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FLOW OF A NON-NEWTONIAN FLUIDWITH HEAT TRANSFER ABOVE AROTATING DISK CONSIDERING THE IONSLIPHazem Ali Attia aa Department of Mathematics , College of Science, Al-QasseemUniversity , Buraidah, Saudi ArabiaPublished online: 25 Jan 2007.

To cite this article: Hazem Ali Attia (2006) FLOW OF A NON-NEWTONIAN FLUID WITH HEAT TRANSFERABOVE A ROTATING DISK CONSIDERING THE ION SLIP, Chemical Engineering Communications, 193:8,976-985, DOI: 10.1080/00986440500351990

To link to this article: http://dx.doi.org/10.1080/00986440500351990

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Flow of a Non-Newtonian Fluid with Heat Transferabove a Rotating Disk Considering the Ion Slip

HAZEM ALI ATTIA

Department of Mathematics, College of Science, Al-Qasseem University,Buraidah, Saudi Arabia

The steady flow and heat transfer of a conducting non-Newtonian fluid due to therotation of an infinite nonconducting disk in the presence of an axial uniform steadymagnetic field are studied considering the ion slip. The governing nonlinear equa-tions are solved numerically using finite differences, and the solution shows thatthe ion slip and the non-Newtonian fluid characteristics give some interesting results.

Keywords Rotating disk flow, Hydromagnetic flow, Ion slip, Non-Newtonianfluid, Heat transfer

Introduction

Hydrodynamic flow due to an infinite rotating disk was first introduced by vonKarman (1921). He formulated the problem in the steady state and used similaritytransformations to reduce the governing partial differential equations to ordinarydifferential equations. Asymptotic solutions were obtained for the reduced systemof ordinary differential equations (Cochran, 1934). Rotating disk flows of conduct-ing fluids have practical applications in many areas such as rotating machinery,lubrication, oceanography, computer storage devices, viscometry, and crystal growthprocesses. The influence of an external uniform magnetic field on the flow due to arotating disk was studied (El-Mistikawy and Attia, 1990; El-Mistikawy et al., 1991)without considering the Hall effect or the ion slip in applying Ohm’s law. In fact, theHall effect is important when the Hall parameter, which is the ratio between theelectron-cyclotron frequency and the electron-atom-collision frequency, is high. Thishappens when the magnetic field is high or when the collision frequency is low(Sutton and Sherman, 1965; Crammer and Pai, 1973). Furthermore, the masses ofthe ions and electrons are different and, in turn, their motions will be different.Usually, the diffusion velocity of electrons is larger than that of ions and, as a firstapproximation, the electric current density is determined mainly by the diffusionvelocity of the electrons. However, when the electromagnetic force is very large (suchas in the case of a strong magnetic field), the diffusion velocity of the ions may not benegligible (Sutton and Sherman, 1965; Crammer and Pai, 1973). If we include thediffusion velocity of ions as well as that of electrons, we have the phenomenon ofion slip. In the above-mentioned work, the Hall and ion slip terms were ignoredin applying Ohm’s law, as they have no marked effect for small and moderate values

Address correspondence to Hazem Ali Attia, Department of Mathematics, College ofScience, Al-Qasseem University, P.O. Box 237, Buraidah 81999, Saudi Arabia. E-mail:ah1113@yahoo.com

Chem. Eng. Comm., 193:976–985, 2006Copyright # Taylor & Francis Group, LLCISSN: 0098-6445 print/1563-5201 onlineDOI: 10.1080/00986440500351990

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of the magnetic field. However, the current trend for the application of magneto-hydrodynamics is towards a strong magnetic field, so that the influence of the elec-tromagnetic force is noticeable under these conditions, and the Hall current as wellas the ion slip are important; they have a marked effect on the magnitude and direc-tion of the current density and consequently on the magnetic-force term (Attia,2003). Aboul-Hassan and Attia (1997) studied the steady hydromagnetic problemtaking the Hall effect into consideration. The effect of uniform suction or injectionon the flow of a conducting fluid due to a rotating disk was studied (Attia andAboul-Hassan, 2001) considering the Hall current with the neglect of the ion slip.

