EFFECT OF HEAT AND MASS TRANSFER ON UNSTEADY MHD POISEUILLE FLOW BETWEEN TWO INFINITE PARALLEL POROUS PLATES IN AN INCLINED MAGNETIC FIELD.

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1 EFFECT OF HEAT AND MASS TRANSFER ON UNSTEADY MHD POISEUILLE FLOW BETWEEN TWO INFINITE PARALLEL POROUS PLATES IN AN INCLINED MAGNETIC FIELD. * Joseph K.MP P., Auba PP P., Nitor L.NP P., and Mohammed S.MP P.. Department of Mathematics, Kaduna State Universit, Kaduna Nigeria. Kano Electricit Distribution Compan, Kano Nigeria ABSTRACT The effect of heat and mass transfer on unstead MHD poiseuille flow between two infinite parallel porous plates in an inclined magnetic field has been investigated, where the lower plate is considered porous. The governing equations of the flow field are solved b perturbation technique and the expression for the velocit u, temperature θ and concentration c were obtained. The effect of parameters such as Hartmann number Ha, Grashof number (Gr and Gc), Radiation N, Prandtl number Pr, Schmidt number Sc and Chemical parameter Kc were studied. The results show that at high Hartmann number Ha, the velocit decreased. Velocit increased due to effect of thermal Grashof number Gr and solutal Grashof number Gc. An increase in Prandtl number Pr decreased temperature. Species concentration reduced with increase in chemical parameter Kc and Schmidt number Sc. Kewords: unstead, MHD, poiseuille flow, porous plate, Heat Transfer, Mass Transfer.. INTRODUCTION Magnetohdrodnamic (MHD) is the stud of the dnamics of electricall conducting fluids under the influence of a magnetic field. Fluid such as mercur, molten iron and ionized gases often called plasma b phsicist of which the solar atmosphere is an example, are but a few electricall conducting fluids. 353

2 The applications of the effect of heat and mass transfer on unstead MHD poiseuille flow between two infinite parallel porous plates in an inclined magnetic field are visible in several fields of engineering technolog. The motion of an electricall conducting fluid placed in a constant magnetic field induces current that creates a force on the fluid. The current generated has been used in the designed MHD generators for electricit generation, MHD devices, nuclear engineering and the possibilit of thermonuclear power that has created an immense practical used for understanding the dnamics of electricall conducting fluid. The effect of magnetic field in viscous incompressible flow of electricall conducting fluid is of use in extrusion of plastics and food. Hannes Alfven (94), a Swedish electrical engineer first initiated the stud of MHD. Shercliff (956) considered the stead motion of an electricall conducting fluid in pipes under transverse magnetic fields. Sparrow and Cess (96) observed that free convection heat transfer to liquid metals ma be significantl affected b the presence of magnetic field. Drake (965) considered flow in a channel due to periodic pressure gradient and solved the resulting equation b separation of variables methods. Singh and Ram (978) studied Laminar flow of an electricall conducting fluid through a channel in the presence of a transverse magnetic field under the influence of a periodic pressure gradient and solved the resulting differential equation b the method of Laplace transform. More to this, Ram et al (984) analzed hall effects on heat and mass transfer flow through porous media. Soundelgekar and Abdulla Ali (986) studied the flow of viscous incompressible electricall conducting isothermal plate. Singh (993) considered the stead MHD fluid flow between two parallel plates. John Moone and Nick Stokes (997) considered the effects of thermal radiation and free convection flow past a moving vertical plate. Al-Hadhrami (3) discussed flow through horizontal channels of porous material and obtained velocit expressions in terms of the Renolds number. Ganesh (7) studied unstead MHD Stokes flow of a viscous fluid between two parallel porous plates. Stamenkovic et al () investigated MHD flow of two immiscible and electricall conducting fluids between isothermal, insulated moving plates in the presence of applied electric and magnetic fields. The matched the solution at the interface and it was found that decrease in magnetic field 354

