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ON THE EFFECTIVNESS OF ION SLIP ANDUNIFORM SUCTION OR INJECTION ONSTEADY MHD FLOW DUE TO A ROTATINGDISK WITH HEAT TRANSFER AND OHMICHEATINGHazem Ali Attia aa Department of Mathematics , College of Science, Al-QasseemUniversity , Buraidah, Saudi ArabiaPublished online: 15 Jun 2007.

To cite this article: Hazem Ali Attia (2007) ON THE EFFECTIVNESS OF ION SLIP AND UNIFORMSUCTION OR INJECTION ON STEADY MHD FLOW DUE TO A ROTATING DISK WITH HEAT TRANSFERAND OHMIC HEATING, Chemical Engineering Communications, 194:10, 1396-1407, DOI:10.1080/00986440701401545

To link to this article: http://dx.doi.org/10.1080/00986440701401545

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On the Effectivness of Ion Slip and UniformSuction or Injection on Steady MHD Flow Due

to a Rotating Disk with Heat Transferand Ohmic Heating

HAZEM ALI ATTIA

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

The steady flow and heat transfer of a conducting fluid due to the rotation of an infi-nite nonconducting porous disk in the presence of an axial uniform steady magneticfield are studied considering ion slip and ohmic heating. A uniform injection or suc-tion is applied through the surface of the disk. The relevant equations are solvednumerically using finite differences, and the solution shows that the inclusion ofion slip and the injection or suction through the surface of the disk gives some inter-esting results. It is found that the influence of the Hall and ion slip parameters on thevelocity components is more pronounced in the injection than in the suction case.Also, ohmic heating has a marked effect on the heat transfer rate and it is of interestto see the reversal of the sign of the heat transfer rate for some values of the ion slipand suction parameters.

Keywords Finite difference; Heat generation; Heat transfer; Hydromagneticflow; Numerical solution; Rotating disk flow

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). The extension of the steady hydro-dynamic problem to the transient state was done by Benton (1966).

The problem of heat transfer from a rotating disk at a constant temperature wasfirst considered by Millsaps and Pohlhausen (1952) for a variety of Prandtl numbersin the steady state. Sparrow and Gregg (1960) studied steady-state heat transferfrom a rotating disk maintained at a constant temperature to fluids at any Prandtlnumber.

The effect of uniform suction or injection through a rotating porous disk on thesteady hydrodynamic flow induced by the disk was investigated (Stuart, 1954;Kuiken, 1971; Ockendon, 1972). The extension of the problem studied in Stuart

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., 194:1396–1407, 2007Copyright # Taylor & Francis Group, LLCISSN: 0098-6445 print/1563-5201 onlineDOI: 10.1080/00986440701401545

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(1954), Kuiken (1971), and Ockendon (1972) to the transient state was done by Attia(2002).

The pioneering study of fluid flows near a bounding surface was carried out byPrandtl in 1904. His attention was directed toward control of the boundary layer onaerodynamic bodies, interest in which has continued up to the present. The controlof either a viscous or thermal boundary layer can be achieved by the use ofelectrically conducting working fluids with applied magnetic field. Therefore, it isof interest to study the influence of the magnetic field on both the flow and heattransfer properties of the boundary layers due to a rotating disk. Also, in recentyears, considerable interest has been shown in the phenomenon of heat transfer inmagnetohydrodynamic (MHD) flows, especially in connection with cooling of tur-bine blades and the skins of high-speed aircraft. It is, therefore, of interest to studythe electromagnetic effect on heat transfer and, in turn, on the cooling process ofsuch devices. These results are needed for the design of the wall and the coolingarrangements.

The influence of an external uniform magnetic field on the flow due to a rotatingdisk was studied (El-Mistikawy and Attia, 1990; El-Mistikawy et al., 1991) withoutconsidering the Hall effect or ion slip. In fact, the Hall effect is important when theHall parameter, which is the ratio between the electron-cyclotron frequency andthe electron-atom collision frequency, is high. This happens when the magneticfield is high or when the collision frequency is low (Sutton and Sherman, 1965).Furthermore, the masses of the ions and electrons are different and, in turn, theirmotions will be different. Usually, the diffusion velocity of electrons is larger thanthat of ions and, as a first approximation, the electric current density is determinedmainly by the diffusion velocity of the electrons. However, when the electromagneticforce is very large (such as in the case of strong magnetic field), the diffusion velocityof the ions may not be negligible (Sutton and Sherman, 1965). If we include thediffusion velocity of ions as well as that of electrons, we have the phenomena 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 valuesof 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 ion slip are important; they have a marked effect on the magnitude and directionof the current density and consequently on the magnetic-force term (Sutton andSherman, 1965).

