EFFECT OF CHEMICAL REACTION ON UNSTEADY MHD FREE CONVECTIVE TWO IMMISCIBLE FLUIDS FLOW

Transcription

1 Science World Journal Vol (No 4) 07 ISSN EFFECT OF CHEMICAL REACTION ON NSTEADY MHD FREE CONVECTIVE TWO IMMISCIBLE FLIDS FLOW Mubarak M., Agaie B.G., Joseph K. M. *, Daniel S. and Ayuba P. Full Length Research Article Department of Mathematical Sciences, Kaduna State niversity Nigeria. * Corresponding Author s address: kpop.moses@kasu.edu.ng ABSTRACT The effect of chemical reaction on unsteady MHD free convective two immiscible fluids flow has been studied. Approximate analytical solutions to the governing equations are found for the coupled and linear differential equations using regular perturbation method. Graphs depicting the effect of chemical reaction parameter K c and others flow parameters on velocity, temperature and concentration profiles are obtained and discussed accordingly. The effect of flow parameters on the coefficient of skin friction, Nusselt number and Sherwood number are also tabulated and discussed appropriately. It was observed that the increase in chemical reaction coefficient/parameter K c suppresses both velocity and concentration profiles. Keywords: Chemical Reaction, MHD, Convective, Immiscible, nsteady INTRODCTION Chemical Reaction is a process that involves rearrangement of the molecular or ionic structure of a substance, as distinct from a change in physical form or a nuclear reaction. There are two types of such reactions namely homogeneous reaction which occurs uniformly throughout a given phase of a flow and heterogeneous reaction which takes place in a particular region or within the boundary of a phase, (mavathi, 04). Satya et al (05) said, The study of heat transfer with chemical reaction is of most realistic significance to engineers and scientists because of its universal incidence in many branches of science and engineering. This phenomenon plays a significant role in chemical industry, power and cooling industry for dying, evaporation, energy transfer in a cooling tower and the flow in a desert cooler, etc. According to umavathi (04), the ever increasing industrial application of combined heat and mass transfer of fluid flow with chemical reaction, such as polymer production and manufacturing of ceramics among others has vested a great deal of importance on this area. Satya et al (05) investigated the effect of chemical reaction and heat source on MHD oscillatory flow in an irregular channel. However the flow is through an irregular channel and zero convection was assumed. He obtained among other results, a decrease in concentration and velocity profiles with increase in chemical reaction, Sherwood number increase with increasing chemical reaction parameter. This shows that the motion of the fluid have negative acceleration due to chemical reaction. Krishnamurthy et al (05) studied the effect of chemical reaction on MHD boundary layer flow & melting heat transfer of Williamson nanofluid in porous medium. He found among other results that an increase in the value of chemical reaction parameter decreases the concentration of species in the boundary layer, whereas the velocity and temperature of the fluid are not affected with the rise of chemical reaction parameter. This is due to the fact that chemical reaction in the study leads to consumption of chemical and thus results to decrease in the concentration profile. The flow is in two dimensions. Srinivasacharya and Reddy (04) studied chemical reaction and radiation effects on mixed convection heat and mass transfer over a vertical plate in power-law fluid saturated porous medium. The fluid is not MHD and the flow is two dimensional. The following result among others were found. The velocity and concentration of the fluid increases and suppresses respectively with increase in the value of the chemical reaction parameter. It is worthwhile to note that all the literatures above are limited to single phase flow and some considered non MHD fluid, but problems in the petroleum sector, magneto fluid dynamics etc are multiphase and MHD flow is a good way to transport fluids that are weak conductors in a microscale system (Dragisa et al, 0). Prathap-Kumar et al (04) investigated chemical reaction effect on mixed convection flow of two immiscible viscous fluids in a vertical channel. The governing equations were solved