MHD MIXED CONVECTION IN MICROPOLAR FLUID WITH CROSS DIFFUSION EFFECTS

Transcription

1 HEFA th International Conference on Heat ransfer, Fluid Mechanics and hermodynamics July 014 Orlando, Florida MHD MIXED CONVECION IN MICROPOLAR FLUID WIH CROSS DIFFUSION EFFECS Srinivasacharya D.* and Upendar M. *Author for correspondence Department of Mathematics, National Institute of echnology Warangal, Andhra Pradesh, , India. ABSRAC his paper analyzes cross diffusion effects on the steady, mixed convection heat and mass transfer along a semi-infinite vertical plate embedded in a micropolar fluid in the presence of traverse magnetic field. he governing nonlinear partial differential equations and their associated boundary conditions are transformed into a system of coupled nonlinear ordinary differential equations using a special form of Lie group transformations, namely, the scaling group of transformations and then solved numerically using the implicit finite difference method. he non dimensional velocity, microrotation, temperature and concentration along ith the non dimensional rate of heat and mass transfer at the plate are presented graphically for different values of coupling number, magnetic parameter (M, mixed convection parameter (R i, Soret number (S r and Dufour number (D f. In addition, the skin-friction coefficient and the all couple stress are shon in a tabular form. NOMENCLAURE B Buoyancy parameter = ( β C ( β B 0 C C p C s C C D f Magnetic field coefficient Concentration of the field Specific heat at constant pressure Concentration susceptibility Free stream concentration Skin-friction coefficient Mass diffusivity D K C D f Dufour number ( = C C ν f g g * Dimensionless stream function Dimensionless microrotation Acceleration due to gravity * 3 Gr Grashof number ( = g β L ν j k Dimensional Micro-inertia density hermal conductivity s p C k 1 k L m Mean absorption coefficient hermal diffusion ratio Characteristic length Wall couple stress M Non-dimensional magnetic parameter = ( σ B L ( µ M Non-dimensional couple stress on the all N κ Coupling number ( = κ + µ N u Dimensionless Nusselt number Pr ν Prandtl number, non-dimensional = α q ( x Heat flux q ( m x Mass flux Re Reynolds number, dimensionless Ri Gr Mixed convection parameter ( = Re s Dimensionless concentration Sc ν Schmidt number, non-dimensional = D Sh Sherood number D K S r Soret number ( = ν C m U( x emperature Mean fluid temperature Free stream temperature Free stream velocity m 0 Re u, v Velocity components in the x - and y -directions respectively. x, y Cartesian coordinates along the plate and normal to it x ˆ, yˆ Lie Group transformations of co-ordinate axes Greek symbols α hermal diffusivity α i s Lie Group transformation parameters (i = 1 to 6 β Coefficient of thermal expansion β C Coefficient of solutal expansion γ Spin-gradient viscosity η Non-dimensional similarity variable θ Dimensionless temperature ˆ θ Lie Group transformation of temperature 1596

2 L ϑ Non-dimensional micro-inertia density ( = j Re κ Vortex viscosity λ Spin-gradient viscosity ( = γ j ρν µ Viscosity of the fluid ν Kinematic viscosity ρ Density of the base fluid σ Electrical conductivity τ Wall shear stress ϕ Dimensionless concentration ˆ φ Lie Group transformation ψ Stream function ψ ˆ Lie Group transformation of stream function ω Microrotation component ˆω Lie Group transformation of micromotion component INRODUCION Mixed convection flos are of great interest because of their various engineering, scientific, and industrial applications in heat and mass transfer. Mixed convection of heat and mass transfer occurs simultaneously in the fields of design of chemical processing equipment, formation and dispersion of fog, distributions of temperature, moisture over agricultural fields, groves of fruit trees, and damage of crops due to freezing and pollution of the environment. Extensive studies of mixed convection heat and