ON VARIABLE LAMINAR CONVECTIVE FLOW PROPERTIES DUE TO A POROUS ROTATING DISK IN A MAGNETIC FIELD

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1 ON VARIABLE LAMINAR CONVECTIVE FLOW PROPERTIES DUE TO A POROUS ROTATING DISK IN A MAGNETIC FIELD EMMANUEL OSALUSI, PRECIOUS SIBANDA School of Mathematics, University of KwaZulu-Natal Private Bag X0, Scottsville 309, Pietermaritzburg, SA Received April 4, 006 The hydromagnetic flow of a steady, laminar conducting viscous fluid due to an impulsively started rotating porous disk is studied taking into account the variable fluid properties (density, ρ, viscosity, μ, and thermal conductivity, κ). These fluid properties are taken to be dependent on temperature. The system of axisymmetric nonlinear partial differential equations governing the MHD steady flow and heat transfer are written in cylindrical polar coordinates and reduced to nonlinear ordinary differential equations by introducing suitable similarity parameters. The resulting steady equations are reduced to an initial valued problem and solved numerically using a shooting method. A parametric study of all parameters involved was conducted, and a representative set of results showing the effect of the magnetic field (M), the uniform suction parameter (W < 0) and the relative temperature difference parameter (ε) on velocities, temperature, skin-friction and Nusselt number are illustrated graphically and tabularly to show typical trends of the solutions.. INTRODUCTION The problem of hydrodynamic stability of flow due to a rotating disk has been the subject of study by several investigators since the pioneering work of von Karman [4]. Research into this type of flow has been spurred on by both theoretical imperatives as well as the practical applications of such flows, for example, in industrial machinery and lately, in computer disk drives, Herrero et al. [6]. The early study by von Karman has since been considerably extended starting with the work of Cochran [4] to include, inter alia, the effects of: () impulsively starting the flow from rest (Benton [3], Roger and Lance []), () an axial magnetic field applied to the fluid without Hall effects (El-Mistikawy et al. [5]), (3) an axial magnetic field with Hall effects (Attia and Aboul-Hassan []) and (4) variable fluid properties (Maleque and Sattar []). The effects of variable properties on laminar boundary layers has been considered by, among others, Herwig [7] and Herwig and Klemp [8]. Maleque and Sattar [] have extended the consideration of the effects of variable fluid Corresponding author: sibandap@ukzn.ac.za Rom. Journ. Phys., Vol. 5, Nos. 9 0, P , Bucharest, 006

2 938 Emmanuel Osalusi, Precious Sibanda properties, namely the density, ρ, the viscosity μ and the thermal conductivity κ to flow due to a porous rotating disk. They found, among other things, that for fixed values of the suction parameter and Prandtl number, the momentum boundary layer increased considerably. Earlier work by Stuart [3] showed that the effect of suction is to thin the boundary layer by decreasing the radial and azimuthal components of the velocity while at the same time increasing the axial flow towards the disk at infinity. The recent study by Attia [] considered the effect of temperature dependent viscosity on the flow and heat transfer along a uniformly heated impulsively rotating disk in a porous medium. In this study we extend the work of Maleque and Sattar [] to include the effects of a magnetic field on flow due to a rotating disk in an electrically conducting fluid with temperature dependent density, viscosity and thermal conductivity.. GOVERNING EQUATIONS The description of the physical problem closely follows that of Maleque and Sattar []. We use a non-rotating cylindrical polar coordinate system, (r, ϕ, z) where z is the vertical axis in the cylindrical coordinates system with r and ϕ as the radial and tangential axes respectively. The homogeneous, electrically conducting fluid occupies the region z > 0 with the rotating disk placed at z = 0 and rotating with constant angular velocity Ω. The fluid velocity components are (u, v, w) in the directions of increasing (r, ϕ, z) respectively, the pressure is P, the density of the fluid is ρ and T is the fluid temperature. The surface of the rotating disk is maintained at a uniform temperature T w. Far away from the wall, the free stream is kept at a constant temperature T and at constant pressure P. The external uniform magnetic field is applied perpendicular to the surface of the disk and has a constant magnetic flux density B 0 which is assumed unchanging with a small magnetic Reynolds number (Rem ). Following Jayaraj [9] (and more recently, Maleque and Sattar []), we assume that the dependency of the fluid properties, viscosity μ and thermal conductivity κ coefficients and density ρ are functions of temperature alone and obey the following laws; a b c μ=μ [ TT / ], κ=κ [ TT / ], ρ=ρ [ TT / ], (.) where the a, b and c are arbitrary exponents, κ is a uniform thermal conductivity of heat, and μ is a uniform viscosity of the fluid. As in Maleque and Sattar [] the fluid under consideration is a flue gas with a = 0.7, b = 0.83, and c =.0. The case c =.0 is that of an ideal gas. The physical model and geometrical coordinates are shown in Fig..

