 21. It is observed that the Nusselt numbers decrease with an increase in Grashof number at both the walls η = 1 and η = Conclusion An analysis is carried out to study the heat and mass transfer flow of couple stress fluids between two parallel channels bounded by porous beds and the density is dependent on temperature. When there is a uniform axial temperature variation along the walls the primary flow shows incipient flow reversal at the upper plate for an increase in temperature along that plate. Similarly flow reversal at the lower plate occurs with a decrease in temperature along that plate. The magnetic field arising as a body couple in the governing equations is shown to increase the axis dispersion coefficient. Acknowledgments The authors are grateful to the learned referees for their useful technical comments and valuable suggestions which led to a improvement of the paper.

16 AAM: Intern. J. Vol. 12 Issue 1 June Figure 2. Plots of velocity U versus η for different values of M Figure 3. Plots of velocity U versus η for different values of Gr Figure 4. Plots of velocity U versus η for different values of a Figure 5. Plots of velocity U versus η for different values of σ

17 556 N.P. Ratchagar & R. Vijayakumar Figure 6. Plots of temperature T versus η for different values of M Figure 7. Plots of T versus η for different values of Gr Figure 8. Plots of T versus η for different values of a Figure 9. Plots of T versus η for different values of σ

18 AAM: Intern. J. Vol. 12 Issue 1 June Figure 10. Variation of K 2 τ P e 2 with τ for different values of M Figure 11. Variation of K 2 τ P e 2 with τ for different values of Gr Figure 12. Variation of K 2 τ P e 2 with τ for different values of a Figure 13. Variation of K 2 τ P e 2 with τ for different values of σ

19 558 N.P. Ratchagar & R. Vijayakumar Figure 14. Plots of θ m versus ξ for different values of Hartmann number M Figure 15. Plots of θ m versus ξ for different values of Grashof number Gr Figure 16. Plots of θ m versus ξ for different values of couple stress parameter a Figure 17. Plots of θ m versus ξ for different values of porous parameter σ

20 AAM: Intern. J. Vol. 12 Issue 1 June Figure 18. Plots of θ m versus ξ for different values of time τ Figure 19. Plots of C f versus M for different values of Gr Figure 20. Plots of Nu versus M at η = 1 for different values of Gr Figure 21. Plots of Nu versus M at η = 1 for different values of Gr REFERENCES Aris R On the dispersion of a solute in a fluid flowing through a tube Proc. Roy. Soc. Lond. A Vol. 235 pp Barton N. G Solute Dispersion and Weak Second-Order Recombination at Large Times in Parallel Flow J. Fluid Mech Vol. 164 pp. 289.

21 560 N.P. Ratchagar & R. Vijayakumar Barton N.G. and Stokes A.N A computational method for shear dispersion in parallel flow Computational Techniques and Applications Vol. 85 pp Beavers G.S. and Joseph D.D Boundary conditions at a naturally permeable wall J. Fluid Vol. 30 pp Bestman A.R Low Reynolds number flow in a heated tube of varying section Australian Mathematical Society Series B. Vol. 25 pp Bhuvaneswari M. and Sivasankaran S Free convection flow in an inclined plate with variable thermal conductivity by scaling group transformations Proceedings of the 21st National Symposium on Mathematical Sciences SKSM21 AIP Conference Proceedings Vol pp Chatwin P. C On the interpretation of some longitudinal dispersion experiments Jour. Fluid Mech. Vol. 48 pp Chinyoka T. and Makinde O.D Unsteady and porous media flow of reactive non- Newtonian fluids subjected to buoyancy and suction/injection International Journal of Numerical Methods in Heat and Fluid Flow Vol. 25 No. 7 pp Cramer K.R. and Pai S.I Magnetofluid dynamics for engineers and applied physicists Scripta Publishing Company Washington D.C. Erdogan E. M. and Chatwin P.C The effects of curvature and buoyancy on the laminar dispersion of solute in a horizontal tube Journal of Fluid Mechanics Vol. 29 pp Gill W. N. and Sankarasubramanian R Exact analysis of unsteady convective diffusion Proc. Roy. Soc. Lond. A Vol. 316 pp Hughes W.F. and Young F.J The electromagnetodynamics of fluids 1st edition Wiley Inc New York. Lighthill M. J Initial development of diffusion in Poiseuille flow J. Inst. Maths. Applics. Vol. 2 pp Makinde O.D. Khan W.A. and Khan Z.H Buoyancy effects on MHD stagnation point flow and heat transfer of a nanofluid past a convectively heated stretching/shrinking sheet International Journal of Heat and Mass Transfer Vol. 62 pp Mazumder B.S Dispersion of solutes in combined free and forced convective flow through a channel Acta Mechanica Vol. 32 No. 4 pp Rashidi M.M. Momoniat E. and Rostami B Analytic approximate solutions for MHD boundary-layer viscoelastic fluid flow over continuously moving stretching surface by Homotopy Analysis Method with two auxiliary parameters Journal of Applied Mathematics Vol pp Rudraiah N. Pal D. and Siddheshwar P. G Effect of couple stresses on the unsteady convective diffusion in fluid flow through a channel Biorheology Vol. 23 No. 4 pp Rundora L. and Makinde O.D Effects of Navier slip on unsteady flow of a reactive variable viscosity non-newtonian fluid through a porous saturated medium with asymmetric convective boundary conditions. Journal of Hydrodynamics Ser. B Vol. 27 No. 6 pp Sankar. R Introduction to partial differential equations Prentice-Hall of India. Sankarasubramanian R. and Gill W. N Unsteady convective diffusion with interphase mass transfer Proceedings of the Royal Society of London. Series A Vol. 333 pp

