Convective Hydromagentic Slip Flow with Variable Properties Due to a Porous Rotating Disk

In this paper we investigate convective heat transfer characteristics of steady hydromagnetic slip flow over a porous rotating disk taken into account the temperature dependent density, viscosity and thermal conductivity in the presence of Hall current, viscous dissipation and Joule heating. Using von-Karman similarity transformations we reduce the governing equations for flow and heat transfer into a system of ordinary differential equations which are highly nonlinear and coupled. The resulting nondimensional equations are solved numerically by applying Nachtsheim-Swigert iteration technique. The results show that when modeling a thermal boundary layer, with temperature dependent fluid properties, consideration of Prandtl number as constant within the boundary layer, produces unrealistic results. Therefore it must be treated as variable throughout the boundary layer. Results also show that the slip factor significantly controls the flow and heat transfer characteristics.


Introduction
n recent years, the flow dynamics due to a rotating disk, originating from the early formulation of von Karman (1921), has been a popular area of research.Since then many researchers (Cochran, 1934;Roger and Lance, 1960;Benton, 1965;Kuiken, 1971;Owen and Rogers, 1989;Herrero et al 1994;Kelson and Desseaux, 2000;Andrsson and Korte, 2002;Takhar et al 2002) have studied and reported results on disk-shaped bodies with or without heat transfer.Flow due to a rotating disk is encountered in many industrial, geothermal, geophysical, technological and engineering applications.A few of them are rotating heat exchangers, rotating disk reactors for bio-fuels production, computer disk drives, and gas or marine turbines.The effects of an applied magnetic field on the steady flow due to the rotation of a disk of infinite or finite extent were studied by El-Mistikawy et al. (1991) and El-Mistikawy and Atia (1990).Atia and Aboul-Hassan (1997) studied steady hydromagnetic flow due to an infinite disk rotating with uniform angular velocity in the presence of an axial magnetic field.In their analysis they neglected the induced magnetic field but considered Hall current.Attia (1998) studied the effects of suction as well as injection in the presence of a magnetic field on the unsteady flow past a rotating porous disk.It was found that the combined effect of a magnetic field with strong injection may stabilize the growth of the boundary layer.Sparrow et al. (1971) studied the flow of Newtonian fluid due to the rotation of a porous-surfaced disk with a set of linear slip-flow conditions.A substantial reduction in torque then occurred as a result of surface slip.Miklavcic and Wang (2004) further revisited the problem of Sparrow et al. and pointed out that the slipflow boundary conditions could also be used for slightly rarefied gases or for flow over grooved surfaces.Arikoglu and Ozkol (2006) studied MHD slip flow over a rotating disk with heat transfer.It is observed that both the slip factor and the magnetic flux decrease the velocity in all directions and thicken the thermal boundary layer.Recently, Osalusi et al. (2008) studied thermal-diffusion and diffusion-thermo effects on MHD slip flow due to a rotating disk.
In classical treatment of thermal boundary layers, fluid properties (such as density, viscosity, thermal conductivity) are assumed to be constant; however, experiments indicate that this assumption only makes sense if temperature does not change rapidly for the application of interest.To predict the flow behavior accurately, it may be necessary to take into account these variable properties.Zakerullah and Ackroyd (1979) investigated free convection flow above a horizontal circular disk considering variable fluid properties.In the case of fully developed laminar flow in concentric annuli, the effect of the variable property has been investigated by Herwig and Klemp (1988).Atia (2006) studied unsteady hydromagnetic flow due to an infinite rotating disk, considering temperature dependent viscosity in a porous medium with Hall and ion-slip currents.Maleque and Sattar (2005a) studied the effect of variable properties on the steady laminar convective flow due to a rotating disk while Maleque and Sattar (2005b) further investigated the same problem in the presence of Hall current.Osalusi and Sibanda (2006) revisited the problem of Maleque and Sattar (2005a), considering magnetic effect.
When fluid properties such as viscosity and thermal conductivity vary with temperature, Prandtl number (see section 2) varies too.All of these afore-mentioned works considered Prandtl number as constant within the boundary layer, although viscosity and thermal conductivity depends on temperature.Hence one of the motivations behind this study is also to investigate how variable Prandtl number affects the flow and heat transfer characteristics.
In the present study we extend the work of Maleque and Sattar (2005b) and analyze the flow and heat transfer characteristics in the presence of viscous dissipation and Joule heating, considering slip flow boundary condition at the surface of a uniformly heated rotating disk.The resulting governing equations are solved numerically applying Nachtsheim-Swigert (1965) iteration technique.Graphical results for non-dimensional velocity and temperature profiles including skin-friction coefficient and the Nusselt number in tabular form are presented for a range of values of the parameters characterizing the flow.The accompanying discussion provides physical interpretations of the results.