3 inclination angle flattens out the velocit and temperature profiles. Rajput and Sahu () studied the effect of a uniform transverse magnetic field in the unstead transient free convection flow of an incompressible viscous electricall conducting fluid between two infinite vertical parallel porous plates with constant temperature and variable mass diffusion. Maonge et al () studied stead MHD poiseuille flow between two infinite parallel porous plates in an inclined magnetic field and discovered that high magnetic field strength decreases the velocit. Heat transfer effects on rotating MHD coquette flow in a channel partiall filled b a porous medium with hall current has been discussed b Singh and Rastogi (). Choudhar and DebH () studied heat and mass transfer for viscoelastic MHD boundar laer flow past a vertical flat plate. Sandeep and Sugunamma (3) analzed the effect of an inclined magnetic field in unstead free convection flow of a dust viscous fluid between two infinite flat plates filled b a porous medium. Joseph et al (4) studied the unstead MHD coquette flow between two infinite parallel porous plates in an inclined magnetic field with heat transfer. The found out that when the magnetic field is high, it reduces the energ loss through the plate. Large Nusselt number corresponds to more active convection and high prandtl number decreases the temperature distribution. The unstead MHD poiseuille flow between two infinite parallel plates in an inclined magnetic field with heat transfer has been studied b Idowu et al (4). In this paper, we considered one dimensional poiseuille flow of an electricall conducting fluid between two infinite parallel porous plates under the influence of magnetic field with heat and mass transfer.. PROBLEM FORMULATION The concept of magneto hdrodnamics phenomenon can simpl be described as follows: consider an electricall conducting fluid moving with velocit V. at right angles to this flow, we appl a magnetic field, the field strength of which is represented b the vector B. we assume that the fluid has attained unstead state conditions. That is, flow variables are dependent of the time t. Because of the interaction of the two 355

4 fields, an electric field vector denoted E is induced at right angles to both V and B. This electric field is given b E = V B () If we assume that the conducting fluid is isotropic/exhibits adiabatic flow in spite of the magnetic field, then we denote the electrical conductivit of the fluid b a scalar σ. B Ohm s law[4], the densit of the current induced in the conducting fluid denoted J is given b J = σe () Or J = σ(v B) (3) Simultaneousl occurring with the induced current is the Lorentz force F given b F = J B (4) This force occurs because, as an electric generator, the conducting fluid cuts the lines of the magnetic field. The vector F is the vector cross product of both J and B and is a vector perpendicular to the plane of both J and B. This induced force is parallel to V but in opposite direction. Laminar flow through a channel under uniform transverse magnetic field is important because of the use of MHD generator, MHD pump and electromagnetic flow meter. We now consider an electricall conducting viscous, unstead, incompressible fluid moving between two infinite parallel plates both kept at a constant distance h between them. Both plates of the channel are fixed with no motion. This is plane poiseuille flow. The equations of motion are the continuit equation. V = (5) and the Navier-Stokes equations ρ t + V. V = f B P + μ V (6) 356

5 Where ρ is the fluid densit, f B is the bod force per unit mass of the fluids, μ is the fluid viscosit and P is the pressure acting on the fluid. If we assume a one dimensional flow so that we choose the axis of the channel formed b the two plates as the x-axis and assume that the flow is in this direction. Observe that u, v and w are the velocit components in x, and z directions respectivel. Then this implies v = w = and u, then the continuit equation is satisfied. From this we infer that u is independent of x. This makes the nonlinear term [(V. )V] in the Navier-Stokes equation vanish. We neglect bod forces f B which are mainl due to gravit in the Navier-Stokes equations and replace them with the Lorentz force and from the assumption that the flow is one dimensional, it means that the governing equation for this flow is u t = ρ P x + υ u + F x ρ (7) Where v = μ ρ is the kinematics viscosit and F x is the component of the magnetic force in the direction of x-axis. Assuming unidirectional flow so that v = w = and B x = B z = since magnetic field is along -direction so that V = iu and B = B o j. Where B o is the magnetic field strength. Now, F x = σ[(iu jb o )] jb o (8) So that we have F x ρ = σ ρ B o u (9) Then (7) becomes u t = ρ P x + υ u σ ρ B o u () From (), when angle of inclination is introduced, we have 357

6 u t = ρ P x + υ u σ ρ B o u sin (α) () Where α is the angle between V and B. Equation () is general in the sense that both fields can be assessed at an angle α for α π. Because of the porosit of the lower plate, the characteristic velocit υ o is taken as a constant so as to maintain the same pattern of flow against suction and injection of the fluid in which it is moving perpendicular to the fluid flow. The origin is taken at the Centre of the channel and x, coordinate axis are parallel and perpendicular to the channel walls respectivel. The governing equations, that is; the momentum equation, the energ equation and the concentration equation are as follows: The momentum equation is given as ρ u u = v t o P + μ u σ B x ρ o u sin (α) + gβ(t T ) + gβ (C C ) () since the flow is isentropic, the energ equation is given as T t = k T + q ρcp ρcp (3) Where k is the thermal conductivit of the fluid, ρ is the densit, c p is the specific heat constant pressure and T is the temperature. The concentration equation is given as C t = D C Kc (C C ) (4) The q in (3) is called the heat flux. It is given b, q = 4α (T T ) (5) The boundar conditions are U (, t) =, T = T ; at t =, U ( L, t ) =, 358