22 AAM: Intern. J. Vol. 12 Issue 1 June Sekar R. and Raju R Effect of magnetic field dependent viscosity on Soret-driven thermoconvective instability of ferromagnetic fluid in the presence of rotating anisotropic porous medium of sparse particle suspension International Journal of Mathematical Sciences Vol. 12 pp Sibanda P. and Makinde O.D On steady MHD flow and heat transfer past a rotating disk in a porous medium with ohmic heating and viscous dissipation International Journal of Numerical Methods for Heat and Fluid Flow Vol. 20 No. 3 pp Siddheshwar P.G. and Thangaraj R.P Dispersion of solute in a fully developed flow of a Boussinesq Stokes suspension Hydrodynamics VI - Theory and applications - Cheng and Yeow eds. Taylor and Francis Group London pp Sivasankaran S Effect of variable thermal conductivity on buoyant convection in a cavity with internal heat generation Nonlinear Analysis: Modelling and Control Vol. 12 No. 1 pp Srinivasacharya D. Srinivasacharyulu N. and Odelu O Flow of couple stress fluid between two parallel porous plates IAENG International Journal of Applied Mathematics Vol. 41 No. 2 pp Kafoussias N.G. Tzirtzilakis E.E. and Raptis A Free-forced convective boundarylayer flow of a biomagnetic fluid under the action of a localized magnetic field Canadian Journal of Physics Vol. 86 pp Tzirtzilakis E.E. Kafoussias N.G. and Raptis A Numerical study of forced and free convective boundary layer flow of a magnetic fluid over a flat plate under the action of a localized magnetic field Journal of Applied Mathematics and Physics ZAMP Vol. 61 No. 5 pp

23 562 N.P. Ratchagar & R. Vijayakumar Appendix a 2 + a 4 4a 2 M 2 2 m 3 = a 2 a 4 4a 2 M 2 2 m 1 = a 3 = e m 1 ασ + m 1 a 4 = e m 1 m 1 ασ a 5 = e m 3 ασ + m 3 a 6 = e m 3 m 3 ασ a 8 = m 2 1e m 1 a 9 = m 2 1e m 1 a 10 = m 2 3e m 3 a 11 = m 2 3e m 3 C 1 = 1 I 3 I 1 a 3 a I 2 a 4 a 2 10 I 1 a 5 a 8 a 10 + I 1 a 6 a 9 a 10 I 2 a 6 a 8 a 10 + I 2 a 5 a 9 a 10 I 1 a 2 11a 3 I 1 a 11 a 6 a 8 + I 1 a 5 a 9 a 11 I 2 a 2 11a 4 I 2 a 11 a 5 a 8 + I 2 a 6 a 9 a 11 C 2 = 1 I 3 I 1 a 4 a I 2 a 3 a 2 10 I 1 a 6 a 8 a 10 + I 1 a 5 a 9 a 10 I 2 a 5 a 8 a 10 + I 2 a 6 a 9 a 10 I 1 a 2 11a 4 I 1 a 11 a 5 a 8 + I 1 a 6 a 9 a 11 I 2 a 2 11a 3 I 2 a 11 a 6 a 8 + I 2 a 5 a 9 a 11 C 3 = 1 I 3 I 1 a 5 a I 2 a 6 a I 1 a 3 a 10 a 8 + I 1 a 4 a 11 a 8 I 2 a 10 a 4 a 8 + I 2 a 3 a 11 a 8 I 1 a 5 a 2 9 I 1 a 10 a 4 a 9 + I 1 a 3 a 9 a 11 I 2 a 6 a 2 9 I 2 a 10 a 3 a 9 + I 2 a 4 a 9 a 11 C 4 = 1 I 3 I 1 a 6 a I 2 a 5 a 2 8 I 1 a 10 a 4 a 8 + I 1 a 3 a 11 a 8 I 2 a 10 a 3 a 8 + I 2 a 4 a 11 a 8 I 1 a 6 a 2 9 I 1 a 10 a 3 a 9 + I 1 a 4 a 9 a 11 I 2 a 5 a 2 9 I 2 a 10 a 4 a 9 + I 2 a 3 a 9 a 11 C 9 = 1 ū C 10 = C 5 = 1 q 2 1 q 2 C 6 = 1 q 2 1 q 2 C1 C 2 cosh m 1 1 ūm 2 + C 1+C 2 sinh m 1 6M 2 m 3 1 I 1 = ασ 1 G m 1 + C 3 C 4 cosh m 3 m 3 G u M 2 p I 2 = ασ G+1 u M 2 p 2M 2 + C 3+C 4 sinh m 3 m 3 3 I 3 =a 2 5a 2 8 a 2 6a 2 8 2a 10 a 3 a 5 a 8 2a 11 a 3 a 6 a 8 + 2a 4 a 6 a 10 a 8 + 2a 4 a 5 a 11 a 8 a 2 11a 2 3 a 2 10a 2 4 a 2 5a a 2 6a a 2 3a a 2 4a a 10 a 4 a 5 a 9 2a 11 a 4 a 6 a 9 + 2a 3 a 6 a 9 a a 3 a 5 a 9 a 11 Ec Pr I 4 = 12a 2 M 4 I 5 = 3C2 1 M 4 a 2 m 2 1 +M 2 +m 4 1 m 2 1 I 6 = 3C2 3 M 4 a 2 m 2 3 +M 2 +m 4 3 m 2 3 I 7 = 24M 4 a 2 M 2 m 1 m 3+m 2 1 m2 3 m 1 m 3 2 I 8 = 24M 4 a 2 m 1 m 3 +M 2 +m 2 1 m2 3 m 1 +m 3 2 I 9 = 6C2 2 M 4 a 2 m 2 1 +M 2 +m 4 1 2m 2 1