Study of the rotating disk problem with the inclusion of the Hall current and ionslip was done by Attia (2003a) and Attia and Aboul-Hassan (2004). Attia (2003b)has also studied the effect of suction and injection on the transient flow of a conduct-ing fluid due to a rotating disk in the presence of an external uniform magnetic fieldneglecting the Hall current and ion slip.

In the present work the steady hydromagnetic flow of a viscous, incompressible,and electrically conducting fluid due to the uniform rotation of an infinite noncon-ducting porous disk in an axial uniform steady magnetic field is studied consideringthe Hall current, ion slip, and ohmic heating. The fluid is subjected to a uniform suc-tion or injection through the disk. The governing nonlinear differential equations aresolved numerically using finite difference approximations. Some interesting effectsfor the Hall current, ion slip, and suction or injection velocity on the velocity andtemperature fields are investigated.

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Basic Equations

The disk is assumed to be insulating and rotating in the z ¼ 0 plane about the z-axiswith a uniform angular velocity x. The fluid is assumed to be incompressible and hasdensity q, kinematic viscosity t, and electrical conductivity r. An external uniformmagnetic field is applied in the z-direction and has a constant flux density Bo. Themagnetic Reynolds number is assumed to be very small, so that the induced magneticfield is negligible. The electron-atom collision frequency is assumed to be relativelyhigh, so that the Hall effect and ion slip cannot be neglected (Sutton and Sherman,1965). A uniform injection or suction is applied at the surface of the disk for theentire range from large injection velocities to large suction velocities. 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 fluid motion is governed by the continuity equation, the Navier-Stokesequation, and the generalized Ohm’s law (Sutton and Sherman, 1965), which arerespectively given by

~rr:~VV ¼ 0

qð~VV : ~rrÞ~VV ¼ � ~rrpþ lr2~VV þ~JJ ^~BBo

~JJ ¼ r ~EE þ ~VV ^ B̂Bo � bð~JJ ^~BBoÞ þbBi

Boð~JJ ^~BBoÞx~BBo

� �

where ~EE is the electric field that results from charge separation and is in thez-direction. The last term in the last equation expresses the ion slip effect, whereb ¼ 1=nq is the Hall factor, n is the electron concentration per unit volume, �q isthe charge of the electron, and Bi is the ion slip parameter (Sutton and Sherman,1965). In cylindrical coordinates, the velocity vector is written as

~VV ¼ u~eer þ n~eeu þ w~eez

and the above set of equations of steady motion takes the form

u;r þu

rþ w;z ¼ 0 ð1Þ

q uu;r þ wu;z �n2

r

� �þ rB2

o

ða2 þ Be2Þ ðau� BenÞ þ p;r ¼ l u;rr þu;rr� u

r2þ u;zz

� �ð2Þ

q un;r þ wn;z þunr

� �þ rB2

o

ða2 þ Be2Þ ðan þ BeuÞ ¼ l n;rr þn;rr� n

r2þ n;zz

� �ð3Þ

qðuw;r þ ww;zÞ þ p;z ¼ lðw;rr þw;r

rþ w;zzÞ ð4Þ

where Be (¼rbBo) is the Hall parameter, which can take positive or negative values,and a ¼ 1þBeBi. The boundary conditions are given as

z ¼ 0; u ¼ 0; n ¼ rx;w ¼ wo ð5aÞ

z!1; u! 0; n ! 0; p! p1 ð5bÞ

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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 from the induced axialcomponent, as indicated in Equation (5b). We introduce von Karman transforma-tions (von Karman, 1921):

u ¼ rxF ; n ¼ rxG;w ¼ffiffiffiffiffiffiffixnp

H; z ¼ffiffiffiffiffiffiffiffiffin=x

pf; p� p1 ¼ �qnxP

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 rBo

2=qx, which represents the ratioof 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� F 2 þ G2 � c

ða2 þ Be2Þ ðaF � BeGÞ ¼ 0 ð7Þ

d2G

df2�H

dG

df� 2FG � c

ða2 þ Be2Þ ðaG þ BeFÞ ¼ 0 ð8Þ

d2H

df2�H

dH

df� dP

df¼ 0 ð9Þ

f ¼ 0;F ¼ 0;G ¼ 1;H ¼ S; ð10aÞ

f!1;F ! 0;G ! 0;P! 0; ð10bÞ

where S ¼ wo=ffiffiffiffiffiffiffixnp

is the uniform suction or injection parameter, which takesconstant negative values for suction and constant positive values for injection. Theabove system of Equations (6)–(8) with the prescribed boundary conditions givenby Equation (10) are sufficient to solve for the three components of flow velocity.Equation (9) can be used to solve for pressure distribution if required.