both numerically by finite difference method and analytically by perturbation. It was found among other results that the Grashof number for mass and heat transfer affect the flow in the two regions with and without chemical reaction, the flow was suppressed by the first order reaction in both regions and viscous dissipation, viscosity ratio, width ratio, conductivity ratio enhance the flow. The two methods used were also found to agree very well for small value of perturbation parameter. It is important to state that although the flow is two phase as well the fluids under consideration are non MHD, viscous dissipation was considered and transport properties of the fluids were assumed to be constant. mavathi et al (00) studied unsteady flow and heat transfer of porous media sandwiched between viscous fluids through a horizontal channel with isothermal wall temperature. Brinkman equation was used to model the flow in the porous region. The governing equations in the three regions were transformed to ordinary differential equation by perturbation where the period and non - periodic terms where collected separately. Boundary conditions were applied to the resulting ordinary differential equations for both region and a closed form solutions were obtained. The effect of physical parameters such as prandtl number, viscosity ratio, conductivity ratio, etc on the flow were computed numerically and presented graphically. It was found among other results that both the, Prandtl number, viscosity ratio & porous parameter have a negative effect on velocity and temperature profiles in the porous region as well as in the clear regions. The flow under consideration is three phase & non MHD and mass transfer is not considered in the study. In line to the above literatures, this paper focuses on the effect of chemical reaction on unsteady MHD free convective two immiscible fluids flow. 63

2 Science World Journal Vol (No 4) 07 ISSN Formulation of Problem We consider two immiscible fluids flow in a horizontal channel with the assumptions that the upper channel is porous and the lower non porous. The fluid is bounded by two infinite horizontal parallel plates X and Z. There are two regions y[0, h] and y[ h, 0] depicted as Region I and Region II on the geometry as depicted below (mavathi et al (00)) Chemical Reaction is a process that involves rearrangement of the molecular or ionic structure of a substance, as distinct from a change in physical form or a nuclear reaction. There are two types of such reactions namely homogeneous reaction which occurs uniformly throughout a given phase of a flow and heterogeneous reaction which takes place in a particular region or within the boundary of a phase, (mavathi, 04). Satya et al (05) said, The study of heat transfer with chemical reaction is of most realistic significance to engineers and scientists because of its universal incidence in many branches of science and engineering. This phenomenon plays a significant role in chemical industry, power and cooling industry for dying, evaporation, energy transfer in a cooling tower and the flow in a desert cooler, etc. According to umavathi (04), the ever increasing industrial application of combined heat and mass transfer of fluid flow with chemical reaction, such as polymer production and manufacturing of ceramics among others has vested a great deal of importance on this area. Satya et al (05) investigated the effect of chemical reaction and heat source on MHD oscillatory flow in an irregular channel. However the flow is through an irregular channel and zero convection was assumed. He obtained among other results, a decrease in concentration and velocity profiles with increase in chemical reaction, Sherwood number increase with increasing chemical reaction parameter. This shows that the motion of the fluid have negative acceleration due to chemical reaction. Krishnamurthy et al (05) studied the effect of chemical reaction on MHD boundary layer flow & melting heat transfer of Williamson nanofluid in porous medium. He found among other results that an increase in the value of chemical reaction parameter decreases the concentration of species in the boundary layer, whereas the velocity and temperature of the fluid are not affected with the rise of chemical reaction parameter. This is due to the fact that chemical reaction in the study