Mathematical Model
Let us consider a steady hydromagnetic laminar flow of an electrically conducting fluid due to a porous rotating disk of infinite extent in the presence of an external uniform magnetic field directed perpendicular to the disk.The fluid properties are taken as strong functions of temperature.A uniform suction or injection through the disk is considered for the whole range of suction or injection velocities.

Basic Equations
The equations governing the steady hydromagnetic laminar convective flow are: Equation of continuity: .( ) 0 Ohm's law for a moving conductor with Hall currents: [ ] Maxwell electromagnetic equations: . 0, , .0 Energy equation: Here q is the velocity vector, B is the magnetic field vector, E denotes the electrical field vector which results from charge separation and is in the z -direction, J is the current density vector, p is the pressure, ρ is the density of the fluid, µ is the viscosity of the fluid, σ is the electrical conductivity of the fluid, κ is the

Governing equations
In non-rotating cylindrical polar coordinates ( , , ) r z φ , let us consider a disk which rotates with constant angular velocity Ω about the z -axis.The disk is placed at 0 z = , and the fluid occupies the region 0 z > , where z is the vertical axis in the cylindrical coordinates system with r and φ as the radial and tangential axes respectively.The components of the flow velocity q are ( , , ) u v w in the directions of increasing ( , , ) r z φ respectively.The surface of the rotating disk is maintained at a uniform temperature w T and far away from the wall, the free stream is kept at a constant temperature T ∞ and at a constant pressure p ∞ .The fluid is assumed to be Newtonian, viscous and electrically conducting.An external uniform magnetic field is applied in the zdirection.The electron-atom collision frequency is assumed to be relatively high so that the Hall effect cannot be neglected.Ion-slip effects are however ignored in the present analysis.
From equation (4), using the relation 0 ∇ ⋅ = B for the magnetic field ( , , ) everywhere in the fluid.This assumption is valid only when the magnetic Reynolds number is very small so that magnetic induction effects can be ignored.For the current density ( , , ) . Hence we consider that the disk is non-conducting and therefore 0 z J = at the disk and hence zero everywhere.Finally we consider the case of a short circuit problem in which the applied electric field = E 0 and also assume that the induced magnetic field is negligible in comparison with the applied magnetic field.
In the absence of electric field E and electron pressure e p equation (3) becomes where We also assume that the fluid properties, viscosity ( µ ), thermal conductivity ( κ ) and density ( ρ ) are functions of temperature alone and obey the following laws (see Jayaraj, 1995;later used by Malek and Sattar, 1995b;Osalusi and Sibanda, 2006) [ ] where a , b and d are arbitrary exponents while µ ∞ , κ ∞ and ρ ∞ are the viscosity, thermal conductivity and density of the ambient fluid respectively.The flow configurations and geometrical coordinates are shown in Figure 1.Due to steady axially symmetric, compressible hydromagnetic laminar flow of a homogeneous fluid the governing equations take the following form (see Malek and Sattar, 1995b): Figure 1.Flow configurations and coordinate system ( ) If the mean free path of the fluid particles is comparable to the characteristic dimensions of the flow field domain, the assumption of continuum media is no longer valid, and as a consequence Navier-Stokes equation breaks down.In the range 0.1 10 Kn < < of Knudsen number, the high order continuum equations (Burnett equations) should be used.For the range of 0.001 0.1 Kn < < , no-slip boundary conditions cannot be used and should be replaced with the following expression (Gad-el-Hak, 1999): where t U is the target velocity, ξ is the target momentum accommodation coefficient and λ is the mean free path.For 0.001 Kn < , the no-slip boundary condition is valid; therefore, the velocity at the surface is equal to zero.In this study the slip and the no-slip regimes of the Knudsen number that lies in the range 0 0.1 Kn < < are considered.