The problem of heat transfer from a rotating at a constant temperature was firstconsidered by Millsaps and Pohlhausen (1952) for a variety of Prandtl numbers inthe steady state. Sparrow and Gregg (1960) studied the steady-state heat transferfrom a rotating disk maintained at a constant temperature to fluids at any Prandtlnumber. Later Attia (1998) extended the problem discussed in Millsaps andPohlhausen (1952) and Sparrow and Gregg (1960) to the unsteady state in the pres-ence of an applied uniform magnetic field. The steady flow of a non-Newtonian fluiddue to a rotating disk with uniform suction was considered by Mithal (1961). Later,the influence of a uniform magnetic field on the flow of a non-Newtonian fluid wasstudied by Attia (2003) neglecting the Hall current and the ion slip.

In the present work the steady hydromagnetic flow of a viscous, incompressible,and electrically conducting non-Newtonian fluid due to the uniform rotation of aninfinite nonconducting disk in an axial uniform steady magnetic field is studied con-sidering the ion slip. The governing nonlinear differential equations are solvednumerically using the finite difference approximations. Some interesting effects forthe Hall current, the ion slip, and the non-Newtonian fluid characteristics arereported.

Basic Equations

The disk is assumed to be insulating and rotating impulsively from rest in the z ¼ 0plane about the z-axis with a uniform angular velocity x. The fluid is assumed to beincompressible and has density q, kinematical viscosity n, and electrical conductivityr. An external uniform magnetic field is applied in the z-direction and has a constantflux density Bo. The magnetic Reynolds number is assumed to be very small, so thatthe induced magnetic field is negligible. The electron-atom collision frequency isassumed to be relatively high, so that the Hall effect and the ion slip cannot beneglected (Sutton and Sherman, 1965; Crammer and Pai, 1973). Due to the axialsymmetry of the problem about the z-axis, the cylindrical coordinates (r,u,z) areused. For the sake of definiteness, the disk is taken to be rotating in the positiveu direction. Due to the symmetry about the z ¼ 0 plane, it is sufficient to considerthe problem in the upper half-space only. The non-Newtonian fluid considered inthe present study is that for which the stress tensor si

j is related to the rate of straintensor ei

j as (Mithal, 1961),

sij ¼ 2lei

j þ 2lceikek

j � pdij; ei

j ¼ 0

where p denotes the pressure, l is the coefficient of viscosity, and lc is the coefficientof cross viscosity. If the Hall and ion slip terms are retained in generalized Ohm’s law,then the equations of steady motion in cylindrical coordinates (Sutton and Sherman,

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1965; Crammer and Pai, 1973) are

@u

@rþ u

rþ @w

@z¼ 0 ð1Þ

q u@u

@rþ w

@u

@z� v2

r

� �þ rB2

o

ða2 þ b2eÞðau� bevÞ ¼

@srr

@rþ @s

zr

@zþ

srr � su

u

rð2Þ

q u@v

@rþ w

@v

@zþ uv

r

� �þ rB2

o

ða2 þ b2eÞðavþ beuÞ ¼

@sru

@rþ@sz

u

@zþ

2sru

rð3Þ

q

�u@w

@rþ w

@w

@z

�¼ @s

rz

@rþ @s

zz

@zþ sr

z

rð4Þ

where u, v, and w are the velocity components in the directions of increasing r, u,and z, be ¼ rbBo is the Hall parameter, which can take positive or negative values,b ¼ 1=nq is the Hall factor, n is the electron concentration per unit volume, �q isthe charge of the electron (Sutton and Sherman, 1965; Crammer and Pai, 1973),a ¼ 1þ bebi, and bi is the ion slip parameter (Sutton and Sherman, 1965; Crammerand Pai, 1973).