leads to consumption of chemical and thus results to decrease in the concentration profile. The flow is in two dimensions. Srinivasacharya and Reddy (04) studied chemical reaction and radiation effects on mixed convection heat and mass transfer over a vertical plate in power-law fluid saturated porous medium. The fluid is not MHD and the flow is two dimensional. The following result among others were found. The velocity and concentration of the fluid increases and suppresses respectively with increase in the value of the chemical reaction parameter. It is worthwhile to note that all the literatures above are limited to single phase flow and some considered non MHD fluid, but problems in the petroleum sector, magneto fluid dynamics etc are multiphase and MHD flow is a good way to transport fluids that are weak conductors in a microscale system (Dragisa et al, 0). Prathap-Kumar et al (04) investigated chemical reaction effect on mixed convection flow of two immiscible viscous fluids in a vertical channel. The governing equations were solved both numerically by finite difference method and analytically by perturbation. It was found among other results that the Grashof number for mass and heat transfer affect the flow in the two regions with and without chemical reaction, the flow was suppressed by the first order reaction in both regions and viscous dissipation, viscosity ratio, width ratio, conductivity ratio enhance the flow. The two methods used were also found to agree very well for small value of perturbation parameter. It is important to state that although the flow is two phase as well the fluids under consideration are non MHD, viscous dissipation was considered and transport properties of the fluids were assumed to be constant. mavathi et al (00) studied unsteady flow and heat transfer of porous media sandwiched between viscous fluids through a horizontal channel with isothermal wall temperature. Brinkman equation was used to model the flow in the porous region. The governing equations in the three regions were transformed to ordinary differential equation by perturbation where the period and non - periodic terms where collected separately. Boundary conditions were applied to the resulting ordinary differential equations for both region and a closed form solutions were obtained. The effect of physical parameters such as prandtl number, viscosity ratio, conductivity ratio, etc on the flow were computed numerically and presented graphically. It was found among other results that both the, Prandtl number, viscosity ratio & porous parameter have a negative effect on velocity and temperature profiles in the porous region as well as in the clear regions. The flow under consideration is three phase & non MHD and mass transfer is not considered in the study. In line to the above literatures, this paper focuses on the effect of chemical reaction on unsteady MHD free convective two immiscible fluids flow. Formulation of Problem We consider two immiscible fluids flow in a horizontal channel with the assumptions that the upper channel is porous and the lower non porous. The fluid is bounded by two infinite horizontal parallel plates X and Z. There are two regions y[0, h] and y[ h, 0] depicted as Region I and Region II on the geometry as depicted below (mavathi et al (00)). Figure : Geometric configuration of the system The concentration and temperature at the walls are Cw, Cw and Tw and Tw. The fluid in Region I is having density ρ, dynamic viscosity µ, thermal conductivity k, thermal diffusivity D, while Region II is having density ρ, dynamic viscosity µ, thermal conductivity k, thermal diffusivity D. The heat transfer is due to the difference in temperature of the plates where Tw is greater than Tw and the density decreases in the direction of gravitational force. It is also assumed that the flow is fully developed and all variables are functions of y and t only since the binding surface is infinite. The magnetic field is assumed very small. Therefore, the governing equations for the flow are given as follows: 64

3 Science World Journal Vol (No 4) 07 ISSN Region I (Porous Region) v 0 () u u P u ( ) t x / V B 0u u K g ( T T ) g ( C C ) () f w c w T T T q ( r c p V ) k (3) t C C C t V D K C Cw ( ) (4) Region II (Clear Region) v 0 (5) u u P u ( ) t x V B 0u g ( T T ) g ( C C ) (6) f w c w T T T q ( r c p V ) k (7) t C C C t V D K C Cw ( ) (8) Assuming the boundary and interface conditions on velocity are no slip. Thus, the boundary and interface conditions on velocity for both fluids are: ( h) 0 ( h) 0 (0) (0) (9) at y 0 The boundary and interface conditions on temperature for both fluids are: T ( h) T w ( h) T w T (0) T (0) (0) T T k k at y 0 The boundary and interface conditions on temperature for both fluids are: C ( h) C w C ( h) C w C (0) C (0) () T T D D at y 0 vary with y, they are function of time only. Thus, assuming V = V = V, we can write the cross velocity as: V = V 0 ( + εae iwt ). Where ω is the frequency parameter, ε is a small positive constant and A is a real positive constant such that εa<<. It is assumed that the transpiration velocity varies periodically with time about a non-zero constant mean (Sturat, 955). When εa=0, the constant transpiration is recovered. Applying the following dimensionless quantities: i y t υ h v -h P Ti -Tw i=, y=, t=, V= V =, P ( ), θ i=, u h h υ v μ u x T -T 0 w w μ D μc ρ k β β α =, γ=, Pr=, τ =, β =, η =, φ =, p f c μ D k ρ k βf βc h υ σh B C -C 4I h K =, Sc=, M =,C =, F=, 0 i w i K D μ Cw -Cw k q r =4 ρ gh β (T -T ) ρ gh β (C -C ) (T -T )I, Gr=, Gc=, f w w c w w i w μu μu ρ K h K h ξ= =, K =, K = ρ τ D D c c To transform the governing equations to dimensionless form REGION I Dividing equation () by ρ we get u u u P Bu 0 u / K ( V ) t x g ( T T ) g ( C C ) () f w c w th P Pu y yh, T ( Tw Tw ) Tw, t,, u x h Introducing the above dimensionless quantities into equation (), we get it ( e ) P ( M K ) Gr GcC (3) t Similarly, by inserting the appropriate dimensionless quantities into equations (3) and (4), we get it F ( e ) (4) t P P it c e c c r r C C C K C ( ) (5) t S S REGION II In similar manner as in Region I by inserting the appropriate dimensionless quantities into equation (6), (7) and (8) we get it ( e ) P M Grm GcC (6) t F ( ) (7) t P P it e r r From equation () and (5), it is clear that the V and V do not 65

4 Science World Journal Vol (No 4) 07 ISSN The boundary and interface conditions for velocity are: C it C C Kc C ( e ) (8) t S c S c () 0 ( ) 0 (0) (0) (9) C at y 0 The boundary and interface conditions for Temperature are: () 0 ( ) (0) (0) (0) at y 0 The boundary and interface conditions for Concentration are: C () 0 C ( ) C (0) C (0) () C C at y 0 Solution of the Problem To solve equation () to (8), we expanding the solutions in the perturbation term ε to separate the periodic and non periodic parts. Assuming ε<<. it ( y, t ) ( y ) e ( y ) 0 ( y, t ) ( y ) e ( y ) it 0 C ( y, t ) C ( y ) e C ( y ) it 0 ( y, t ) ( y ) e ( y ) it 0 ( y, t ) ( y ) e ( y ) it 0 C ( y, t ) C ( y ) e C ( y ) it 0 We get: Region I The period and non-periodic parts for the velocity: 0 : ( M K ) P G G C () r 0 c 0 : ( M K i) 0 Gr GcC (3) The period and non-periodic parts for the Temperature: 0 : Pr F 0 (4) i F 0 : Pr ( Pr ) Pr (5) The period and non-periodic parts for the Concentration: 0 : C ScC K C 0 (6) 0 0 c 0 C ScC isc Kc C ScC0 : ( ) (7) Region II The period and non-periodic parts for the velocity: P Grm0 GcC 0 : 0 (8) M i 0 Grm GcC : ( ) (9) The period and non periodic parts for the temperature: 0 Pr0 F0 : 0 0 (30) Pr ipr F Pr0 : ( ) (3) The period and non periodic parts for the concentration: 0 ScC 0 : C 0 K c C 0 0 (3) ScC isc ScC 0 : C ( K c ) C (33) The corresponding boundary and interface conditions become: Region I: Non Periodic Part () ( ) 0 (0) (0) (34) C 0 0 () 0 ( ) at y 0 (0) (0) (35) 0 0 C () 0 C ( ) at y 0 C (0) C (0) (36) C C Periodic Part: () 0 ( ) 0 at y 0 (0) (0) (37) C () 0 ( ) at y 0 (0) (0) (38) at y 0 66

5 Science World Journal Vol (No 4) 07 ISSN C () 0 C ( ) C (0) C (0) (39) C C at y 0 Consider equation (4) Pr F m Pr m F 0 Pr Pr 4F Pr Pr 4F,& m m Thus ( ) m y y C e C e (46) 0 0 iwt Nu ( L ) e y y y y c m e +c m e e (c m e +c m e +k m e +k m e ) m m iwt m m m m Sherwood number: The ratio of convective to diffusive mass transfer at the upper and lower plates is given as follows: C0 iwt C Sh( ) e y y c m e +c m e e (c m e +c m e +k m e +k m e ) m 3 m 4 iwt m 9 m0 m 3 m C 0 iwt C Sh( L ) e y y y y c m e +c m e e (c m e +c m e +k m e +k m e ) m m iwt m m m m Consider (6) DATA PRESENTATION AND DISCSSION OF RESLTS C0 ScC0 K cc0 0 Numerical evaluation of the analytical solutions obtained by regular perturbation method has been carried out where epsilon is m Scm K c 0 taking as the perturbation parameter, and the graphical Sc Sc 4Kc Sc Sc 4Kc representation of the result obtained has been computed as m3, & m4 depicted on figure 7. This is to explore the important features of the governing parameters in the velocity profile, temperature, profile, and concentration profile. Throughout the computation, we my 3 my 4 C0 C3e C 4 e (47) employ constant values to the following governing parameters Gr=5, Gc=5, Pr=, α=, β=, γ=, ξ=, η=, Sc=0.78, F=3, K=, Consider (3) M=, P=, Kc=0, except the varying ones in the respective 0 0 ( M K ) 0 P G r0 GcC0 figures. m3y m m V 3 P Gr ( C m y m y e C e ) Gc ( C m 3e C 4e 4y ) In section 4., graphs and tables depicting relationships between m5y m6y various flow parameters and the flow are presented and 0 C 5e C 6e (48) Complementary discussed. Table demonstrates the effect of flow parameters in Sherwood For particular solution we assume number. It is observed that increase in Schmidt number leads to a my my my 3 m4y 0 K Particular K e K 3e K 4e K 5e decrease in Sherwood number at the upper plate and a decrease Differentiating and substituting into the original equation to solve at the lower plate. An opposite behavior is observed due to for K s, we get the following general solution increase in Chemical reaction parameter, while increase in m5y m6y my my m3y m4y 0 C 5e C 6e K K e K 3e K 4e K 5 e (49) conductivity ratio leads to an increase in Sherwood number at both plates. Consider (5) Table depicts the effect of flow parameters on Nusselt number Pr ( i Pr F) Pr at both plates. Increase in Prandtl number and conductivity ratio 0 all cause decrease in Nusselt number at the upper plate with m m PrV Pr 0 (50) opposite behavior noticed at the lower plate. The Radiation parameter (F) has no effect on the Nusselt number at both plates. Substituting equation (46) into (50) and solving the homogeneous Table 3 shows the effect of flow parameter on the Coefficient of part skin friction. Increase in Grashof number for heat and mass m7y m8y C 7e C 8 e (5) transfer, Prandtl number, and Schmidt number all yield a Complementary decrease in the coefficient of skin friction at the upper plate and a corresponding increase at the lower plate. It is also observed that Now let increase in Radiation Parameter, Hartman number, chemical my my K 6e K 7 e (5) Particular reaction parameter and viscosity ratio results in an increase in the coefficient of skin friction at the upper plate and a corresponding Differentiating and substituting into the original equation to solve decrease at the lower plate, and increase in porosity parameter for unknown expressions, we get the following general solution increase the coefficient of skin friction at the upper plate only, m7y m8y my my C 7e C 8e K 6e K 7 e (5) while increase in conductivity ratio and diffusivity ratio increase both and decrease both the upper and lower plates respectively. The effect of first order chemical reaction on concentration and Nusselt number: The rates of heat transfer in the upper and lower velocity profile is depicted on figure and 3 respectively. These plates are given as follows: figures show that an increase in the chemical reaction parameter 0 iwt 0 Nu ( ) e (Kc) leads to a decrease in both the concentration and velocity y y profiles. This result is similar to what Prathap, mavathi and m m iwt m 7 m 8 m m cme +cme e (c7m7e +c8m8e +k 6me +k 7me ) Shreedevi obtained for mixed convective flow of two immiscible viscous fluids in a vertical channel. 67

6 Science World Journal Vol (No 4) 07 ISSN Ideally, taking the coefficient of chemical reaction to zero, the effect of all other flow parameter should be the same as in the original work. However, the porosity term in the original work was taking to Region II instead of Region I and although velocity no slip is assumed, the boundary conditions for temperature and concentration was as well interchange, these problems translated to the result originally obtained. The effect of Grashof number for heat, and mass transfer on the velocity profiles are depicted on the figure 4 and 5 respectively. In both figures it is observed that as Grashof numbers increase, which literally means increase in buoyancy force over viscous force, the velocity increases in both regions with almost equal strength. However, the Grashof number for mass transfer increases the velocity more than the Grashof number for mass transfer. Figure 6 and 7 depict the effect of Prandtl number (Pr) and Porosity Parameter (K) on the velocity profile. It is observed in figure 3 that as molecular diffusion on momentum over molecular diffusion of heat increases, the velocity increases in both regions. It is also observed from figure 4 that the velocity tends to its maximum as K tends to zero. This implies that an increase in K leads to a decrease in velocity in Region I (porous region) & also in Region II. However, the velocity drag in Region I is very large for large value of K. A similar result was observed by mavathi, Liu, Prathap and Shaik-Meera for porous media sandwiched between viscous fluids. Figure 8, 9, and 0 shows the effect of viscosity ratio (α), Radiation parameter (F), and Hartman number (M) on the velocity profile. It is found on figure 8 that as viscosity ratio increases the velocity decreases. Figure 9 shows that increase in the radiation parameter results to a decrease in the velocity profile in both regions. Increase in the Hartman number also results to a decrease in the velocity profile in both regions as depicted on figure 0. As for Hartman number, similar result was obtained by Muteen for two phase MHD fluid flow and Krishnamurthy et al for MHD boundary layer flow. Figure and depict the effect of Schmidt number (Sc) and Diffusivity ratio (γ) on the velocity profile. It is observed on figure that domination of viscous diffusion rate over molecular diffusion rate leads to an increase in the flow velocity profile in both regions. A similar behavior is observed on figure as increase in diffusivity ratio leads to increate in velocity profile. Figure 3, 4 and 5 portray the effect Prandtl number (Pr), Radiation parameter (F) and Conductivity ratio on the temperature profile. It is observed that increase in both Prandtl number and conductivity ratio lead to an increase in the temperature profile. However, increase in the Radiation parameter cause decrease in the temperature profile as shown on figure 4. Figure 6 and 7 demonstrate the effect of Schmidt number (Sc) and Diffusivity ration (γ) on the concentration profile. It is observed on figure 6 that as viscous diffusion rate dominates the molecular diffusion rate, the concentration profile increase in both regions. A similar result is observed on figure 7 where increase in Diffusivity ratio causes an increase in the concentration profile in both regions. When the chemical parameter K c = 0 and ratio of diffusivity γ = 0, then our results will be in total agreement to the results of Joseph et al (05) Table : Effect of Flow Parameters on Sherwood Number ε Sc L Kc Sh() Sh(L) Table : Effect of Flow Parameters on Nusselt Number E Pr F B Nu(L) Nu() Table 3: Effect of Flow Parameters on Coefficient of Skin Friction Figure : Effect of Chemical Reaction Parameter (Kc) on Concentration Profile Figure 3: Effect of Chemical Reaction Parameter (Kc) on Velocity Profile 68

7 Science World Journal Vol (No 4) 07 ISSN Figure 4: Effect of Grashof Number (Gr) for Heat Transfer on Velocity Field Figure 8: Effect of Viscosity Ratio on Velocity Profile Figure 5: Effect of Grashof Number (Gc) for Mass Transfer on Velocity Field Figure 9: Effect of Radiation Parameter (F) on Velocity Profile Figure 6: Effect of Prandtl Number (Pr) on Velocity Field Figure 0: Effect of Hartman Number (M) on Velocity Profile Figure 7: Effect of Porosity Parameter on Velocity Field Figure : Effect of Schmidt Number on velocity Profile 69

8 Science World Journal Vol (No 4) 07 ISSN Figure : Effect of Diffusivity Ratio of Velocity Profile Figure 6: Effect of Schmidt Number (Sc) on Concentration Profile Figure 3: Effect of Prandtl Number (Pr) on Temperature Profile Figure 4: Effect of Radiation Parameter (F) on Temperature Profile Figure 7: Effect of Diffusivity Ratio on Concentration Profile Summary and Conclusion In this paper, the effect of first order chemical reaction on unsteady MHD free convective two immiscible fluid flows with heat and mass transfer has been studied. The governing equations of the flow were converted to dimensionless form using appropriate non dimensional parameters. The resulting partial differential equations were transformed to ordinary differential equation by regular perturbation method. The velocity, concentration and temperature fields were obtained from the resulting ODEs. Graphs depicting the effect of chemical reaction and other flow parameters on the velocity, temperature and concentration profiles were obtained. The effect of flow parameters on coefficient of skin friction, Nusselt number and Sherwood number were also studied and tabulated appropriately. It was concluded that the increase in chemical reaction coefficient/parameter K c suppresses both velocity and concentration profiles. Figure 5: Effect of Conductivity Ratio on Temperature Profile Nomenclature T w, T w Wall Temperature C w, C w Wall Concentration, V Velocity Components t Time υ Kinematic Viscosity P Pressure K Permeability of Porous Medium α Viscosity Ratio B o Coefficient of Electromagnetic Field M Hartman Number θ Dimensionless Temperature F Radiation Parameter 70

9 Science World Journal Vol (No 4) 07 ISSN Pr C p G r G c Sc γ A g ε ω β Prandtl Number Specific Heat at Constant Pressure Grashof Number for Heat Transfer Grashof Number for Mass Transfer Schmidt Number Ratio of Diffusivity Positive Real Number Acceleration due to gravity Coefficient of periodic parameter Frequency Parameter Thermal Conductivity V = F + iωpr V = iωsc + KC V 3 = M + K V 30 = M + K + iω V 4 = Pr (B E ) V 5 = F V B 6 = Sc V L 60 = (Kc ) V 7 = (a E ) (FE V +Priω) 9 V (B E ) 0 = (Sciω) V L+Kc = (E M +iω) (a E ) m = Pr+ Pr +4F m = Pr Pr +4F m 4 = Sc Pr +4Kc m 6 +4V 3 m 9 = Sc+ Pr +4V m = + +4V 30 m 4 = V 4 V 4 +4V V m 3 5 m 7 = Pr+ Pr +4V m 0 = Sc Pr +4V V 8 = M a m 3 = Sc+ Pr +4Kc m 8 = Pr Pr +4V m = +4V 30 m 3 = V 4+ V 4 +4V 5 m 5 = V 6+ V 6 +4V 60 m 6 = V 6 V 6 +4V 60 m 7 = V 7 + V 7 +4V 8 m 9 = V 4+ V 4 +4V 9 V 6 + V 6 +4V 0 m 8 = V 7 V 7 +4V 8 m 0 = V 4 V 4 +4V 9 m = m = V 6 V 4 +4V 0 m 3 = V 7+ V 7 +4V m 3 = V 7 V 7 +4V r = e m 4 r = e (m 4 m 3 ) r 3 = e (m m ) r 4 = B m 4 e m 4 r 5 = m m e m m r 6 = B m 4 e (m 4 m 3 ) (B m 3 ) r 7 = e m 6 r 8 = e m 3 m 4 r 9 = e (m 6 m 5 ) r 0 = Lm 6 e m 6 r = m 3 m 4 e m 3 m 4 r = Lm 6 e (M 6 M 5 ) LM 5 r 3 = e (m 8 m 7 ) r 4 = e m 5 m 6 r 5 = m 5 m 6 e m 5 m 6 r 6 = a m 8 e (m 8 m 7 ) a m 7 r 7 = e (m 0 m 9 ) r 8 = e m 7 m 8 r 9 = m 7 m 8 e m 7 m 8 r 0 = B m 0 e (m 0 m 9 ) B m 9 r = e m 9 m 0 r = e (m m ) r 3 = m 9 m 0 e m 9 m 0 r 4 = Lm e (M M ) LM r 5 = e (m 4 m 3 ) r 6 = e m m r 7 = a m 4 e (m 4 m 3 ) a m 3 r 8 = m m e m m c 3 = r 3r 4 r r 5 c r 3 r 6 r 3 r = r c 3 r 5 r 3 c = c e (m m ) c 3 = r 9r 0 r 7 r c r 9 r r 8 r 4 = c 3 e m 3 m 4 c 4 = e m 4 c 5 e m 4 m 3 c 5 = r 7 c 3 r 8 r 9 c 6 = e m 6 c 5 e m 6 m 5 K = P V 3 REFERENCES Daniel, S., Tella, Y. and Joseph, K.M. (04), Slip Effect on MHD Oscillatory Flow of Fluid in a Porous Medium with Heat and Mass Transfer and Chemical Reaction. Asian Journal of Science and Technology. 5 (3): Joseph K.M, Peter A., Peter E.A, A, sman S. (05), nsteady MHD Free Convective Two Immiscible Fluid Flow with Heat and Mass Transfer. International Journal of Mathematics and Computer Research, 3(5): Krishnamurthy M. R., Prasannakumara B. C., Gireesha B. J and Gorla R. S. R. (05), Effect of chemical reaction on MHD boundary layer flow and melting heat transfer of Williamson nanofluid in porous medium. Engineering Science and Technology, an International Journal, -8. Kumar J. P., mavathi J.C and Kalyan S, (04), Chemical Reaction Effect on Mixed Convection Flow of Two Immiscible Viscous Fluids in a Vertical Channel. Open Journal of Heat and Momentum Transfer, (): Satya P.V., Venkateswarlu B., Devika B. (05), Chemical Reaction and Heat Source Effect on MHD Oscillatory Flow in an Irregular Channel. Ain Shams Engineering Journal. 0. Sharma P.R., Sharma K. (04), nsteady MHD Two- and Heat Transfer Through a Horizontal Channel. International Journal of Engineering Science Invention Research & Development. (3): Srinivasacharya D. and Reddy G. S. (05), Chemical reaction and radiation effects on mixed convection heat and mass transfer over a vertical plate in power-law fluid saturated porous medium. Journal of the Egyptian Mathematical Society. -8. mavathi J.C., Liu I.C, Kumar J.P., and Meera S, (00), nsteady Flow and Heat Transfer of Porous Media Sandwiched between viscous fluids. Applied Mathematics and Mechanics. 3 (): mavathi, J. C., Chamkha, A. J., Mateen, A., & Kumar J. P. (008), nsteady Magnetohydrodynamic Two and heat transfer in a horizontal channel. Heat and Technology Journal. 6 (): 3. 7