Boundary conditions
By using equation ( 15), the appropriate boundary conditions for our model are (i) On the surface of the disk ( (ii) Matching with the quiescent free stream ( z → ∞ ): (16c)

Transformation of the model
To obtain the solutions of the governing equations ( 10)-( 14) together with the boundary conditions ( 16) we introduce a dimensionless normal distance from the disk, ( ) along with the von-Karman transformation ( ) where υ ∞ is the kinematic viscosity of the ambient fluid and 17) into ( 10)-( 14) we obtain the following nonlinear ordinary differential equations where ( ) is the relative temperature difference parameter, which is positive for a heated surface, negative for a cooled surface and zero for uniform properties.Thus by using (17 is the slip factor and represents a uniform suction when 0 s w < and uniform injection when 0 s w > at the surface of the disk.

Particular cases
A number of special cases can be derived from the full transformed momentum and energy equations ( 18)-( 21

Variable Prandtl Number
The Prandtl number is a function of viscosity and as viscosity varies across the boundary layer, the Prandtl number varies, too.The assumption of constant Prandtl number inside the boundary layer may produce unrealistic results.Therefore, Prandtl number related to the variable viscosity is defined by ( 1) Pr (1 ) Pr (1 ) ( 1) At the surface ( 0 η = ) of the disk, this can be written as Pr Pr (1 ) From equation ( 23 (1 ) Pr( 1) (25) Equation ( 25) is the corrected non-dimensional form of the energy equation in which Prandtl number is treated as variable.It is mentionable that this correction does not appear in the literature.

Parameters of engineering interest
The parameters of engineering interest for the present problem are the skin-friction coefficient (Cf ) and the Nusselt number ( Nu ) which indicate physically wall shear stress and rate of heat transfer respectively.The action of the variable properties in the fluid adjacent to the disk sets up a tangential shear stress, which opposes the rotation of the disk.As a consequence, it is necessary to provide a torque at the shaft to maintain a steady rotation.The radial shear stress r τ and tangential shear stress t τ are defined by: Hence the skin-frictions ( (1 ) Re (0 The rate of heat transfer from the disk surface to the fluid is computed by the application of Fourier's law as given below Hence the Nusselt number ( ) is obtained as is the rotational Reynolds number.Thus from equations ( 28), ( 29) and ( 31) we see that skin-friction coefficient and Nusselt number are proportional to the numerical values of (0) F′ , (0) G′ and (0) θ ′ − which are calculated in the process of integration when solving the corresponding differential equations.

Method of solutions
The set of equations ( 18)-( 20) and ( 25) are highly nonlinear and coupled and therefore the system cannot be solved analytically.The system of transformed governing equations ( 18)-( 20) and ( 25) with boundary conditions ( 12) is solved numerically using shooting method similar to that described by Nachtsheim-Swigert (1965).In equation ( 22 0), (0) , 1, 2 6, where The last three of these represents asymptotic convergence criteria. Choosing and expanding in a first-order Taylor's series after using equations (26) yields where subscript 'C' indicates the value of the function at max η determined from the trial integration.Solution of these equations in a least-square sense requires determining the minimum value of 6 2 Now differentiating ∏ with respect to i g we obtain Substituting equation ( 27) into (29) after some algebra we obtain ., ; , 1,2,3.
Now solving the system of linear equations (30) we obtain the missing (unspecified) values of i g as Thus adopting this numerical technique aforementioned, a computer program was set up for the solutions of the governing non-linear ordinary differential equations ( 18)-( 20) and ( 25) of our problem where the integration technique was adopted as a sixth-order Runge-Kutta method of integration.The velocity and temperature are determined as a function of the coordinate η and displayed graphically.

Numerical experiment
In this paper, the effects of Hall current, viscous dissipation and Joule heating on a steady hydromagnetic convective slip flow of a viscous, Newtonian, electrically conducting fluid with variable properties over a rotating porous disk have been investigated numerically by using Nachtsheim-Swigert shooting iteration technique.It can be seen that the solutions are affected by the seven parameters, namely suction (or injection) parameter s w , magnetic field parameter (or Hartmann number) Ha , Hall current parameter m , relative temperature difference parameter γ , Prandtl number Pr , Eckert number Ec and slip parameter ε .Since experimental data of the physical parameters are not available, in the numerical simulations the choice of the values of the parameters was dictated by the values chosen by the previous investigators.