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

� �¼ k

@2T

@z2þ @

2T

@r2þ 1

r

@T

@r

� �þ rB2

o

a2 þ Be2ðu2 þ n2Þ ð11Þ

where T is the temperature of the fluid, cp is the specific heat at constant pressure ofthe fluid, and k is the thermal conductivity of the fluid. The last term in Equation (11)represents the ohmic heating. The boundary conditions for the energy problem arethat the temperature, by continuity considerations, equals Tw at the surface of thedisk. At large distances from the disk, T ! T1; where T1 is the temperature of theambient fluid. In terms of the nondimensional variable T ¼ ðT � T1Þ= ðTw � T1Þ

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and using von Karman transformations Equation (11) takes the form (the bar will bedropped):

1

Pr

@2T

@f2�H

@T

@fþ cEc

a2 þ Be2ðF2 þ G2Þ ¼ 0 ð12Þ

where Pr is the Prandtl number given by Pr ¼ cpl=k and Ec ¼ x2r2=cpðTw � T1Þ isthe Eckert number. The boundary conditions for the energy problem, in terms of Tand von Karman transformations, are expressed as

Tð0Þ ¼ 1; f!1 : T ! 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

rdTð0Þ

df

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

ffiffiffiffiffiffiffiffiffix=n

p=kðTw � T1Þ the last equation becomes

Nu ¼ � dTð0Þdf

The system of nonlinear ordinary differential equations (6)–(8) and (12) is solvedunder the conditions given by Equations (10) and (13) for the three components ofthe flow velocity and temperature distribution, using the Crank-Nicolson method(Ames, 1977). The resulting system of difference equations has to be solved in theinfinite domain 0 < f <1:A finite domain in the f-direction can be used instead,with fchosen large enough to ensure that the solutions are not affected by imposingthe asymptotic conditions at a finite distance. The independence of the results fromthe length of the finite domain and the grid density was ensured and successfullychecked by various trial-and-error numerical experiments. Computations were car-ried out for f1 ¼ 12, and it was found that a value of f1 ¼ 10 is adequate for theranges of the parameters studied here, which are chosen based on the selected valuesof similar parameters in previous work (Millsaps and Pohlhausen, 1952; Sparrowand Gregg, 1960; Sutton and Sherman, 1965; Attia, 2003a; Attia and Aboul-Hassan,2004). It should be mentioned that the results obtained herein reduce to thosereported by Sparrow and Gregg (1960) when S ¼ 0 and c ¼ 1 and also to those givenby Attia and Aboul-Hassan (2004) by setting Bi ¼ 0 in the present results. Thesecomparisons give validity to the present solution and lend confidence to the correct-ness of the solutions presented in this article.

Results and Discussion

The three velocity components F, G, and H are obtained at different values of f.These velocity components have some general characteristics that can be predicted

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from the basic equations. The value of H at a given f decreases as F in the regionbelow it increases. This follows from the continuity equation. The suction decreasesH (increases the axial flow towards the disk). It decreases F and G because as theaxial flow towards the disk is faster, the time during which the fluid is acquiringthe radial and azimuthal velocities by means of the centrifugal and viscosity forcesis shorter. Injection has the opposite effect; it increases the three velocity compo-nents. Neglecting the Hall and ion slip effects (Be ¼ 0, Bi ¼ 0), the applied uniformmagnetic field (defined in terms of the parameter c) represents the single effect of themagnetic field on the flow. It is clear from Equations (7) and (8) that the magneticfield has a damping effect on the radial and azimuthal velocity components due tothe magnetic resistive forces. Equation (6) then shows that the magnetic field has,in turn, a damping effect on the axial flow towards the disk. In the case of large suc-tion, the magnetic field has no significant effect on H, because F is already very smalland any change in its value due to magnetic damping cannot change H significantly.