The boundary conditions are given as

z ¼ 0; u ¼ 0; v ¼ rx; w ¼ wo ð5aÞ

z!1; u! 0; v! 0; p! p1 ð5bÞ

Equation (5a) indicates the no-slip conditions of viscous flow applied at the surfaceof the disk and ensures that the convective velocity normal to the surface of the diskspecifies the mass injection or withdrawal. Due to the uniform suction or injection,the vertical velocity component takes a constant nonzero value at z ¼ 0. Far fromthe surface of the disk, all fluid velocities must vanish aside the induced axial compo-nent as indicated in Equation (5b). We introduce von Karman transformations(1921):

u ¼ rxF ; v ¼ rxG; w ¼ffiffiffiffiffiffixvp

H; z ¼ffiffiffiffiv

x

rf; p� p1 ¼ �qvxP

where f is a nondimensional distance measured along the axis of rotation and F, G,H, and P are nondimensional functions of the modified vertical coordinate f. Wedefine the magnetic interaction number c by c ¼ rB2

o=qx, which represents theratio of the magnetic force to the fluid inertia force. With these definitions,Equations (1)–(5) take the form

dH

dfþ 2F ¼ 0 ð6Þ

d2F

df2�H

dF

df� F2 þ G2 � c

ða2 þ b2eÞðaF � beGÞ

� 1

2K

dF

df

� �2

þ3dG

df

� �2

þ2Fd2F

df2

!¼ 0 ð7Þ

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d2G

df2�H

dG

df� 2FG � c

ða2 þ b2eÞðaG þ beFÞ þ K

dF

dfdG

df� F

d2G

df2

� �¼ 0 ð8Þ

d2H

df2�H

dH

df� 7

2K

dH

dfd2H

df2þ dP

df¼ 0 ð9Þ

f ¼ 0; F ¼ 0; G ¼ 1; H ¼ 0 ð10aÞ

f!1; F ! 0; G ! 0; P! 0 ð10bÞ

where K is the parameter that describes the non-Newtonian behavior, K ¼ lc=lx.Due to the difference in temperature between the wall and the ambient fluid,

heat transfer takes place. The energy equation, by neglecting the dissipation terms,takes the form (Millsaps and Pohlhausen, 1952; Sparrow and Gregg, 1960)

qcp u@T

@rþ w

@T

@z

� �� k

@2T

@z2¼ 0 ð11Þ

where k and cp are thermal conductivity and the specific heat at constant pressure ofthe fluid. The boundary conditions for the energy problem are that the temperature,by continuity considerations, equals Tw at the surface of the disk. At large distancesfrom the disk, T ! T1 where T1 is the temperature of the ambient fluid.

In terms of the non dimensional variable h ¼ ðT � T1Þ=ðTw � T1Þ and usingvon Karman transformations, Equation (11) takes the form

1

Pr

d2h

df2þH

dhdf¼ 0 ð12Þ

where Pr is the Prandtl number given by Pr ¼ cpl=k. The boundary conditions interms of h are expressed as

f ¼ 0 : h ¼ 1; f!1; h! 0 ð13Þ

The heat transfer from the disk surface to the fluid is computed by application ofFourier’s law:

q ¼ �kdT

dz

� �w

Introducing the transformed variables, the expression for q becomes

q ¼ �kðTw � T1Þffiffiffiffixn

rdhð0Þ

df

By rephrasing the heat transfer results in terms of a Nusselt number defined asNu ¼ q

ffiffiffiffiffiffiffiffiffin=x

p=kðTw � T1Þ, the last equation becomes

Nu ¼ �dhð0Þ

df

The system of nonlinear ordinary differential equations (6)–(8) and (12) issolved under the conditions given by Equations (10) and (13) for the three compo-nents of the flow velocity and temperature distribution, using the Crank-Nicolsonmethod (Ames, 1977). The resulting system of difference equations has to be solvedin the infinite domain 0 < f <1. A finite domain in the f-direction can be used

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instead with f chosen large enough to ensure that the solutions are not affected byimposing the asymptotic conditions at a finite distance. The independence of theresults from the length of the finite domain and the grid density was ensured andsuccessfully checked by various trial-and-error numerical experimentations. Compu-tations are carried out for f1 ¼ 10, which is adequate for the ranges of the para-meters studied here. It should be pointed out that the steady-state solutionsreported by Attia and Aboul-Hassan (2001) are reproduced by setting bi ¼ 0in the present results. These comparisons lend confidence in the correctness of thesolutions presented in this article.