Code verification
To assess the accuracy of the present code, we reproduced the values of (0) F′ , (0) G′ , ( ) H ∞ and (0) θ ′ for constant property models of Kelson and Desseaux (2000) (herein and after referred as KD2000) (see case-ii in section 3.1) and Arikoglu and Ozkol (2006) (herein and after referred as AO2006) (see case-iv in section 3.1).Tables 2-4 show the comparisons of the data produced by the present code and those of KD2000 and AO2006.In fact the results show a close agreement, and hence justify the use of the present code for the current model. .In each case, we found excellent agreement among the results.It was also found that 001 .0 = ∆η provided sufficiently accurate (error less than 6 10 − ) results and further refinement of the grid size was therefore not warranted.

Results and discussion
For the purpose of discussing the results, the numerical calculations are presented in the form of nondimensional velocity (radial, tangential and axial) and temperature profiles.In the calculations the values of the parameters namely suction (or injection) parameter s w , magnetic field parameter (or Hartmann number) Ha , Hall current parameter m , relative temperature difference parameter γ , Prandtl number Pr , Eckert number Ec and slip parameter ε are varied.The influence of the magnetic field parameter (Hartmann number) Ha on F , G , and H − distributions is depicted in Figures 3(a)-(c).An increase in Ha induces a significant decrease in radial and tangential velocity profiles throughout the boundary layer; this is due to fact that imposition of a magnetic field to an electrically conducting fluid creates a drag force called the Lorentz force that has a tendency to slow down the flow around the disk at the expense of increasing its temperature.From Figure 3(c) it is also apparent that inward axial velocity decreases substantially with the increase of the Hartmann number.An increase in Hartmann number increases temperature profiles and hence increases the thermal boundary layer as can be seen from Figure 3 Maleque and Sattar (2005b), and Osalusi and Sibanda (2006).Studying a limited set of parameter values such as 0 γ = , 0.5 (Maleque and Sattar, 2005b) and 0 γ = , 0.01 (Osalusi and Sibanda, 2006) and considering Prandtl number as constant within the boundary layer, they concluded that an increase in γ does not change the thickness of the thermal boundary layer.Here 0 ε = represents no-slip condition at the surface of the disk.From Figure 6(a) we see that the radial boundary layer decreases very rapidly with the increase of the slip factor.The thickness of the radial boundary layer is higher for no-slip flow compared to the slip flow.Fig. 6(a) further indicates that for large values of ε i.e. ε → ∞ , the rotating disk does not cause rotation of the fluid particles.Because in this range of ε the flow becomes entirely potential, there will be no motion in the fluid.This can be further explained as follows: the centrifugal force acting on the rotating disk (as like a centrifugal fan) will throw out the fluid that sticks to it.On the other hand, the flow in the axial direction will come forward to compensate for this thrown fluid.But increasing the slip on the surface of the disk reduces the amount of fluid that can stick on it; as a consequence the efficiency of the rotating disk is reduced substantially and is unable to transfer its circumferential momentum to the fluid particles.A reduction in the circumferential velocity results in a reduction in the centrifugal force which in turn decreases the inward axial velocity substantially as can be seen from Figure 6(c).From Figure 6(d) we see that the thermal boundary layer increases as slip factor ε increases.Figure 6(e) shows a decreasing effect of ε on the variable Prandtl number throughout the boundary layer.
In Table 5 we present skin-friction in radial and tangential directions and rate of heat transfer for various values of the pertinent parameters for a fixed value of Pr .It can be seen that skin-friction in the radial direction decreases while skin-friction in the tangential direction increases with the increase of the suction parameter ( 0) s w < .On the other hand, the rate of heat transfer increases with the increase of the suction parameter.An opposite effect is observed for the case of injection ( 0) The effects of the Hall current parameter on the radial and tangential skin-frictions and the rate of heat transfer can be seen from Table 5. Skin-friction in the radial direction increases within the range of 0 1 m ≤ ≤ .Outside of this range of m an opposite behavior is observed.Tangential skin-friction decreases when m increases within the range of 0 1 m ≤ ≤ , and outside this range of m tangential skin-friction increases with the further increase of m .The rate of heat transfer increases with the increase of m for some The effect of increasing Eckert number Ec has a decreasing effect on the radial skin-friction and on the rate of heat transfer whereas it has a very minor increasing effect on the tangential skin-friction as can be seen from Table 5.
The variation of the radial and tangential skin-frictions and the rate of heat transfer for some selected values of the slip factor ε are shown in Table 5.From here we see that skin-friction in both directions decreases with the increase of the slip factor.The largest skin-friction is found for the case of no-slip at the surface.On the other hand the rate of heat transfer increases with the increase of slip factor within the range of 0 1 ε ≤ ≤ .But outside of this range of ε , the rate of heat transfer decreases with the further increase of the slip factor.Thus the rate of heat transfer can be strongly controlled by controlling the slip on the disk.C ) is tabulated in Table 6.From this table we see that in both cases the rate of heat transfer from the surface of the disk to the fluid decreases for all increasing values of γ .We also see that rate of heat transfer for the variable property case is lower than the constant property case and the relative error between them increases significantly with the increase of γ .Therefore, consideration of Prandtl number as constant within the boundary layer for variable property is unrealistic.It is also mentionable that for our studied parameter values the relationship between the relative temperature difference parameter and the variable Prandtl number is an inverse relationship.So, the effect of Pr on the radial and tangential skin-frictions and on the rate of heat transfer is just the reverse of the effect of γ on them.
thermal conductivity of the fluid, p C is the specific heat of the fluid, T is the temperature of the fluid, and Φ is the viscous dissipation function.In equation (3factor, e − is the charge of electron, e n is the electron concentration per unit volume and e p is the electronic pressure.In equation (5) the term 2 J σ represents Joule heating whereas µΦ is the viscous dissipation or frictional heating effects.
) with the boundary conditions (22) which are as follows: ) there are three asymptotic boundary conditions and hence three unknown surface conditions Nachtsheim-Swigert developed an iteration technique to overcome the difficulties of determining the guess values of the unknown surface boundary conditions required for the shooting method.Within the context of the initial value method and the Nachtsheim-Swigert shooting iteration technique the outer boundary conditions may be functionally represented by