mass transfer of a non-isothermal vertical surface under boundary layer approximation have been undertaken by several authors. he majority of these studies dealt ith the traditional Netonian fluids. It is ell knon that most fluids hich are encountered in chemical and allied processing applications do not satisfy the classical Neton s la and are accordingly knon as non-netonian fluids. Due to the important applications of non-netonian fluids in biology, physiology, technology, and industry, considerable efforts have been directed toards the analysis and understanding of such fluids. A number of mathematical models have been proposed to explain the rheological behaviour of non-netonian fluids. Among these, the fluid model introduced by Eringen [1] exhibits some microscopic effects arising from the local structure and micro motion of the fluid elements. Further, they can sustain couple stresses and include classical Netonian fluid as a special case. he model of micropolar fluid represents fluids consisting of rigid, randomly oriented (or spherical particles suspended in a viscous medium here the deformation of the particles is ignored. Micropolar fluids have been shon to accurately simulate the flo characteristics of polymeric additives, geomorphologic sediments, colloidal suspensions, haematological suspensions, liquid crystals, lubricants etc. he heat and mass transfer in micropolar fluids is also important in the context of chemical engineering, aerospace engineering and also industrial manufacturing processes. he problem of free convection, heat and mass transfer in the boundary layer flo along a vertical surface submerged in a micropolar fluid has been studied by several investigators. In recent years, several simple boundary layer flo problems have received ne attention ithin the more general context of magneto hydrodynamics (MHD. Several investigators have extended many of the available boundary layer solutions to include the effects of magnetic fields for those cases hen the fluid is electrically conducting. he study of magneto-hydrodynamic flo for an electrically conducting fluid past a heated surface has important applications in many engineering problems such as plasma studies, petroleum industries, MHD poer generators, cooling of nuclear reactors, the boundary layer control in aerodynamics, and crystal groth. In addition, there has been a reneed interest in studying MHD flo and heat transfer in porous media due to the effect of magnetic fields on flo control and on the performance of many systems using electrically conducting fluids. he problem of MHD mixed convection heat and mass transfer in the boundary layer flo along a vertical surface submerged in a micropolar fluid has been studied by several investigators. Seddeek [] investigated the analytical solution for the effect of radiation on flo of a magneto-micropolar fluid past a continuously moving plate ith suction and bloing. Mahmoud [3] analyzed the effects of slip and heat generation/absorption on MHD mixed convection flo of a micropolar fluid over a heated stretching surface. zirtzilakis et al. [4] studied the action of a localized magnetic field on forced and free convective boundary layer flo of a magnetic fluid over a semi-infinite vertical plate. Hayat [5] studied the effects of heat and mass transfer on the mixed convection flo of a MHD micropolar fluid bounded by a stretching surface using Homotopy analysis method. Das [6] considered the effects of partial slip on steady boundary layer stagnation point flo of an electrically conducting micropolar fluid impinging normally toards a shrinking sheet in the presence of a uniform transverse magnetic field. In most of the studies related to heat and mass transfer process, Soret (mass fluxes can also be created by temperature gradients and Dufour (energy flux caused by a concentration gradient effects ere neglected on the basis that they are of a