3 3 On variable laminar convective flow 939 Fig.. The flow configuration and the coordinate system. The equations governing the motion of the MHD laminar flow of the homogeneous fluid take the following form ( ρ ru) + ( ρ rw) = 0, r z ( ) ( ) ( ) u u w u P u u u σ ρ ν + u r r z + = μ + μ + μ, r r r r r z z ρ v uν v v v v ( ) ( ) ( ) ( ) B o σ ρ u + + w = μ + μ + μ v, r r z r r r r z z ρ ( u w w w P w ) ( ) ( w) w ( ) ρ + + = μ + μ + μ, r z z r r r r z z T T T κ T T ( ) ( ) ( ), ρ Cp u + w = κ + + κ r z r r r r z z B o (.) (.3) (.4) (.5) (.6) where σ is the electrical conductivity and C p is the specific heat at constant pressure. The appropriate boundary conditions for the flow induced by an infinite disk (z = 0) which is started impulsively into steady rotation with constant angular velocity Ω and a uniform suction/injection W through the disk are given by u= 0, v=ω r, w= W, T = Tw, at z= 0 (.7) u 0, v 0, T T, P P as z. w

4 940 Emmanuel Osalusi, Precious Sibanda 4 3. SIMILARITY TRANSFORMATION The solutions of the governing equations are obtained by introducing a dimensionless normal distance from the disk, η= z ( Ω/ν ) / along with the von-karman transformations, u=ωrf( η, ) v=ωrg( η, ) w= ( Ων ) H( η) P P = μ Ωp( η) and T T =ΔTθ( η ), (3.8) where ν is a uniform kinematic viscosity of the fluid and Δ T = Tw T. Substituting these transformations into equations (.) (.6) gives the nonlinear ordinary differential equations, H + F+ chθ ( + εθ ) ε = 0, (3.9) F + aε( + εθ) θ F [ F G + HF + MF( + εθ ) ]( + ε θ ) c a= 0, (3.0) G + aεg ( + εθ) [ FG+ HG + MG( + εθ ) ]( + ε θ ) c a= 0, (3.) θ + bεθ ( +εθ) Pr Hθ ( + ε θ ) c b= 0, (3.) where Pr =μ Cp/ k is the Prandtl number, M =σ B 0 /ρ Ω is the magnetic interaction parameter that represents the ratio of the magnetic force to the fluid inertia and ε =Δ TT / is the relative temperature difference parameter, which is positive for a heated surface, negative for a cooled surface and zero for uniform properties. These equations differ from those in Maleque and Sattar [] by way of the additional terms that involve the magnetic parameter M. The transformed boundary conditions are given by; F = 0, G =, H = W, θ=, at η= 0 (3.3) F = G =θ= p= 0, at η, where W = w/ ν Ω represents a uniform suction (W < 0) or injection (W > 0) at the surface. The skin friction coefficients and the rate of heat transfer to the surface are given by the Newtonian formulas: and v w τ ( ) a t = μ + =μ + Re Ω G (0), z r φ ε u w ( ) z= 0 τ ( ) a r = μ + =μ + ε Re Ω F (0). z r z= 0

5 5 On variable laminar convective flow 94 and Hence the tangential and radial skin-frictions are respectively given by Fourier s law a Cf G ( + ε ) Re = (0), (3.4) a Cf F t ( + ε ) Re = (0). (3.5) T ( ) r ( ) b q= κ = κ Δ T + ε Ω θ (0), z z= 0 ν is used to calculate the rate of heat transfer from the disk surface to the fluid. The Nusselt number Nu is obtained as b ( + ε ) Re Nu = θ (0), (3.6) where Re ( =Ω r/ν ) is the rotational Reynolds number. 4. METHOD OF SOLUTION Equations (3.9) (3.) are solved numerically using a shooting method for different values of suction, W < 0 and parameters Pr, ε and M. To reduce the equations to first order equations we set F = y, G = y, H = y3, θ=y4, F = y 5, G = y6, θ= y7 to get; y = y5, y(0) = 0, y = y, y (0) =, y = y cy y ( +εy ) ε, y (0) = W, y = y, y (0) =, = ε 7 5( +ε 4) + [ ( +ε 4) ]( +ε 4) c a (5) 5(0) = s, 6 = ε 7 6( +ε 4) + [ ( +ε 4) ]( +ε 4) c a, (6) 6(0) = s, 7 = ε 7( +ε 4) + Pr 3 7( +ε 4) c b, 7(0) = (7), y ay y y y y y y My y y y y a y y y y y y y My y y y y b y y y y y y s (4.7) where s (5), s (6) and s (7) are determined such that y 5 ( ) = 0, y6( ) = 0 and y7( ) = 0. The essence of this method is to reduce the boundary value problem to an initial value problem and then use a shooting numerical technique to guess

6 94 Emmanuel Osalusi, Precious Sibanda 6 s (5), s (6) and s (7) until the boundary conditions y 5 ( ) = 0, y6( ) = 0 and y7( ) = 0 are satisfied. The resulting differential equations are then easily integrated using the initial value solvers lsode and fsolve available in GNU Octave. To establish the validity of our numerical code, a comparison of our calculated results with those of Maleque and Sattar [] and Kelson and Desseaux [0] for M = 0 is shown in Table. The three set of results compare favourable for W < 0. Table Comparison of current and recent numerical values of the radial and tangential skin -friction coefficients and the rate of heat transfer coefficient obtained for Pr = 0.7, M 0 and ε = 0. Present ( M 0) Maleque & Sattar [] Kelson & Desseaux [0] W F (0) G (0) θ (0) F (0) G (0) θ (0) F (0) G (0) θ (0) RESULTS AND DISCUSSIONS Following Maleque & Sattar [], the numerical solutions displayed in Tables 3 and Figs. 4 are relevant for a flue gas, that is, when Pr = We have confined our analysis to the case when we have suction velocity only, that is, when W < 0. Table shows the effect of increasing the magnetic field strength on the radial and tangential skin-friction coefficients and the rate of heat transfer for variable property ε = 0. and fixed suction coefficient W =. For moderate increases in the strength of the magnetic field the effect is a gradual monotonic decrease in the values of F, G and θ. Table 3 shows that cooling the surface while holding constant the magnetic field strength and the suction parameter has the effect of increasing the radial and tangential skin-friction and the rate of heat transfer coefficients. Table Numerical values of the radial and tangential skin-friction coefficients and the rate of heat transfer coefficient obtained for ε = 0., W = and Pr = 0.64 M F (0) G (0) θ (0)

7 7 On variable laminar convective flow 943 Table 3 Numerical values of the radial and tangential skin-friction coefficients and the rate of heat transfer coefficient obtained for M = 0., W = and Pr = 0.64 ε F (0) G (0) θ (0) Figs. (a) (d) show the effects of ε on the velocity (radial, tangential and axial) and temperature profiles. The primary purpose of these figures is to give a comparison between the constant property and variable property solutions when a magnetic field is present. The results are qualitatively similar to those given by Maleque and Sattar [] except that the effect of the magnetic field is depress the motion of the fluid. In Fig. (a), it is seen that due to the existence of a centrifugal force the radial velocity attains a maximum value close to the disk for all values of ε. However, in contrast to the observations in Maleque and Sattar [], where the maximum velocity is larger at the surface of the disk in the case of constant property, the effect of a moderate increase in ε is not seen to depress the fluid motion at the disk surface. For most part of the boundary layer, at any fixed position η, the radial velocity increases with the increase of the relative temperature difference parameter ε. The tangential velocity, as observed in Fig. (b) is found to increase with increasing values of ε at a fixed point of the boundary layer while in Fig. (c) that axial velocity decreases with an increase in the relative temperature differences ε. The results Fig. (d) shows that the non-dimensional temperature increases with increasing values of ε, but the rate of increase is very small and hence confirming the findings in Maleque and Sattar [] that the thermal boundary layer does not vary with ε. The effects of fluid suction ( Ws ) for ε= M = 0 and Pr = 0.64 on the radial, the tangential, the axial velocity profiles and temperature profiles are shown in Fig. 3(a) (d). Strong suction has a stabilizing effect on the axial velocity; the radial velocity attains its maximum near the surface while the tangential velocity and temperature decay rapidly away from the surface. The radial velocity reaches its maximum very close to the surface and decreases monotonically away from the boundary layer. Similar effects of W are also observed in case of the tangential velocity. It is noticed from Fig. 3(d) that the thermal boundary layer increases gradually for a decrease in suction velocity. Fig. 4 shows the effect of the magnetic field on velocity and temperature profiles. Imposition of a magnetic field generally creates a drag force that has the tendency to slow down the flow around the disk at the same time increasing fluid temperature. This is shown by the decreases in the radial, tangential and axial

8 944 Emmanuel Osalusi, Precious Sibanda 8 velocity profiles as M increases as shown in Fig. 4(a) (d). The increases in the temperature profiles as M increases are accompanied by a corresponding increase in the thermal boundary layer. (a) (b) Fig.. (a), (b).

9 9 On variable laminar convective flow 945 (c) (d) Fig.. (a) Effect of ε on the radial velocity profiles, (b) effect of ε on the tangential velocity profiles, (c) effect of ε on the axial velocity profiles (d) effect of ε on the temperature profiles: for M = 0, W =, Pr = 0.64, a = 0.7, b = 0.83, c =.

10 946 Emmanuel Osalusi, Precious Sibanda 0 (a) (b) Fig. 3. (a), (b).

11 On variable laminar convective flow 947 (c) (d) Fig. 3. (a) Effect of W on the radial velocity profiles, (b) effect of W on the tangential velocity profiles, (c) effect of W on the axial velocity profiles (d) effect of W on the temperature profiles: for M = 0, ε = 0, Pr = 0.64, a = 0.7, b = 0.83, c =.

12 948 Emmanuel Osalusi, Precious Sibanda (a) (b) Fig. 4. (a), (b).

13 3 On variable laminar convective flow 949 (c) (d) Fig. 4. (a) Effect of M on the radial velocity profiles, (b) effect of M on the tangential velocity profiles, (c) effect of M on the axial velocity profiles (d) effect of M on the temperature profiles: for M = 0, W =, Pr = 0.64, a = 0.7, b = 0.83, c =.

14 950 Emmanuel Osalusi, Precious Sibanda 4 6. CONCLUSION In this paper we have extended the work of Maleque and Sattar [] on the effects of variable properties on the problem of a steady laminar flow due to a rotating disk to include the effects of an applied magnetic field. The study confirms that the radial velocity reaches its maximum value close to the surface of the disk. In the presence of a magnetic field the radial velocity increases at constant rate for all values of ε so that the largest maximum value is not obtained for constant property ε = 0 close to the disk surface. The effect of the magnetic field for fixed suction rate and Prandtl number is to accelerate the rate of decrease of the radial and tangential skin-friction coefficients and the rate of heat transfer coefficient. Acknowledgement. This material is based upon work supported in part by the National Research Foundation (NRF). REFERENCES. H. A. Attia, Unsteady flow and heat transfer of viscous incompressible fluid with temperaturedependent viscosity due to a rotating disc in a porous medium, J. Phys. A: Math. Gen., 39, , H. A. Attia and A. L. Aboul-Hassan, On hydromagnetic flow due to a rotating disk, Appl. Math. Modelling, 8, , E. R. Benton, On the flow due to a rotating disk, J. Fluid Mech., 7, , W. G. Cochran, The flow due to a rotating disk, Proc. Camb. Philos. Soc., 30(3), , T. M. A. El-Mistikawy, H. A. Attia, and A. Megahed, The rotating disk flow in the presence of a weak magnetic field, In Proc. 4th Conf. on Theoret. and Appl. Mech., Cairo, Egypt, November 5 7, 69 8, J. Herrero, J. A. C. Humphrey, and F. Gilralt, Comparative analysis of coupled flow and heat transfer between co-rotating disks in rotating and fixed cylindrical enclosures, ASME J. Heat Transfer, 300,, H. Herwig, The effect of variable properties on momentum and heat transfer in a tube with constant heat flux across the wall, Int. J. Heat Mass Transfer, 8, 44 44, H. Herwig and K. Klemp, Variable property effects of fully developed laminar flow in concentric annuli, ASME J. Heat Transfer, 0, 34 30, S. Jayaraj, Thermophoresis in laminar flow over cold inclined plates with variable properties, Heat Mass Transfer, 40, 67 74, N. Kelson and A. Desseaux, Note on porous rotating disk flow, ANZIAM J., 4(E), C847 C855, (000).. A. K. Maleque and A. M. Sattar, Steady laminar convective flow with variable properties due to a prous rotating disk, J. Heat Transfer, 7, , M. G. Rogers, and G. N. Lance, The rotationally symmetric flow of a viscous fluid in presence of infinite rotating disk, J. Fluid Mech., 7, 67 63, J. T. Stuart, On the effect of uniform suction on the steady flow due to a rotating disk, Q. J. Mech. Appl. Math., 7, , T. von Karman, Uber laminare und turbulente Reibung, Z. Angew. Math. Mech.,, 33 55, 9.