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ANALYTICAL SOLUTION FOR FLOW OF ADUSTY FLUID IN A CIRCULAR PIPE WITHHALL AND ION SLIP EFFECTSHazem 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): ANALYTICAL SOLUTION FOR FLOW OF A DUSTY FLUID INA CIRCULAR PIPE WITH HALL AND ION SLIP EFFECTS, Chemical Engineering Communications, 194:10,1287-1296

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Analytical Solution for Flow of a Dusty Fluid ina Circular Pipe with Hall and Ion Slip Effects

HAZEM ALI ATTIA

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

Unsteady flow of a dusty electrically conducting incompressible viscous fluid in a cir-cular pipe is studied considering ion slip. A constant pressure gradient in the axialdirection and a uniform magnetic field directed perpendicular to the flow directionare applied. The particle phase is assumed to behave as a viscous nonconductingfluid. A series solution for the governing equations of both fluid and particle phasesis obtained for the whole range of physical parameters.

Keywords Dusty fluid; Hall effect; Ion slip; MHD; Particle phase

Introduction

The flow of a dusty and electrically conducting fluid through a circular pipe in thepresence of a transverse magnetic field has important applications such as magneto-hydrodynamic (MHD) generators, pumps, accelerators, and flow meters. The per-formance and efficiency of these devices are influenced by the presence ofsuspended solid particles in the form of ash or soot as a result of the corrosionand wear activities and=or the combustion processes in MHD generators and plasmaMHD accelerators. When the particle concentration becomes high, mutual particleinteraction leads to higher particle-phase viscous stresses and can be accounted forby endowing the particle phase with the so-called particle-phase viscosity. Therehave been many studies dealing with theoretical modeling and experimental mea-surements of the particle-phase viscosity in a dusty fluid (Soo, 1969; Gidaspowet al., 1989; Grace, 1982; Sinclair and Jackson, 1989).

The flow of a conducting fluid in a circular pipe has been investigated by manyresearchers (Gadiraju et al., 1992; Dube and Sharma, 1975; Ritter and Peddieson,1977; Chamkha, 1994). Gadiraju et al. (1992) investigated steady two-phase verticalflow in a pipe. Dube and Sharma (1975) and Ritter and Peddieson (1977) reportedsolutions for unsteady dusty-gas flow in a circular pipe in the absence of a magneticfield and particle-phase viscous stresses. Chamkha (1994) obtained exact solutionsthat generalized the results reported by Dube and Sharma (1975) and Ritter andPeddieson (1977) by the inclusion of magnetic and particle-phase viscous effects.Attia (2003) extended the problem discussed by Chamkha (1994) to the case ofnon-Newtonian fluid. It should be noted that in the above studies the ion slip effectwas ignored.

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., 194:1287–1296, 2007Copyright # Taylor & Francis Group, LLCISSN: 0098-6445 print/1563-5201 onlineDOI: 10.1080/00986440701399780

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In the present study, the unsteady flow of a dusty viscous incompressibleelectrically conducting fluid in a circular pipe is investigated considering ion slip.The carrier fluid is assumed viscous, incompressible, and electrically conducting.The particle phase is assumed to be incompressible, pressureless, and electricallynonconducting. The flow in the pipe starts from rest through the application of aconstant axial pressure gradient. Series solution for the governing momentumequations for both the fluid and particle phases is obtained for the whole range ofphysical parameters.

Governing Equations

Consider the unsteady laminar axisymmetric horizontal flow of a dusty conductingfluid through an infinitely long pipe of radius d driven by a constant pressure gradient.A uniform magnetic field is applied perpendicular to the flow direction. The Hallcurrent and ion slip are taken into consideration, and the magnetic Reynolds numberis assumed to be very small; consequently, the induced magnetic field is neglected(Sutton and Sherman, 1965; Crammer and Pai, 1973). We assume that both phasesbehave as viscous fluids and that the volume fraction of suspended particles is finiteand constant (Chamkha, 1994).