smaller order of magnitude than the effects described by Fourier s and Fick s las. But these effects are considered as second order phenomena and may become significant in areas such as hydrology, petrology, geosciences, etc. he Soret effect, for instance, has been utilized for isotope separation and in mixture beteen gases ith very light molecular eight and of medium molecular eight. he Dufour effect as recently found to be of order of considerable magnitude so that it cannot be neglected [7]. Alam and Rahman [8] have investigated the Dufour and Soret effects on mixed convection flo past a vertical porous flat plate ith variable suction. Chamkha [9] has focused on the numerical modelling of the effects of Soret, Dufour and radiation on heat and mass transfer by MHD mixed convection from a semi-infinite, isothermal, vertical and permeable surface immersed in a uniform porous medium. Raat and Bhargava [10] presented a mathematical model for the steady thermal convection heat and mass transfer in a micropolar fluid saturated Darcian porous medium in the presence of significant Dufour and Soret effects and viscous heating. Srinivasacharya and RamReddy [11] studied Soret and Dufour effects on the mixed convection from a semi-infinite vertical plate embedded in a stable micropolar fluid ith uniform and constant heat and mass flux conditions. he Lie group analysis, also called symmetry analysis, has been developed by Sophius Lie to find point transformations 1597

3 hich map a given differential equation to itself. his method reduces the number of independent variables by one and consequently the governing partial differential equations are transformed into ordinary differential equations ith the associated boundary conditions. his method has dran the attention of several researchers [1-18] to analyze various convective phenomena subject to various flo configurations arising in fluid mechanics, aerodynamics, plasma physics, meteorology and some branches of engineering. Inspired by the investigations mentioned above, the aim of this paper is to consider the effects of transverse magnetic field, Soret and Dufour effects on the mixed convection heat and mass transfer along a vertical plate ith variable heat and mass flux conditions embedded in a micropolar fluid. he governing nonlinear partial differential equations of boundary layer problem are transformed into a system of coupled nonlinear ordinary differential equations using a scaling group of transformations. he Keller-box method given in Cebeci and Bradsha [19] is employed to solve this nonlinear system of equations. he results thus obtained are compared that of the existing results and found to be in good agreement. MAHEMAICAL FORMULAION Consider a steady, laminar, incompressible, todimensional mixed convective heat and mass transfer along a semi infinite vertical plate embedded in a free stream of electrically conducting micropolar fluid ith velocity U ( x, temperature and concentration C. Choose the co-ordinate system such that x axis is along the vertical plate and y axis normal to the plate. he plate is maintained ith variable heat flux q ( x and mass flux qm ( x. A uniform magnetic field of magnitude B 0 is applied normal to the plate. he magnetic Reynolds number is assumed to be small so that the induced magnetic field can be neglected in comparison ith the applied magnetic field. he Boussinesq approximation is invoked for the fluid properties to relate density changes, and to couple in this ay the temperature and concentration fields, ρ = ρ β β C C to the flo field. In ( 1 ( C ( addition, the Soret and Dufour effects are considered. Using the Boussinesq and boundary layer approximations, the governing equations for the micropolar fluid are given by u v + = 0 x u u du ( x u ρ u + v = ρ U ( x + ( µ + κ + x dx ω κ ρ β β 0 ( ( ( ( C ( * + g + C C + σ B U x u u ρ j u ω v ω γ ω κ + = ω + x DK C u + v = α + x C C