24 AAM: Intern. J. Vol. 12 Issue 1 June I 10 = 6C2 4 M 4 a 2 m 2 3 +M 2 +m 4 3 2m 2 3 I 11 =6a 2 2M 4 C 1 C 2 m C 3 C 4 m C1 C 2 + C 3 C 4 M 6 + G 2 + M M 4 C 1 C 2 m C 3 C 4 m3 4 I 12 = 24a2 M 2 m 3 1 I 13 = 24a2 M 2 I 14 = m 3 EcP 3 r 12M 4 a 2 24a 2 C 2 e m 1 M 2 G 2 m 1 M 2 m 2 q 1 =I 1 m 1 M 2 14 a 2 G 2 M 2 4a 2 GM 2 m I 14a 2 C 1 e m 1 M 2 G m 1 2 M 2 + m 2 1 m 1 M 2 m I 14C 4 e m 3 24a 2 M 2 G 2 m 3 M 2 m 2 3 m 3 M 2 m I 14C 3 e m 3 24a 2 M 2 G m 3 2 M 2 + m 2 3 m 3 M 2 m I 14 e 2m 1+m 3 M 4 C2 2e 2m 3 a 2 m M 2 + m 4 1 C2 4e 2m 1 a 2 m M 2 + m 4 3 2m 2 1 2m 2 3 I 14 I 8 C1 C 3 e m 1 m 3 + C 2 C 4 e m 1+m 3 I7 I 14 C2 C 3 e m 1 m 3 + C 1 C 4 e m 3 m 1 + I 5 I 14 e 2m 1 + I 5 I 14 e 2m 3 I 11 I 14 q 2 = 1 + I 14 a 2 G 2 M 2 + 4a 2 GM 2 + I 5 e 2m I 14a 2 C 1 e m 1 M 2 G m 1 2 M 2 + m 2 1 m 1 M 2 m I 14a 2 C 2 e m 1 M 2 G m M 2 m 2 1 m 1 M 2 m 3 1 I 7 I 14 C2 C 3 e m 3 m 1 + C 1 C 4 e m 1 m 3 I8 I 14 C1 C 3 e m 1+m 3 + C 2 C 4 e m 1 m 3 + I 14 I 6 e 2m 3 + I 14C 3 e m 3 24a 2 M 2 G m 3 2 M 2 + m 2 3 m 3 M 2 m I 14C 4 e m 3 24a 2 M 2 G m M 2 m 2 3 m 3 M 2 m 3 3 I 11 I I 14 e 2m 1+m 3 M 4 C2 2e 2m 3 a 2 m M 2 + m 4 1 C2 4e 2m 1 a 2 m M 2 + m 4 3 2m 2 1 2m 2 3.