The Hall parameter Be appears in the magnetic force terms, and its contribution,neglecting ion slip (Bi¼0 or Be¼0), is proportional to ðF � BeGÞ=ð1þ Be2Þ orðG þ BeFÞ=ð1þ Be2Þ. For small values of Be, the effect of Be on the numerator isstronger than its effect on the denominator. A small positive value of Be decreasesthe magnetic damping on F and increases the magnetic damping on G, thus increas-ing F and decreasing both H and G. A small negative value of Be decreases F andincreases both H and G. For large positive values of Be, the factor ðF � beGÞ mayturn out to be negative and the magnetic field has a propelling effect on F, whichmay exceed its hydrodynamic value, and thus the value of H is below its hydrody-namic value. For such large values of Be, the effect on G is due mainly to the factor1=ð1þ Be2Þ, which becomes very small and produces an increase in G. For largenegative values of Be, the argument is reversed. The magnetic damping on F isreduced due to the decrease in 1=ð1þ Be2Þ. Thus F increases but is still less thanits hydrodynamic value, and consequently H decreases but is more than its hydrody-namic value. The factor(G þBeF) may become negative and this pushes G above itshydrodynamic value, and thus the magnetic field has a propelling effect on G. Forvery large positive or negative values of Be, the magnetic force term decreases greatlyand the limit Be!1 or �1 corresponds to the hydrodynamic limit.

Considering ion slip, the parameter a appears in the magnetic force terms andits contribution is proportional to ðaF �BeGÞ=ða2þBe2Þ or ðaGþBeFÞ=ða2þBe2Þ.For small positive values of a, the effect of a on the numerator is stronger than itseffect on the denominator. Then, small positive a decreases F and G but increasesH. Large or small negative a increases F and G but decreases H. It should be pointedout that the value of adepends on the magnitude and sign of both parameters Beand Bi.

Figure 1 presents the profile of the radial velocity component F for variousvalues of the ion slip parameter Bi and the suction parameter S and for Be� 0and Be� 0, respectively. In these figures c ¼ 1. Figure 1 shows that for Be¼�0.5and Bi ¼ 0, the radial flow reverses its direction at a certain distance from the disk.Increasing Bi, for Be < 0, leads to a negative F for all f as a result of increasing theeffective conductivity (¼c=(a2þBe2), which increases the damping force on F.Figure 1(a) indicates also that for BeBi>0, increasing the magnitude of Bi increasesF due to a decrease in the effective conductivity, which decreases the damping forceon F. Large values of Bi reduce the effective conductivity more, which correspondsto the hydrodynamic case. Figure 1(b) indicates that when Be > 0 and Bi>0,

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increasing Bi decreases F for some f. This may be attributed to the fact that in themagnetic force term in Equation (7), the effect of Bi on the numerator is strongerthan its effect on the denominator, which increases the damping force on F andconsequently decreases F for some f. Also, for Bi < 0, increasing the magnitudeof Bi increases F for small f and then decreases it for larger f. This accountsfor a crossover in the F� f chart with Bi. It is of interest to see that the variationof F with Bi depends on f.

The effect of suction, as shown in Figures 1(c) and 1(d), is to decrease the mag-nitude of the velocity component F for all values of Be and Bi. This is due to the factthat suction provides an easier path for the flow through the wall than that in theradial direction and consequently decreases the radial velocity component. On theother hand, with injection, the radial velocity must carry away not only the incomingaxial flow, but also the injected fluid. Thus, the radial flow increases with injection.

Figure 1. Steady-state profile of F for various values of Be, Bi, and S: (a) S ¼ 0, Be� 0; (b)S ¼ 0, Be� 0; (c) S¼�1, Be� 0; (d) S¼�1, Be� 0; (e) S¼1, Be� 0; and (f) S¼1, Be� 0.

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Figure 1(d) shows that in the case of suction, the crossover in the F profiles occurs atlarger distances from the disk for all values of Bi. The effect of injection is to increaseF for all Be and Bi, as shown in Figures 1(e) and 1(f). It is clear from Figure 1(f) thatthe crossover point in the profiles of F occurs at smaller f.

Figure 2 presents the profile of the azimuthal velocity component G for variousvalues of the ion slip parameter Bi and the suction parameter S and for Be� 0 andBe� 0, respectively. In these figures, c ¼ 1. As shown in Figure 2(a), small negativevalues of Be increase G as a result of decreasing the magnetic damping. Increasing Bi,with Be < 0, decreases G, due to the increase in effective conductivity. Figure 2(a)shows also that for negative values of Bi, increasing the magnitude of Bi increasesG due to the decrease in the damping force on G. Figure 2(b) describes the same find-ings. For BeBi > 0, increasing Bi increases G, while for BeBi < 0, increasing themagnitude of Bi decreases G. Figures 2(c) and 2(d) show that the suction decreases

Figure 2. Steady-state profile of G for various values of Be, Bi, and S: (a) S¼0, Be� 0; (b) S¼0,Be� 0; (c) S¼�1, Be� 0; (d) S¼�1, Be � 0; (e) S¼1, Be� 0; and (f) S¼1, Be� 0.

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G for all Be and Bi as a result of decreasing F. However, injection increases G for allBe and Bi due to the corresponding increase in F.

Figure 3 presents the profile of the axial velocity component H for variousvalues of the ion slip parameter Bi and the suction parameter S and Be� 0 andBe� 0, respectively. In these figures, c ¼ 1. As shown in Figure 3(a), for Be ¼ �0.5and Bi ¼ 0, the axial velocity H reverses its direction at a certain f. Increasing Biincreases H, as a result of decreasing F, and reverses its direction for all f. Figure3(b) shows that for Be > 0, increasing the magnitude of Bi, in general, decreasesH as a result of increasing F. It is also shown in Figure 3(b) that the axial flow isalways towards the disk for all values of Bi. It is shown in Figures 3(c) and 3(d) thatsuction decreases H for all Be and Bi. Suction provides an easier path for theincoming fluid than the radial direction, which increases the axial flow towardsthe disk and more of the inflow goes directly into the porous disk. As a consequence,

Figure 3. Steady-state profile of H for various values of Be, Bi, and S: (a) S¼0, Be� 0;(b) S¼0, Be� 0; (c) S ¼ �1, Be� 0; (d) S¼�1, Be� 0; (e) S¼1, Be� 0; and (f) S¼1, Be� 0.

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H tends to become almost constant with f. Injection increases H and results in apositive H for all Be and Bi. This can be attributed to the fact that, with injection,the incoming flow towards the disk finds itself actively retarded by the out-flowingstream of injected fluid. The result is a decrease in the axial flow towards the disk,which means increasing H. The effect of Be and Bi on H is more pronounced forinjection than for suction.

Tables I and II present the variation of the Nusselt number Nu with the Hallparameter Be, the ion slip parameter Bi, and the suction parameter S for Pr ¼ 0.7and Pr ¼ 10, respectively, and for c ¼ 1. In these tables, we consider two cases:Ec ¼ 0 (neglecting ohmic heating) and Ec ¼ 0.2 (considering ohmic heating). It isclear that in the presence of ohmic heating (Ec ¼ 0.2), the heat transfer rate decreasesas compared with the case Ec ¼ 0 as a result of increasing temperature of the fluid

Table I. Variation of steady-state value of Nu with Be, Bi, and S for Pr ¼ 0.7

Ec ¼ 0 Ec ¼ 0.2

Nu S ¼ �1 S ¼ 0 S ¼ 1 S ¼ �1 S ¼ 0 S ¼ 1

Be ¼ 0, Bi ¼ 0 0.7409 0.2232 0.0262 0.6992 0.1646 �0.0396Be ¼ �0.5, Bi ¼ 0 0.7014 0.1692 0.0129 0.6647 0.1136 �0.0482Be ¼ �0.5, Bi ¼ 0.5 0.6769 0.1318 0.0059 0.6216 0.0474 �0.0860Be ¼ �0.5, Bi ¼ 1 0.6358 0.0819 0.0015 0.5466 �0.0556 �0.1471Be ¼ �0.5, Bi ¼ �0.5 0.7172 0.1963 0.0206 0.6913 0.1573 �0.0225Be ¼ �0.5, Bi ¼ �1 0.7281 0.2162 0.0279 0.7090 0.1875 �0.0040Be ¼ 0.5, Bi ¼ 0 0.7777 0.2777 0.0498 0.7445 0.2321 �0.0023Be ¼ 0.5, Bi ¼ 0.5 0.7732 0.2758 0.0503 0.7494 0.2425 0.0123Be ¼ 0.5, Bi ¼ 1 0.7708 0.2762 0.0516 0.7529 0.2509 0.0229Be ¼ 0.5, Bi ¼ �0.5 0.7859 0.2839 0.0512 0.7372 0.2184 0.0236Be ¼ 0.5, Bi ¼ �1 0.8009 0.2984 0.0567 0.7258 0.1996 0.0564