s p (1 ( (3 (4 C C C DK u + v = D + x m here u and v are the components of velocity along x - and y - directions respectively, ω is the component of microrotation hose direction of rotation lies normal to the xy - plane, g* is the gravitational acceleration, is the temperature, C is the concentration, β is the coefficient of thermal expansions, β c is the coefficient of solutal expansion. C p is the specific heat capacity, B 0 is the coefficient of the magnetic field, µ is the dynamic coefficient of viscosity of the fluid, κ is the vertex viscosity, j is the micro-inertia density, γ is the spin gradient viscosity, σ is the magnetic permeability of the fluid, υ is the kinematic viscosity of the fluid, α is the thermal diffusivity, D is the molecular diffusivity, K is the thermal diffusion ratio, C s is the concentration susceptibility, m is the mean fluid temperature. Equations (1 - (3 represent the conservation of mass, conservation of momentum and conservation of angular momentum, respectively. he last term of the Equation ( stands for the Lorenz force term. he term ( B σ ρ U( x represents the imposed pressure force in the 0 inviscid region of the conducting fluid and ( σ B ρ 0 represents the Lorentz force imposed by a transverse magnetic field to an electrically conducting fluid. he Equations (4 and (5 denote energy and concentration equations respectively. he boundary conditions for the velocity, microrotation, temperature and concentration distributions for the present problem can be considered as: u = 0, v = 0, 0 ω =, k = q ( x, D = q ( x (5 u C m at y = 0 (6a u U( x, ω = 0,, C C as y (6b he boundary condition ω = 0 in Eq. (6a, represents the case of concentrated particle flos in hich the microelements close to the all are not able to rotate, due to the no-slip condition. Introduce the folloing non dimensional variables x y Re u v Re U L x =, y =, u =, v =, Re =, L L U U ν ω L ω =, U ( x = U x U Re q ( x q ( x = + θ, = x Re, Lk Re k q ( x q ( x m m C = C + ϕ, = C x Re LD Re D and the stream function ψ in vie of (1 through u v ψ = in to Eqs. ( (5, e get x (7 ψ =, y 1598

4 3 ψ ψ ψ ψ 1 ψ = x x x 1 N N ω ψ + R x i ( θ + Bϕ + M x 1 N = λ ϑ ω + x x 1 N ψ ω ψ ω ω N ψ ψ θ ψ θ ψ x θ ϕ x x + θ = + x D f x x Pr (8 (9 (10 ψ ϕ ψ ϕ ψ x ϕ θ x x + ϕ = + x S (11 r x x Sc he transformed boundary conditions are ψ ψ θ ϕ = 0, = 0, ω = 0, = 1, = 1, at y = 0 (1a x y y ψ x, ω 0, θ 0, ϕ 0 as y (1b κ here N =, (0 N < 1 is the Coupling number, κ + µ Gr R i = is the mixed convection parameter, Re * 3 Gr = g β L ν, is the Grashof number, Re is the Reynolds number, M ( σ B L ( µ λ = γ ( j ρν is the spin-gradient viscosity, ϑ = L ( j Re the micro-inertia density, D f ( D K C ( Cs C p ν Dufour number and S ( D K ( ν C number. = 0 Re is the magnetic field parameter, = is the = is the Soret r m SIMILARIY SOLUIONS VIA LIE GROUP ANALYSIS We no introduce the one-parameter scaling group of transformations hich is a simplified form of Lie group transformation Γ: ε α1 ε α ε α3 xˆ = xe, yˆ = ye, ψˆ = ψ e, ˆ ω ω ˆ θ θ ˆ ϕ ϕ ε α4 ε α5 ε α6 = e, = e, = e (13 hereα 1, α, α 3, α 4, α 5, α 6 are transformation parameters and ε is a small parameter. his scaling group of transformations transform co-ordinates (x, y,ψ,ω,θ,ϕ to ( ˆ, ˆ, ˆ, ˆ, ˆ, ˆ x y ψ ω θ ϕ. Equations (8 to (11 and boundary conditions Eq. (1 are invariant under the point transformations (13. Substituting the transformations (13 into Eqs. (8 (11 and boundary conditions (1, e get ε ( α1 + α α ψˆ ψˆ ψˆ ψˆ 3 ε α1 e = e xˆ + ˆ xˆ ˆ xˆ ˆ 3 ε ( 3α α3 1 ψˆ ε ( α α4 N ˆ ω e + e R ˆ 3 i x 1 N yˆ + (14 1 N ˆ ε ( α1 α5 ( ( 3 ˆ ( e ˆ ε α α ε α ε α α ψ θ + Be ˆ ϕ + M e xˆ e ˆ is ε ( α1 + α α3 α4 ˆ ˆ ˆ ˆ ε ( α α4 ˆ = λe e ψ ω ψ ω ω ˆ xˆ xˆ ˆ ˆ N ε α ε 4 ( α α3 ψˆ ϑ e ˆ ω + e 1 N ˆ ˆ ˆ ε ( α α3 α5 ψˆ θ ψˆ θ ε ( α α3 α5 e ˆ ˆ x x e yˆ xˆ xˆ yˆ + ε ( α1 + α α5 ˆ ˆ ψˆ e xˆ θ ε ( α1 + α α6 ˆ ϕ θ = + xd ˆ f e ˆ Pr ˆ ˆ ε ( α α3 α6 ψˆ ˆ ϕ ψˆ ˆ ϕ ε ( α α3 α6 e