Results and Discussion

Figures 1(a)–(d) present the profile of the radial velocity component F for variousvalues of the ion slip parameter bi, the Hall parameter be, and the non-Newtonian

Figure 1. Steady-state profile of F for various values of be, bi, and K. (a) k ¼ 0, be� 0, (b)K ¼ 0, be� 0, (c) K ¼ 1, be� 0, (d) K ¼ 1, be� 0.

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parameter K. In these figures c ¼ 1 and K ¼ 0 and 1 (Mithal, 1961; Attia, 2003).Figure 1 shows that for be ¼ �0:5 and bi ¼ 0, the radial flow reverses its directionat a certain distance from the disk. Increasing bi, for be < 0, leads to a negative Ffor all f as a result of increasing the effective conductivity ½¼ c=ða2 þm2Þ�, whichincreases the damping force on F. Figure 1(a) indicates also that for bebi > 0,increasing the magnitude of bi increases F due to the decrease in the effective con-ductivity, which decreases the damping force on F. Large values of bi reducethe effective conductivity more, which corresponds to the hydrodynamic case.Figure 1(b) indicates that when be > 0 and bi > 0, increasing bi decreases F forsome f. This may be attributed to the fact that in the magnetic force term in Equa-tion (7), the effect of bi on the numerator is stronger than its effect on the denomi-nator, which increases the damping force on F and consequently decreases F forsome f. Also, for bi < 0, increasing the magnitude of bi increases F for small f

Figure 2. Steady-state profile G for various values of be, bi, and K. (a) K ¼ 0, be� 0, (b)K ¼ 0, be� 0, (c) K ¼ 1, be� 0, (d) K ¼ 1, be� 0.

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and then decreases it for larger f. This accounts for a crossover in the F-f chartwith bi. It is of interest to see that the variation of F with bi depends on f. The effectof the parameter K is shown in Figures 1(c)–1(d). It is clear in the figures thatthe effect of the parameter K is to reverse the direction of F for thecase bi ¼ be ¼ 0. For be < 0, as shown in Figure 1(c), the parameter K reversesthe direction of F for all values of bi. In Figure 1(d), for be > 0, K reduces themagnitude of F while keeping its direction. The value at which the crossover inthe F-f charts occur increases with K.

Figures 2(a)–(d) present the profile of the azimuthal velocity component Gfor various values of the ion slip parameter bi, the Hall parameter be, and thenon-Newtonian parameter K. In these figures c ¼ 1 and K ¼ 0 and 1. As shownin Figure 2(a), small negative values of be increases G as a result of decreasingthe magnetic damping. Increasing bi, with be < 0, decreases G, due to increase in

Figure 3. Steady-state profile of H for various values of be, bi, and K. (a) K ¼ 0, be� 0, (b)K ¼ 0, be� 0, (c) K ¼ 1, be� 0, (d) K ¼ 1, be� 0.

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the effective conductivity. Figure 2(a) shows also that for negative values of bi,increasing the magnitude of bi increases G due to decrease in the damping forceon G. Figure 2(b) describes the same findings. For bebi > 0, increasing bi increasesG, while for bebi < 0, increasing the magnitude of bi decreases G. Figures 2(c)–(d)show that increasing K increases G for all be and bi. The effect of bi on G is morepronounced for K > 0.