7 U (L, t ) = v L, T = T w ; at t >, C = C at t =, C = C w; at t > (6) In order to solve equations (), (3) and (4) subject to the boundar conditions (6), we introduce the following dimensionless parameters: u = uv L, t = tl v, = L, P = pρ v L, x = xl, θ = T T w T T T = θ(t w T ) T = θ(t w T ) + T, Gr = ρl gβ(t w T ), C = C C C μv C = C w C C(C w C ) C = C(C w C ) + C, Gc = ρl gβ (C w C ), Pr = μcp Cp = μv k Prk, μ N = 4α L k α = N k q, x = xl, = L, = 4L 4α (T T ), Sc = V D D = V Sc Equations (), (3) and (4) now become u t = A u p x + u M u + Grθ + GcC (7) Where M = m sinα and m = LB o σ μ p x. Re = Ha, A =. Since it is poiseuille flow, ρ Pr θ t = θ + N θ (8) C = C t Sc K c C (9) The boundar conditions in dimensionless form are U(, t) =, θ(, t) =, C(, t) = at t = U(, t) =, θ(, t) =, C(, t) = () 3. METHOD OF SOLUTION/SOLUTION OF THE PROBLEM 359

8 The momentum equation, energ equation and concentration equation can be reduced to the set of ordinar differential equations, which are solved analticall. This can be done b representing the velocit, temperature and species concentration of the fluid in the perturbation series as follows U(, t) = U o () + εu ()e iωt + (ε ) () θ(, t) = θ o () + εθ ()e iωt + (ε ) () C(, t) = C o () + εc ()e iωt + (ε ) (3) Substituting equations (), () and (3) into equations (7), (8) and (9). Equating the coefficients of harmonic and non-harmonic term and neglecting the coefficients of higher order of ε, we get: U o () + AU o () M U o () = Q Grθ o () GcC o () (4) U () + AU () bu () = Grθ () GcC () (5) Where b = M + iω, Q = p x (constant) θ o () + N θ o () = (6) θ () a θ () = (7) Where a = iωpr N C o () ScKcC o () = (8) C () ScLC () = (9) The corresponding boundar conditions become U o (, t) =, θ o (, t) =, C o (, t) = U o (, t) =, θ o (, t) =, C o (, t) = U (, t) =, θ (, t) =, C (, t) = 36

9 U (, t) =, θ (, t) =, C (, t) = (3) We now solve equations (4) (9) under the relevant boundar conditions for the mean flow and unstead flow separatel. The mean flows are governed b the equations (4), (6) and (8) where U o, θ o and C o are called the mean velocit, mean temperature and mean concentration respectivel. The unstead flows are governed b equations (5), (7) and (9) where U, θ and C are the unstead components. These equations are solved analticall under the relevant boundar conditions (3) as follows; Solving equations (4), (6) and (8) subject to the corresponding relevant boundar conditions in (3), we obtain the mean velocit, mean temperature and mean concentration as U o () = C 5 e m 5 + C 6 e m 6 + K + K cosn + K 3 sinn + K 4 e m 3 + K 5 e m 4 (3) θ o () = c cosn + c sinn (3) C o () = c 3 e m 3 + c 4 e m 4 (33) Similarl, solving equations (5), (7) and (9) under the relevant boundar conditions in (3), the unstead velocit, unstead temperature and unstead concentration becomes U () = C e m + C e m + K 6 e m 7 + K 7 e m 8 + K 8 e m 9 + K 9 e m (34) θ () = C 7 e m 7 + C 8 e m 8 (35) C () = C 9 e m 9 + C e m (36) Therefore, the solutions for the velocit, temperature and species concentration profiles are U(, t) = C 5 e m 5 + C 6 e m 6 + K + K cosn + K 3 sinn + K 4 e m 3 + K 5 e m 4 + ε[c e m + C e m + K 6 e m 7 + K 7 e m 8 + K 8 e m 9 + K 9 e m ]e iwt (37) 36