xˆ xˆ ˆ ˆ ˆ ˆ + e x x ε ( α1 + α α6 ψˆ e xˆ ˆ ϕ ˆ ε ( α1 + α α6 θ ˆ ϕ = + xd ˆ f e ˆ Sc ˆ ˆ he boundary conditions (1 becomes ψˆ ψˆ = 0, = 0, ˆ θˆ ( 5 ω = 0, 1, ŷ ˆx yˆ e ε α α ϕˆ = yˆ e ε α e ε ( α α ψ y e εα x 3 1 ˆ, ( α6 = 1, (15 (16 (17 at ye ˆ εα = 0 (18a ˆ ω 0, ˆ θ 0, ˆ ϕ 0 ŷe εα as (18b Since the group transformations (13 keep the system invariant, e then have folloing relationships among the parameters α1 + α α 3 = α1 = 3α α 3 = α α 4 = α1 α5 = α1 α 6 = α1 = α α 3 α1 + α α3 α 4 = α α 4 = α 4 = α α 3 (19 α α3 α5 = α1 + α α5 = α1 + α α 6 α α3 α 6 = α1 + α α 6 = α1 + α α 5 Solving the linear system Eq. (19, e have the folloing relationship among the exponents α1 = α3 = α4 and α = α5 = α6 = 0 (0 Hence, the set of transformations Γ reduces to ˆ ε α ˆ ψˆ ε α 1 1 x = xe, y = y, = e, ε α 1 ωˆ ω e, θˆ θ, ϕˆ ϕ ψ (1 = = = Expanding the Lie Group of point transformations in one parameter using the aylor s series in poers of ε, keeping the terms up to the first degree (neglecting higher poers of ε, e get xˆ x = xεα ˆ ˆ 1, y y = 0, ψ ψ = ψεα1, ˆ ( θ θ = 0, ˆ ϕ ϕ = 0 he corresponding characteristic equations of ( are given by dx dy dψ dω dθ dϕ = = = = = (3 εα 0 εα εα Solving the above characteristic equations, e have folloing similarity transformations yˆ = η, ψˆ = xf ˆ ( η, ˆ ω = xg ˆ ( η, ˆ θ = θ( η, ˆ ϕ = ϕ( η (4 1599

5 Substitute the Eq. (4 into Eqs. (14 to (17, e get 1 N f ''' + ff '' + g ' ( f ' + 1 N 1 N R θ + Bϕ + 1 f ' M + 1 = 0 i ( ( (5 N λ g ϑ ( g f '' f g fg 0 1 N + + = (6 1 θ + f θ f θ + D f ϕ '' = 0 Pr (7 + + =0 (8 f (0 = 0, f ( 0 = 0, g ( 0 = 0, θ ' 0 = 1, ϕ ' 0 = 1 at η = 0 (9a ( ( ( g ( ( ( f 1, 0, θ 0, ϕ 0 at η (9b he all shear stress and the all couple stress u ω are τ = ( µ + κ + κω, m = γ.he y = 0 y = 0 τ dimensionless all shear stress C f =, all couple ρu m stress M LρU =, are given by Re ( 0 λ M = g x J and ( 0 C f = f x 1 N x here x =. he non-dimensional rate L Lq of heat transfer, called the Nusselt number Nu = and k ( rate of mass transfer, called the Sherood number Lqm Nu 1 Sh 1 Sh = are given by = and =. D C Re θ (0 Re ϕ(0 ( C RESULS AND DISCUSSION he governing Equations (5 to (8 have been solved numerically using the Keller-box method [19]. his method has a second - order accuracy, unconditionally stable and is easy to be programmed. he calculations are repeated until some convergent criterion is satisfied and the calculations are stopped hen δ f ''(0 10-8, δ g '(0 10-8, δ θ '( and δ ϕ '( In the present study, the boundary conditions for η at are replaced by a sufficiently large value of η here the velocity approaches 1, hereas the temperature and concentration approach zero. In order to see the effects of step size ( η e ran the code for our model ith three different step sizes as η = 0.001, η = 0.01 and η = 0.05 and in each case e found very good agreement beteen them on different profiles. After some trials e imposed a maximal value of η at as 6 and a grid size of η as In order to study the effects of the coupling number N, magnetic field parameter M, Prandtl number Pr, Schmidt number Sc, Dufour number D f and Soret number S r on the physical quantities of the flo, the remaining parameters are fixed as B = 1, λ = 1, Pr =0.7, Sc=0. and ϑ = 0.1. In the absence of coupling number N, Magnetic parameter M, Soret number S r, Dufour number D f and buoyancy parameter B ith R i = 1, λ = 1, ϑ = 0, and Sc = 0. the results of non-dimensional skin-friction f ''(0 and Nusselt number Nu have been compared ith the values of Ramachandran et al. [0], and it as found that they are in good agreement, as shon in table 1. Fig. depict that the variation of coupling number (N on the profiles of non-dimensional velocity, microrotation, temperature and concentration ith similarity variable η. he coupling number N characterizes the coupling of linear and rotational motion arising from the micromotion of the fluid molecules. Hence, N signifies the coupling beteen