Figures 3(a)–(d) present the profile of the axial velocity component H forvarious values of the ion slip parameter bi, the Hall parameter be, and the non-Newtonian parameter K. In these figures c ¼ 1 and K ¼ 0 and 1. As shown inFigure 3(a), for be ¼ �0:5 and bi ¼ 0, the axial velocity H reverses its direction ata certain f. Increasing bi increases H, as a result of decreasing F, and reverses itsdirection for all f. Figure 3(b) shows that for be > 0, increasing the magnitudeof bi, in general, decreases H as a result of increasing F. It is also shown inFigure 3(b) that the axial flow is always towards the disk for all values of bi.Figures 3(c)–(d) present the effect of the parameter K in reversing the direction ofH for the case be ¼ bi ¼ 0. For be < 0, as shown in Figure 3(c), the parameter Kreverses the direction of H for all values of bi. In Figure 3(d), for be > 0, K reducesthe magnitude of H while keeping its direction.

Tables I and II present the variation of the Nusselt number Nu with the Hallparameter be, the ion slip parameter bi, and the non-Newtonian parameter K forPr ¼ 0.7 (suitable for gas) and 10 (suitable for liquid) (Sparrow and Gregg, 1960),respectively and for c ¼ 1. It is clear from Tables I and II that, for be < 0, increasingbi decreases Nu for all K as a result of decreasing the incoming axial flow at nearambient temperature to the disk, which decreases the heat transfer from the surfaceof the disk, i.e., Nu. From Table I, for be > 0, and small bi, increasing bi decreasesNu, but increasing bi more increases Nu for all K. As shown in Table II, forK ¼ 0 (Newtonian case) and be > 0, increasing bi decreases Nu. On the otherhand, for non zero values of K (non-Newtonian case) and for be > 0 and small bi,increasing bi decreases Nu, however, increasing bi more increases Nu. The influenceof bi on Nu becomes more pronounced for be < 0 than for be > 0 for all values of Kand Pr.

Table I. Variation of the steady-state value of Nu with be, bi, and K for Pr ¼ 0.7 andc ¼ 1

Nu K ¼ 0 K ¼ 0.5 K ¼ 1

be ¼ 0, bi ¼ 0 �0.2232 �0.1787 �0.1413be ¼ �0:5, bi ¼ 0 �0.1692 �0.1218 �0.0858be ¼ �0:5, bi ¼ 0:5 �0.1318 �0.0864 �0.0545be ¼ �0:5, bi ¼ 1 �0.0819 �0.0444 �0.0225be ¼ �0:5, bi ¼ �0:5 �0.1963 �0.1491 �0.1118be ¼ �0:5, bi ¼ �1 �0.2162 �0.1697 �0.1325be ¼ 0:5, bi ¼ 0 �0.2777 �0.2353 �0.1987be ¼ 0:5, bi ¼ 0:5 �0.2758 �0.2332 �0.1966be ¼ 0:5, bi ¼ 1 �0.2762 �0.2335 �0.1971be ¼ 0:5, bi ¼ �0:5 �0.2839 �0.2421 �0.2059be ¼ 0:5, bi ¼ �1 �0.2894 �0.2575 �0.2219

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Conclusion

The steady magnetohydrodynamic flow of a non-Newtonian fluid due to an infiniterotating disk in the presence of a uniform magnetic field is studied with heat transferconsidering the Hall effect and the ion slip. The inclusion of the Hall effect, the ionslip, and the non-Newtonian fluid characteristics reveals some interesting phenom-ena, and it is found that the signs of the Hall and ion slip parameters are important.It is of interest to find, in the Newtonian case, that the radial and axial flows reversedirection with f for be < 0 and all values of bi. The effect of the non-Newtonianparameter K is to reverse the axial and radial flow for be ¼ bi ¼ 0. Also, in thenon-Newtonian case, both the axial and radial flows reverse direction for all f forbe < 0 and all values of bi. For be > 0, the variation of the velocity components withthe ion slip depends on f. The heat transfer at the surface of the disk is found todepend on the magnitude and the sign of the Hall and ion slip parameters. The effectof bi on the heat transfer at the surface of the disk is more apparent for be < 0 thanfor be > 0. On the other hand, the effect of non-Newtonian parameter K on the heattransfer is more clear for be > 0 than for be < 0.