10 θ(, t) = c cosn + c sinn + ε[c 7 e m 7 + C 8 e m 8 ]e iwt (38) C(, t) = c 3 e m 3 + c 4 e m 4 + ε[c 9 e m 9 + C e m ]e iwt (39) 4. DISCUSSION OF RESULTS To discuss the effect of heat and mass transfer on unstead MHD poiseuille flow between two infinite parallel porous plates in an inclined magnetic field. The velocit profile u, the temperature distribution θ and the species concentration C are shown graphicall against using matlab for different values of the following parameters such as Hartmann number Ha, thermal Grashof number Gr, modified Grahof number Gc, Radiation parameter N, Prandtl number Pr, Schmidt number Scand chemical parameter Kc. Figures, and 3 present the effect of Hartmann number Ha on velocit u. It is inferred from these figures that an increase in the Hartmann number decreases the fluid velocit. Figures 4, 5 and 6 show the effect of thermal Grashof number Gr on velocit u. It is observed that an increase in thermal Grashof number Gr and the angle of inclination on velocit profile u increases the velocit. Figures 7, 8 and 9 depict the effect of modified Grashof number Gc on velocit u. It is shown that the velocit increases as the modified Grashof number Gc increases. Figure describes the effect of Prandtl number Pr on temperature distribution θ. It is simulated from the figure that an increase in Prandtl number Pr leads to the decrease in temperature. Figures and illustrate the effect of chemical parameter Kc and Schmidt number Sc on the species concentration. It is seen that an increase in the chemical parameter Kc and Schmidt number Sc decreases the species concentration. Table depicts variation of skin frictions τ and τ, Nusselt numbers Nu and Nu and Sherwood numbers Sh and Sh with time t. It is observed that, the time is constant and does not affect the values. 36

11 3.5 Effect of Hartmann number Ha on velocit profile u with α= Velocit u.5 Ha=,, 3, Figure : Effect of Hartmann number Ha on velocit profile u with α = 5, B =, t =.5, ε =., Gr =, Gc =, N = and ω =. 363

12 8 Effect of Hartmann number Ha on Velocit profile u with α= Velocit u 4 3 Ha=,, 3, Figure : Effect of Hartmann number Ha on velocit profile u with α = 3, B =, t =.5, ε =., Gr =, Gc =, N = and ω =. Effect of Hartmann number Ha on Velocit profile u with α=45 - Velocit u Ha=,, 3,

13 Figure 3: Effect of Hartmann number Ha on velocit profile u with α = 45, B =, t =.5, ε =., Gr =, Gc =, N = and ω =. Effect of Grashof number Gr on velocit profile u with α=5.5.5 Gr=,,3,4 velocit u Figure 4: Effect of Grashof number Gr on velocit profile u with α = 5, B =, t =.5, ε =., Gc =, Ha =, N = and ω =.. Effect of Grashof number Gr on velocit profile u with α=3.8 velocit u.6.4. Gr=,,3, Figure 5: Effect of Grashof number Gr on velocit profile u with α = 3, B =, t =.5, ε =., Gc =, Ha =, N = and ω =. 365

14 .5 Effect of Grashof number Gr on velocit profile u with α=45.5 Gr=,,3,4 velocit u Figure 6: Effect of Grashof number Gr on velocit profile u with α = 45, B =, t =.5, ε =., Gc =, Ha =, N = and ω =. Effect of Grashof number Gc on velocit profile u with α=5.5.5 velocit u -.5 Gc=,,3, Figure 7: Effect of Grashof number Gc on velocit profile u with α = 5, B =, t =.5, ε =., Gr =, Ha =, N = and ω =. 366

15 3 Effect of Grashof number Gc on velocit profile u with α=3.5 velocit u.5.5 Gc=,,3, Figure 8: Effect of Grashof number Gc on velocit profile u with α = 3, B =, t =.5, ε =., Gr =, Ha =, N = and ω =..5 Effect of Grashof number Gc on velocit profile u with α=45.5 velocit u.5 Gc=,,3, Figure 9: Effect of Grashof number Gc on velocit profile u with α = 45, B =, t =.5, ε =., Gr =, Ha =, N = and ω =. 367

16 Effect of Prandtl number Pr on temperature distribution T.5 Temperature T Pr=.,.3,.5, Figure : Effect of Prandtl number Pr on temperature distribution θ with B =, t =.5, ε =., Gr =, Ha =, N = and ω =.. Effect of Chemical parameter K c on Species Concentration C K c =.8 K c = K c =3 K c =4 Species Concentration C Figure : Effect of Chemical parameter Kc on Species Concentration C with B =, t =.5, ε =., Gr =, Sc =, N = and ω =. 368