the Netonian and rotational viscosities. As N 1, the effect of microstructure becomes significant, hereas ith a small value of N the individuality of the substructure is much less pronounced. As κ 0 i.e. N 0, the micropolarity is lost and the fluid behaves as nonpolar fluid. Hence, N 0 corresponds to viscous fluid. It is observed from Fig. that the velocity decreases ith the increase of N. he maximum of velocity decreases in amplitude and the location of the maximum velocity moves farther aay from the all ith an increase of N. he velocity in case of micropolar fluid is less than that in the viscous fluid case. It is seen from Fig. 3 that the microrotation component decreases near the vertical plate and increases far aay from the plate ith increasing coupling number N. he microrotation tends to zero as N 0 as is expected. It is noticed from Fig. 4 that the temperature increases ith increasing values of coupling number. It is clear from Fig. 5 that the non-dimensional concentration increases from Netonian case to non-netonian case. he non-dimensional velocity, microrotation, temperature and concentration profiles, for different values of magnetic parameter M is illustrated from Figs It is observed from Fig. 6 that momentum boundary layer thickness decrease i.e., velocity increase as the magnetic parameter (M increases. From Eq. ( hen U ( x u (i.e. imposed pressure force dominates Lorentz force imposed by a transverse magnetic field normal to the flo direction, the effect of the magnetic interaction parameter is to increase the velocity. Similarly, U ( x u, the effect of the magnetic interaction parameter is to decrease the velocity. From Fig. 7, it is clear that the microrotation component increases near the plate and deceases far aay from the plate for increasing values of M. It is noticed from Fig. 8 that the temperature decreases ith increasing values of magnetic parameter. It is clear from Fig. 9 that the non-dimensional concentration decreases ith increasing values of M. he magnetic field gives rise to a motive force to an electrically conducting fluid, this force makes the fluid experience acceleration by decreasing the friction beteen its layers and thus decreases its temperature and concentration. Figures 10 to 13 depict that the effect of mixed convection parameter Ri on the profiles of non-dimensional velocity, microrotation, temperature and concentration. Fig. 10 explains that the dimensionless velocity rises as Ri increases. he higher value of Ri leads to the greater buoyancy effect in mixed convection flo, hence it accelerate the flo. he non- 1600

6 dimensional velocity of the fluid is increasing from pure forced convection (as Ri 0 to pure free convection (Ri > 1. It is seen from Fig. 11, that the magnitude of the microrotation increases ith an increase in mixed convection parameter Ri. From Fig. 1, it is observed that the non-dimensional temperature of the fluid flo is decreasing from pure forced convection case (Ri i 0 to the pure free convection case (Ri > 1. It is clear from Fig. 13 that the non-dimensional concentration decrease as Ri increases. he effect of Soret number S r and Dufour number D f on the non-dimensional velocity, microrotation, temperature and concentration is shon in Figures 14 to 17. It is observed from Figure 14 that the velocity decreases ith the increase of Dufour number D f (or decrease of Soret number Sr. It is noticed from Fig. 15 that the microrotation component decrease near the vertical plate and increase far aay from the plate ith increasing Dufour number (or decreasing of Soret number, shoing a reverse rotation near the to boundaries. he reason is that the microrotation field in this region is dominated by a small number of particles spins that are generated by collisions ith the boundary. It is clear from Figure 16 that the temperature of the fluid increases ith the increase of Dufour number (or decrease of Soret