References

Aboul-Hassan, A. L. and Attia, H. A. (1997). The flow due to a rotating disk with Hall effect,Phys. Lett. A, 228, 286–290.

Ames, W. F. (1977). Numerical Methods in Partial Differential Equations, 2nd ed., AcademicPress, New York.

Attia, H. A. (1998). Unsteady MHD flow near a rotating porous disk with uniform suction orinjection, Fluid Dyn. Res., 23, 283–290.

Attia, H. A. (2003). Unsteady flow of a non-Newtonian fluid above a rotating disk with heattransfer, Int. J. Heat Mass Transfer, 46(41), 2695–2700.

Attia, H. A. and Aboul-Hassan, A. L. (2001). Effect of Hall current on the unsteady MHDflow due to a rotating disk with uniform suction or injection, Appl. Math. Model.,25(12), 1089–1098.

Cochran, W. G. (1934). The flow due to a rotating disk, Proc. Camb. Philos. Soc., 30(3),365–375.

Table II. Variation of the steady-state value of Nu with be, bi, and K for Pr ¼ 10 andc ¼ 1

Nu K ¼ 0 K ¼ 0.5 K ¼ 1

be ¼ 0, bi ¼ 0 �0.8811 �0.6846 �0.5287be ¼ �0:5, bi ¼ 0 �0.7117 �0.5309 �0.4098be ¼ �0:5, bi ¼ 0:5 �0.5811 �0.4069 �0.3005be ¼ �0:5, bi ¼ 1 �0.3945 �0.2520 �0.1754be ¼ �0:5, bi ¼ �0:5 �0.7992 �0.6221 �0.4964be ¼ �0:5, bi ¼ �1 �0.8595 �0.6887 �0.5631be ¼ 0:5, bi ¼ 0 �1.0579 �0.8737 �0.7116be ¼ 0:5, bi ¼ 0:5 �1.0427 �0.8646 �0.7106be ¼ 0:5, bi ¼ 1 �1.0372 �0.8653 �0.7186be ¼ 0:5, bi ¼ �0:5 �1.0921 �0.9027 �0.7322be ¼ 0:5, bi ¼ �1 �1.1613 �0.9703 �0.7934

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Crammer, K. R. and Pai, S.-I. (1973). Magnetofluid Dynamics for Engineers and AppliedPhysicists, McGraw-Hill, New York.

El-Mistikawy, T. M. A. and Attia, H. A. (1990). The rotating disk flow in the presence ofstrong magnetic field, Proceedings of the 3rd International Congress of Fluid Mechanics,Cairo, Egypt, vol. 3, 1211–1222.

El-Mistikawy, T. M. A., Attia, H. A., and Megahed, A. A. (1991). The rotating disk flow inthe presence of weak magnetic field, Proceedings of the 4th Conference on Theoretical andApplied Mechanics, Cairo, Egypt, 69–82.

Millsaps, K. and Pohlhausen, K. (1952). Heat transfer by laminar flow from a rotating disk,J. Aeronaut. Sci., 19, 120–126.

Mithal, K. G. (1961). On the effects of uniform high suction on the steady flow of a non-Newtonian liquid due to a rotating disk, Q. J. Mech. Appl. Math., 14, 401–410.

Sparrow, E. M. and Gregg, J. L. (1960). Mass transfer, flow, and heat transfer about arotating disk, J. Heat Transfer, 82, 294–302.

Sutton, G. W. and Sherman, A. (1965). Engineering Magnetohydrodynamics, McGraw-Hill,New York.

von Karman, T. (1921). Uber laminare und turbulente reibung, Z. Angew. Math. Mech., 1(4),233–235.

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