17 Effect of Schmidt number Sc on Species Concentration C. Sc= Sc= Sc=3 Sc=4 Species Concentration C Figure : Effect of Schmidt number Sc on Species Concentration C with Gr=,Gc=, B =, t =.5, ε =., Gr =, Kc =, N = and ω =. Table : Variation of skin frictions τ and τ, Nusselt numbers Nu and Nu and Sherwood numbers Sh and Sh with time t. t τ τ Nu Nu Sh Sh

18 SUMMARY AND CONCLUSION In this section, we studied the effect of heat and mass transfer on unstead MHD poiseuille flow between two inifinite parallel porous plates in an inclined magnetic field. The governing equations, that is, the momentum, energ and species concentration equations have been written in dimensionless form using the dimensionless parameters. Perturbation method was emploed to evaluate and solve the velocit profile u, temperature distribution θ, the species concentration C, Skin frictions τ and τ, Nusselt numbers Nu and Nu and Sherwood numbers Sh and Sh. The investigation of this research work leads to the following conclusions: At high Hartmann number Ha, the velocit decreased. Velocit increased due to effect of thermal Grashof number Gr and solutal Grashof number Gc. An increase in Prandtl number Pr decreased temperature. Species concentration reduced with increase in chemical parameter Kc and Schmidt number Sc. 37

19 This work can be applied in electric power generator, extrusion of plastics in the manufacture of raon and nlon, extrusion of food in the manufacture of macaroni etc. Appendix CONSTANTS m = in, m = in, m 3 = ScKc, m 4 = ScKc, m 5 = A+ A +4M, m 6 = A A +4M, m 7 = a, m 8 = a ; a = iωpr N, m 9 = ScL, m = ScL; L = Kc iω, m = A+ A +4b, m = A A +4b, T = K + K cos N K 3 sin N + K 4 e m 3 + k 5 e m 4, T = K + K cos N + K 3 sin N + K 4 e m 3 + k 5 e m 4, T 3 = K 6 e m 7 + k 7 e m 8 + K 8 e m 9 + k 9 e m, T 4 = K 6 e m 7 + k 7 e m 8 + K 8 e m 9 + k 9 e m, K = Q M, K = Gr C AN C (M +N ) ((M +N ) A N ) K 5 =, K 3 = Gr(C (M +N ) C AN), K 4 = A N +(M +N ) Gc C 4 m 4 +Am 4 M, K 6 = GrC 7 m 7 + Am 7 b, K 7 = GrC 8 m 8 +Am 8 b, K 8 = GcC 9 m 9 +Am 9 b, K 9 = GcC, m +Am b C =, C cos N =, C sin N 3 = C 4e m4, C e m 3 4 = C 5 = T C 6 e m 6 e m 5 C 8 = e m 7, C 6 = T e m 5 e m 5 T e m 5 (e m 5 m 6 e m 6 m 5) (e m 7 m8 e m 8 m7), C 9 = C e m e m 9, C = e m 3 (e m 3 m4 e m 4 m3),, C 7 = C 8e m8, e m 7 e m 9 (e m 9 m e m m9), GcC 3 m 3 +Am 3 M 37

20 C = T 4 C e m e m, C = T 3e m e m T 4 e m, e m m e m m τ = m 5 C 5 e m 5 + m 6 C 6 e m 6 + NK sinn + NK 3 cosn + m 3 K 4 e m 3 + m 4 K 5 e m 4 + ε(m C e m + m C e m + m 7 K 6 e m 7 + m 8 K 7 e m 8 + m 9 K 8 e m 9 + m K 9 e m )e iωt, τ = m 5 C 5 e m 5 + m 6 C 6 e m 6 NK sinn + NK 3 cosn + m 3 K 4 e m 3 + m 4 K 5 e m 4 + ε(m C e m + m C e m + m 7 K 6 e m 7 + m 8 K 7 e m 8 + m 9 K 8 e m 9 + m K 9 e m )e iωt, Nu = NC sinn + NC cosn + ε(m 7 C 7 e m 7 + m 8 C 8 e m 8)e iωt, Nu = NC sinn + NC cosn + ε(m 7 C 7 e m 7 + m 8 C 8 e m 8)e iωt, Sh = m 3 C 3 e m 3 + m 4 C 4 e m 4 + ε(m 9 C 9 e m 9 + m C e m )e iωt, Sh = m 3 C 3 e m 3 + m 4 C 4 e m 4 + ε(m 9 C 9 e m 9 + m C e m )e iωt 37

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