number. Figure 17 explains that the non-dimensional concentration of the fluid decreasing ith increase of Dufour number Df (or decrease of Soret number Sr. able shos that the effects of the coupling number N, Soret number S r and Dufour number D f, mixed convection parameter R i and the magnetic parameter M on the skin friction C f, the dimensionless all couple stress M, heat transfer rate (Nusselt number Nu and mass transfer rate (Sherood number Sh. It is seen from this table that the skin friction, the all couple stress heat and mass transfer rates decrease ith the increasing coupling number N. For increasing value of N, the effect of microstructure becomes significant; hence the all couple stress decreases. It is clear that skin friction, Nusselt number and Sherood number are increasing as mixed convection parameter Ri is increasing, here as the all couple stress decrease. As explained earlier, increasing effect of convection cooling as a result of the great buoyancy effect for large Ri. he effect of magnetic parameter is to increase the skin friction coefficient, Nusselt number, Sherood number and decrease the all couple stress. Further, it is observed that the skin friction coefficient and Nusselt number are decreasing and that of all couple stress and Sherood number are decreasing ith the increasing of Dufour number D f (or ith the decrease of Soret number S r. able 1 Comparison of results for a vertical plate in viscous fluids ithout stratification case Ramachandran et al. [0] for Ri = 1.0 f ''(0 Nu Pr [13] Present [13] Present able Effect of N,Ri, M, and S r, D f on skin friction, all couple stress, Heat and Mass ransfer rates. N Ri M D f S r - f''(0 - g''(0 1/θ(0 1/φ( Figure Profile for various values of coupling parameter N Figure 3 Microrotation Profile for various values of coupling parameter N 1601

7 Figure 4 emperature Profile for various values of coupling parameter N Figure 7 Microrotation Profile for various values of magnetic parameter M Figure 5 Concentration Profile for various values of coupling parameter N Figure 8 emperature Profile for various values of magnetic parameter M Figure 6 Velocity Profile for various values of magnetic parameter M Figure 9 Concentration Profile for various values of magnetic parameter M 160

8 Figure 10 Velocity Profile for various values of mixed convection parameter Ri Figure 13 Concentration Profile for various values of mixed convection parameter Ri Figure 11 Microrotation Profile for various values of mixed convection parameter Ri Figure 14 Velocity Profile for various values of Soret (S r and Dufour (D f numbers Figure 1 emperature Profile for various values of mixed convection parameter Ri Figure 15 Microrotation Profile for various values of Soret (S r and Dufour (D f numbers 1603

9 Figure 16 emperature Profile for various values of Soret (S r and Dufour (D f numbers Figure 17 Concentration Profile for various values of Soret (S r and Dufour (D f numbers CONCLUSION In this paper, a boundary layer analysis for mixed convection heat and mass transfer in an electrically conducting micropolar fluid over a vertical plate ith variable heat and mass flux conditions in the presence of a uniform magnetic field, Soret and Dufour effects is considered. Numerical results indicates that the higher values of the coupling number N (i.e., the effect of microrotation becomes significant result in loer skin friction, all couple stress, and also the loer nondimensional heat and mass transfer coefficients in the boundary layer compared to that of Netonian fluid case (N = 0. As Ri increase the skin friction coefficient heat and mass transfer rates increases hile the all couple stress decreases. he increase in the magnetic parameter M increases the skin friction coefficient, all couple stress, non-dimensional heat and mass transfer coefficients (Nu and Sh. he decrease in Soret number (S r (or increase in Dufour number D f