Table II. Variation of steady-state value of Nu with Be, Bi, and S for Pr ¼ 10

Ec ¼ 0 Ec ¼ 0.2

Nu S ¼ �1 S ¼ 0 S ¼ 1 S ¼ �1 S ¼ 0 S ¼ 1

Be ¼ 0, Bi ¼ 0 10.033 0.881 0.0034 9.4416 0.2346 �0.1728Be ¼ �0.5, Bi ¼ 0 10.010 0.712 0.00339 9.4849 0.1115 �0.1408Be ¼ �0.5, Bi ¼ 0.5 9.994 0.581 0.00339 9.2003 �0.3415 �0.2171Be ¼ �0.5, Bi ¼ 1 9.968 0.394 0.00339 8.6787 �1.1095 �0.3534Be ¼ �0.5, Bi ¼ �0.5 10.019 0.799 0.00337 9.6507 0.3841 �0.0967Be ¼ �0.5, Bi ¼ �1 10.026 0.859 0.00339 9.7541 0.5574 �0.0695Be ¼ 0.5, Bi ¼ 0 10.059 1.058 0.00338 9.5884 0.5659 �0.1381Be ¼ 0.5, Bi ¼ 0.5 10.054 1.043 0.00338 9.7165 0.6872 �0.0952Be ¼ 0.5, Bi ¼ 1 10.052 1.037 0.00339 9.7989 0.7702 �0.0687Be ¼ 0.5, Bi ¼ �0.5 10.066 1.092 0.0034 9.3785 0.3791 �0.2115Be ¼ 0.5, Bi ¼ �1 10.081 1.161 0.00339 9.0231 0.0827 �0.3411

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due to Joule dissipation. It is of interest to see the reversal of the sign of the heattransfer rate for some values of the parameters Bi and S. This is because of the effectof ohmic heating in increasing the temperature of the fluid, which exceeds the tem-perature of the plate. It is clear that, for Be < 0 and all values of S, increasing Bidecreases Nu as a result of decreasing the incoming axial flow at near-ambient tem-perature to the disk, which decreases the heat transfer from the surface of the disk,i.e., Nu. On the other hand, for Ec ¼ 0.2, Be > 0 and for all values of S, increasing Biincreases the incoming flow and then increases Nu. However, for Ec ¼ 0 and Be > 0the effect of Bi on Nu depends on the parameter S. For the suction case (S ¼ �1),increasing Bi decreases Nu, but for the injection case (S ¼ 1) increasing Bi increasesNu. The influence of Bi on Nu is clearer for Be > 0 than for Be < 0. Increasing suc-tion velocity increases Nu, but increasing injection velocity decreases Nu. This isexpected since suction brings the fluid at near-ambient temperature to the disk,which increases Nu, while injection has the reverse effect.

Conclusion

Steady MHD flow due to an infinite rotating disk in the presence of a uniform suc-tion or injection perpendicular to its plane is studied with heat transfer consideringthe Hall effect, ion slip, and ohmic heating. The inclusion of the Hall effect, ion slip,and uniform suction or injection reveals some interesting phenomena, and it is foundthat the signs of the Hall and ion slip parameters are important. It was found thatthe three velocity components reverse direction for certain values of the magneticfield and the Hall and ion slip parameters. The variation of the velocity componentswith ion slip depends on f, especially for positive values of the ion slip parameter.The influence of the Hall and ion slip parameters on the velocity components is morepronounced in the injection than in the suction case. The effect of the ion slip para-meter is more apparent for positive values of the Hall parameter than negativevalues. The heat transfer at the surface of the disk is found to depend on themagnitude and the sign of the Hall and ion slip parameters as well as the suctionor injection velocity. Also, ohmic heating has a marked effect on the heat transferrate, and it is of interest to see the reversal of the sign of the heat transfer rate forsome values of the ion slip and suction parameters.

References

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Attia, H. A. (2002). On the effectiveness of uniform suction-injection on the unsteady flow dueto a rotating disk with heat transfer, Int. Commun. Heat Mass Transfer, 29 (5), 653–661.

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El-Mistikawy, T. M. A., Attia, H. A., and Megahed, A. A. (1991). The rotating disk flow inthe presence of weak magnetic field, in Proceedings of the 4th Conference on Theoreticaland Applied Mechanics, Cairo, Egypt, Nov. 5–7, 69–82.

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