Figure 2 .
Figure 2. Variation of (a) radial velocity, (b) tangential velocity, (c) axial velocity, (d) temperature profile, and (e) variable Prandtl number for several values of s w .

For
the present investigation we considered our working fluid as flue gas.For flue gases (ambient Prandtl number, Pr 0.64 ∞ = ) the values of the exponents a , b and d are taken as 0

Figure 3 .
Figure 3. Variation of (a) radial velocity, (b) tangential velocity, (c) axial velocity, (d) temperature profile, and (e) variable Prandtl number for several values of Ha .

Figure 4 .
Figure 4. Variation of (a) radial velocity, (b) tangential velocity, (c) axial velocity, (d) temperature profile, and (e) variable Prandtl number for several values of m .

Figure 5
Figure 5(d) depicts that temperature profile increases significantly with the increase of γ .Quantitatively, at 8.0 η =the value of θ increases by 6026.8% when the value of γ increases from 0 to 1.0.Thus the thickness of the thermal boundary layer increases markedly with the increase of γ which is a direct contradiction to the findings ofMaleque and Sattar (2005b), andOsalusi and Sibanda (2006).Studying a limited set of parameter values such as
heat transfer decreases with the further increase of m .

table we
see that for a positive value of γ , Prandtl number at the surface of the disk Pr w decreases as γ increases.The opposite effect is observed when γ is negative.It must be noted that for ) it can be seen that for 0 γ → , the variable Prandtl number Pr equals the ambient Prandtl number Pr ∞ .For η → ∞ that is outside the boundary layer, ( ) θ η becomes zero.Therefore Pr equals Pr ∞ regardless of the values of γ .Table1shows the variation of the Prandtl number at the surface of the disk for several values of γ for a fixed value of the ambient Prandtl number Pr 0.64 ∞ =

Table 1 .
Values of Pr versus γ for Pr 0.64

Table 2 .
Numerical values of

Table 5
also shows that skin-friction in the radial direction increases for all increasing values of the In this range of Ha , radial skin-friction decreases as Ha increases.Tangential skin-friction increases while the rate of heat transfer decreases for all increasing values of the Hartmann number.

Table 5 .
Numerical values ofHa , m , Ec , and ε Finally, the significance of the relative temperature difference ( γ ) on the rate of heat transfer for both variable Prandtl number ( Pr V ) and constant Prandtl number ( Pr 2 s w