results in loer skin friction and heat transfer rates and higher all couple stress and mass transfer rates. REFERENCES [1] Eringen, A. C., heory of Micropolar Fluids, Journal of Mathematics and Mechanics, Vol. 16, 1966, 1-18 [] Seddeek, M. A., Odda, S. N., Akl, M. Y., and Abdelmeguid, M. S., Analytical Solution for the Effect of Radiation on Flo of a Magneto-Micropolar Fluid Past a Continuously Moving Plate With Suction and Bloing, Computational Materials Science, Vol. 46, 009, [3] Mahmoud, M., and Waheed, S., Effects of Slip and Heat Generation/Absorption on MHD Mixed Convection Flo of a Micropolar Fluid over a Heated Stretching Surface, Mathematical Problems in Engineering, Vol. 010, 010, 0 [4] zirtzilakis, E. E., Kafoussias, N. G., and Raptis, A., Numerical Study of Forced and Free Convective Boundary Layer Flo of a Magnetic Fluid over a Plate under the action of a Localized Magnetic Field, Zeitschrift fur Angeandte Mathematik und Physik (ZAMP, Vol. 61, 010, [5] Hayat,., Mixed Convection Flo of a Micropolar Fluid ith Radiation and Chemical Reaction, International Journal for Numerical Methods in Fluids, Vol. 67, 011, [6] Das, K., Slip Effects on MHD Mixed Convection Stagnation Point Flo of a Micropolar Fluid toards a Shrinking Vertical Sheet, Computers and Mathematics ith Applications, Vol. 63, 01, [7] Eckert, E. R. G., and Drake, R. M., Analysis of Heat and Mass ransfer, 197, McGra Hill [8] Alam, M. S., and Rahman, M. M., Dufour and Soret Effects on Mixed Convection Flo Past a Vertical Porous Flat Plate ith Variable Suction, Nonlinear Analysis: Modelling and Control, Vol. 11(1, 006, 3-1 [9] Ali Chamkha, J., and Abdullatif Ben-Nakhi, MHD Mixed Convection Radiation interaction along a Permeable Surface immersed in a Porous Medium in the presence of Soret and Dufour Effects, Heat and Mass ransfer, Vol. 44, 008, [10] Raat, M., and Bhargava, R., Finite Element Study of Natural Convection Heat and Mass ransfer in a Micropolar Fluid Saturated Porous Regime ith Soret/Dufour Effects, International Journal of Applied Mathematics and Fluid Mechanics, Vol. 5, 009, [11] Srinivasacharya, D., and RamReddy, Ch., Effects of Soret and Dufour on Mixed Convection Heat and Mass ransfer in a micropolar fluid ith heat and mass fluxes, Journal of Heat ransfer, Vol.133, 010, [1] Hibiki,., and Ishii, M., Development of one-group interfacial area transport equation in bubbly flo systems. Int. J. Heat Mass ransfer., Vol. 45, 00, [13] anthanuch,., and Meleshko, S. V., On definition of an admitted lie group for functional differential equations, Commun. Nonlinear Sci. Numer. Simul., Vol. 9, 004, [14] Bluman, G., Broadbridge, P., King, J. R., and Ward, M. J., Similarity: generalizations, applications and open problems, J. Eng. Math., Vol. 66, 010, 1 9. [15] Hamad, M. A. A., Analytical solution of natural convection flo of a nanofluid over a linearly stretching sheet in the presence of magnetic field. Int. Commun. Heat Mass ransfer., Vol. 38, 011, [16] Salama, F. A., Lie group analysis for thermophoretic and radiative augmentation of heat and mass transfer in a Brinkman- Darcy flo over a flat surface ith heat generation, Acta Mech. Sin. Vol. 7(4, 011, [17] Rosmila A.B, Kandasamy R, Muhaimin I, Lie symmetry group transformation for MHD natural convection flo of nanofluid over linearly porous stretching sheet in presence of thermal stratification, Appl. Math. Mech. -Engl. Ed., Vol. 33(5, 01,

10 [18] N. Vishnu Ganesh et al., Lie symmetry group analysis of magnetic field effects on free convective flo of a nanofluid over a semi-infinite stretching sheet, Journal of the Egyptian Mathematical Society (013, j.joems [19] Cebeci,., and Bradsha, P., Physical and Computational Aspects of Convective Heat ransfer, 1984, Springer Verlin [0] Ramachandran, N., Chen,. S., and Armaly, B. F., Mixed Convection in Stagnation Flos adjacent to Vertical Surfaces, Journal of Heat ransfer, Vol. 110, 1988, (OSI ID: