Magnetohydrodynamic (MHD) Plane Poiseuille Flow With Variable Viscosity and Unequal Wall Temperatures

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1 Iranian Journal of Cheical Engineering Vol. 11, No. 1 (Winter), 014, IAChE Resea rch note Magnetohydrodynaic (MHD) Plane Poiseuille Flow With Variable Viscosity and Unequal Wall Teperatures A. Kuar Jhankal 1, M. Kuar 1- Birla Institute of Technology, Extension Center-Jaipur, Jaipur (INDIA) - Arya College of Engineering & Research Center, Kukas, Jaipur (INDIA) Abstract The plane Poiseuille flow with unequal wall teperatures of an incopressible fluid having teperature dependent viscosity in the presence of transverse agnetic field is studied. The coupled differential equations of oentu and energy are solved nuerically by RKF45 (Runge-Kutta-Fehlberg fourth-fifth) ethod. Nuerical results for the diensionless velocity profiles, the teperature profiles and the heat transfer coefficient are presented and discussed graphically for various paraeters. The study provides quantitative inforation of interest; in general, it is observed that the axiu velocity does not occur in the iddle of the channel but oves towards the upper wall as the agnetic field increases. The teperature and heat transfer coefficient increases with the increasing value of agnetic field and EPr. Keywords: Poiseuiller Flow, Variable Viscosity, Unequal Wall Teperature, Nuerical Study 1. Introduction The agnetohydrodynaic (MHD) flow between two parallel walls is a classical proble whose solution has any applications in agnetohydrodynaic power generators, agnetohydrodynaic pups, accelerators, aerodynaic heating, electrostatic precipitation, polyer technology, petroleu industry, purification of oil and fluid droplets and sprays, etc. The subject of agnetohydrodynaic is largely perceived to have been initiated by Swedish electrical engineer Hannes Alfvén [1] in 194. If an electrically conducting fluid is placed in a constant agnetic field, the otion of the fluid induces currents which create forces on the fluid. The production of these currents has led to the design of, aong other devices, the agnetohydrodynaic generators for electricity production. The equations which describe agnetohydrodynaic flow are a cobination of continuity equation and Navier-Stokes equations of fluid dynaics and Maxwell s equations of electroagnetis. The governing equations are differential equations that have to be solved either analytically or nuerically. Nahe [] considered the plane Couette flow for fluid having teperature dependent Corresponding author: anujjhankal@yahoo.co 63

2 Magnetohydrodynaic (MHD) Plane Poiseuille Flow With Variable Viscosity and Unequal Wall Teperatures viscosity, Hausenblas [3] considered the plane Poiseuille flow taking a siplified for of the viscosity and teperature relation keeping both walls at the sae teperature, the sae proble was considered by Bansal and Jain [4] taking the walls at unequal teperature. Shercliff [5] studied the steady otion of an electrically conducting fluid in pipes under transverse agnetic fields. Drake [6] considered flow in a channel due to periodic pressure gradient and solved the resulting equation by separation of variables ethod. Singh & Ra [7] considered lainar flow of an electrically conducting fluid through a channel in the presence of a transverse agnetic field under the influence of a periodic pressure gradient and solved the resulting differential equation by the ethod of Laplace transfor. Ra et al. [8] have analyzed hall effects on heat and ass transfer flow through porous edia. Shioura [9] discussed agnetohydrodynaics turbulent channel flow under a unifor transverse agnetic field. Singh [10] considered steady agnetohydrodynaic fluid flow between two parallel plates. Kazuyuki [11] discussed inertia effects in two diensional agnetohydrodynaic channel flow and Al-Hadhrai [1] considered flow of fluids through horizontal channels of porous aterials and obtained velocity expressions in ters of the Reynolds nuber. Ganesh [13] studied unsteady agnetohydrodynaic Stokes flow of a viscous fluid between two parallel porous plates. They considered fluid being withdrawn through both walls of the channel at the sae rate. In this paper we consider the proble of plane Poiseuille flow with unequal wall teperatures of an incopressible electrically conducting fluid under the influence of transverse agnetic field. The boundary layer equations governing the otion are studied by Bansal and Jain [4], in the absence of agnetic field. In the present study, the influence of transverse agnetic field on plane Poisenille flow with unequal wall teperature of an incopressible electrically conducting fluid is investigated. The governing boundary layer equation of oentu and energy are solved nuerically using Runge-Kutta-Fehlberg Fourth-Fifth order ethod and the effect of various paraeters is presented and discussed.. Forulation of the proble Consider a steady two-diensional lainar electrically conducting flow with variable viscosity between two parallel walls, taking x-axis along the central line of the channel and y-axis is perpendicular to it. A agnetic field of unifor strength B o is applied in the y-direction, which produces the agnetic field in the x-direction. We consider the case of a short circuit proble in which the applied electric field is E = 0, and also assue that the induced agnetic field is sall copared to the external applied agnetic field B o. This iplies a sall agnetic Reynolds nuber. The boundary layer equations governing the otion are (Bansal and Jain [4]): d μ du dp σeb 0u =. dy dy dx dt k dy du + μ = 0 dy (1) () 64 Iranian Journal of Cheical Engineering, Vol. 11, No. 1

3 Kuar Jhankal, Kuar Along with the boundary conditions are: y = h:u = 0,T = T 1 y = h:u = 0,T = T (WhereT > T ) (3) Where h is the distance between the walls, and the otion is due to the constant pressure gradient dp dx along the axis of the channel. Let us introduce the following nondiensional quantities: u y μ T T u =, η =, μ =,T = u h μ T T Then equations (1) and () take the fors: d du μ Mu = dη dη dt du + μepr = 0 dη dη (4) (5) (6) And the corresponding boundary conditions reduce to: η = 1:u = 0,T = 1 η = 1:u = 0,T = 0 (7) Where Pr = μ0c p / k (Prandtl nuber), and E = u / C (T T ) (Eckert nuber) where p 1 0 h dp u ( = ) is the axiu μ dx 0 velocity in the iddle of the channel in the plane Poiseuille flow with constant fluid properties (Schilichting [14]), μ 0 is the viscosity of the fluid at teperature T 0 and M = σeb0h / μ0 is the non-diensional agnetic paraeter. 3. Nuerical solution The differential equations (5) and (6) subject to boundary condition (7) are solved taking an epirical relation between viscosity and teperature 1/ μ = 1+ αt (Hausenblas [3]) nuerically using Runge-Kutta-Fehlberg fourth-fifth order ethod. To solve these equations we adopted sybolic algebra software Maple. Maple uses the well known Runge-Kutta-Fehlberg Fourth-fifth order (RFK45) ethod to generate the nuerical solution of a boundary value proble. The effect of various paraeters on velocity profile, teperature profile and the coefficient of heat transfer in ters are shown in Figs. 1 to Results and discussion When the walls of the channel are at different teperatures, nuerical coputations are perfored for various values of the physical paraeters involved in the equations viz. the agnetic paraeter M, α (where α 1 1 = 1 ( μ ) ) and EPr (where E is the Eckert nuber and Pr is the Prandtl nuber). The calculated results are presented in Figs 1-5 to understand the effects of paraeters on the flow and teperature field. The ipacts of agnetic paraeter M on the velocity and teperature profiles are very significant fro a practical point of view. Figs. 1 and show the effects of agnetic paraeter (M) on the velocity profile for the different values of α respectively. It is observed that in the present case axiu velocity does not occur in the iddle of the channel but oves towards the upper wall as the value of M and α increases. Moreover, the rise in agnitude of the velocity is quite Iranian Journal of Cheical Engineering, Vol.11, No. 1 65

4 Magnetohydrodynaic (MHD) Plane Poiseuille Flow With Variable Viscosity and Unequal Wall Teperatures significant in the present case showing that the volue rate of flow at a section increases with increase in M and α. This happens due to the Lorentz force arising fro the interaction of agnetic and electric field during the otion of the electrically conducting fluid. in the teperature distribution leads to an iportant conclusion, that in the present case the transfer of heat at the lower wall will be ore when copared with the constant viscosity case. The variation of the teperature gradient T (0) which is significant in evaluating the rate of heat transfer is presented in Fig. 5, against agnetic paraeter M for the different values. It is observed that T (0) increases with the increasing value of M and EPr. This heat transfer is very iportant in production engineering to iprove the quality of the final product. Figure 1. Velocity profile, against the perpendicular distance fro central line, for different values of agnetic paraeter M (when α=0.5). Figure 3. Teperature profile against the perpendicular distance fro central line for different values of agnetic paraeter M (when EPr =0.7). Figure. Velocity profile, against the perpendicular distance fro central line, for different values of agnetic paraeter M (when α=0.7). On the other hand, Figs. 3 and 4 show the effects of agnetic paraeter (M) on the teperature profile for the different values of respectively. Fro the plot it is observed that teperature increases with the increasing value of M and EPr. This change Figure 4. Teperature profile against the perpendicular distance fro central line for different values of agnetic paraeter M (when EPr = 1.0). 66 Iranian Journal of Cheical Engineering, Vol. 11, No. 1

5 Kuar Jhankal, Kuar Figure 5. Coefficient of heat transfer, against agnetic paraeter for different values of EPr. 5. Conclusions A atheatical odel has been presented for the proble of plane Poiseuille flow with unequal wall teperatures of an incopressible fluid having teperature dependent viscosity in the presence of transverse agnetic field. The governing boundary layer equations were solved nuerically using the Runge-Kutta-Fehlberg ethod using Maple software. We infer fro this study that the axiu velocity does not occur in the iddle of the channel but oves towards the upper wall as the value of M and α increases. The teperature and T( 0) increases with the increasing value of M and EPr, this heat transfer is very iportant in production engineering to iprove the quality of the final product. Acknowledgent The authors wish to express their sincere thanks to the learned referee for his valuable suggestions. Noenclature B 0 Constant applied agnetic field (Wb ) 1 C p Specific heat at constant pressure 1 1 (Jkg K ) E Eckert nuber h Distance of the walls fro the iddle of the channel () M Diensionless agnetic field paraeter p Pressure (Pa) Pr Prandtl nuber T Teperature (K) u Velocity coponent in x direction (s -1 ) u Maxiu velocity (s -1 ) v Velocity coponent in y direction (s -1 ) x Diensional horizontal coordinate () y Diensional vertical coordinate () Greek sybols η Diensionless space variable σ e Current density (C - s -1 ) x Theral conductivity ( s -1 ) μ Dynaic viscosity (Pa s) Superscript - Diensional quantities Subscript 1 Condition at the upper wall 0 Condition at the lower wall References [1] Alfvén, H., "Existence of electroagnetic-hydrodynaic waves", Nature, 150(3805), 405, (194). [] Nahe, R., "Beiträge zur hydrodynaschen Theorie der Lagerreibung", Ing. Arch., 11, 191, (1940). [3] Hausenblas, H. "Die nicht isothere Ströung einer zähen Flüssigkeit durch enge Spalte und Kapillarröhren", Ing. Arch., 18, 151, (1950). [4] Bansal, J.L. and Jain, N.C., "Variable viscosity plane poiseuille flow with unequal wall teperature", Indian J. Iranian Journal of Cheical Engineering, Vol.11, No. 1 67

6 Magnetohydrodynaic (MHD) Plane Poiseuille Flow With Variable Viscosity and Unequal Wall Teperatures pure appl. Math., Vol. 6(7), 800, (1975). [5] Shercliff, J.A., "Entry of conducting and non-conducting fluids in pipes", Journal of Matheatical Proc. of the Cabridge Philosophical Soc., 5, 573, (1956). [6] Drake, D.G., "On the flow in a channel due to a periodic pressure gradient", Quart. J. of Mech. and Appl. Maths., 18(1), 1, (1965). [7] Singh, C.B. and Ra, P.C., "Unsteady agnetohydrodynaic fluid flow through a channel", Journal of Scientific Research., XXVIII(), (1978). [8] Ra, P.C., Singh, C.B. and Singh, U., "Hall effects on heat and ass transfer flow through porous ediu", Astrophysics Space Science, 100, 45, (1984). [9] Shioura, Y., "Large eddy siulation of agnetohydrodynaic turbulent channel flow under unifor agnetic field", Physics Fluids, A3(1), 3098, (1991). [10] Singh, C.B., "Magnetohdrodynaic steady flow of liquid between two parallel plates", In: Proc. of First Conference of Kenya Matheatical Society, 4, (1993). [11] Kazuyuki, U., "Inertia effects on two diensional agnetohydrodynaic channel flow under travelling sine wave agnetic field", Phys. Fluids A3(1), 3107, (1991). [1] Al-Hadhrai, A.K., Elliot, L., Ingha, M.D. and Wen, X., "Flow through horizontal channels of porous aterials", International Journal of Energy Research, 7, 875, (003). [13] Ganesh, S. and Krishnabal, S., "Unsteady agnetohydrodynaic Stokes flow of viscous fluid between two parallel porous plates", Journal of Applied Sciences, 7, 374, (007). [14] Schlichting, H., Boundary Layer Theory, McGraw-Hill Book Co., Inc., New York, (1968). 68 Iranian Journal of Cheical Engineering, Vol. 11, No. 1