To formulate the governing equations for this investigation, the balance laws ofmass and linear momentum are considered along with information about interfacialand external body forces and stress tensors for both phases. The balance laws ofmass (for the fluid and particulate phases, respectively) can be written as

@t/� ~rr � ðð1� /Þ~VVÞ ¼ 0; ð1aÞ

@t/þ ~rr � ð/~VVpÞ ¼ 0 ð1bÞ

where t is time, / is the particulate volume fraction, ~rr is the gradient operator, ~VV isthe fluid-phase velocity vector, and ~VVp is the particulate-phase velocity vector. Thetrue densities for both phases are assumed constant.

The balance laws of linear momentum (for the fluid and particulate phases,respectively) can be written as

qð1� /Þð@t~VV þ ~VV � ~rr~VVÞ ¼ ~rr � r$ �~ff þ~bb; ð2aÞ

qp/ð@t~VVp þ ~VVp � ~rr~VVpÞ ¼ ~rr � r$p �~ff þ~bbp ð2bÞ

where q is the fluid-phase density, qp is the particle-phase density, r$ is the fluid-phase stress tensor, ~ff is the interphase force per unit volume associated with therelative motion between the fluid and particle phases,~bb is the fluid-phase body forceper unit volume, and ~bbp is the particle-phase body force per unit volume.

Along with Equations (1) and (2), the following constitutive equations are used:

r$ ¼ ð1� /Þð�PI

$ þ lð ~rr~VV þ ~rr~VVT ÞÞ; ð3aÞ

r$

p ¼ /lpð ~rr~VVp þ ~rr~VVTp ÞÞ; ð3bÞ

f ¼ Nqp/ðV � VpÞ; ð3cÞ~bb ¼ �1=qð~JJ ^~BBoÞ; ð3dÞ

~bbp ¼~00 ð3eÞ

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where P is the fluid pressure, I$

is the unit tensor, l fluid dynamic viscosity, lp is theparticle-phase dynamic viscosity, N is the inverse relaxation time (the inverse timerequired by a particle to reduce its velocity relative to that of the fluid by e�1 fromits initial value (Chamkha, 1994)), ~JJ is the electric current density vector, ~BBo is theuniform applied magnetic induction field vector, and a transposed T denotes thetranspose of a second-order tensor.

If the Hall term is retained, the current density ~JJ is given by (Sutton andSherman, 1965: Crammer and Pai, 1973):

~JJ ¼ e ~VV ^~BBo � bð~JJ ^~BBoÞ þbBi

Boð~JJ ^~BBoÞ ^~BBo

� �ð4Þ

where e is the electric conductivity of the fluid, b is the Hall factor (Sutton andSherman, 1965), and Bi is the ion slip parameter (Sutton and Sherman, 1965). Equa-tion (4) may be solved in ~JJ to obtain the electromagnetic Lorentz force in the form(Sutton and Sherman, 1965):

~JJ �~BBo ¼ �eB2

oBi

Bi2 þ Be2V~kk ð5Þ

where Be ¼ e b Bo is the Hall parameter and ~kk is a unit vector along the z-direction.Solving Equation (5) for~JJ and substituting in Equations (1)–(3) with expanding yields

q@V

@t¼ � @P

@zþ l

r

@

@rr@V

@r

� �þ

qp/

1� /NðVp � VÞ � rB2

oBi

Bi2 þ Be2V ð6Þ

qp

@Vp

@t¼

lp

r

@

@rr@Vp

@r

� �þ qpNðV � VpÞ ð7Þ

where r is the distance in the radial direction and @P=@z is the fluid pressure gradient. Inthis work, q, qp, and / are all constants. In reality the volume fraction of particles in thesuspension is not constant. The assumption of constant and uniform distribution of / isan idealization and is applicable in limited situations. This is done herein so as to allowthe governing equations to be solved in closed form. It is hope that the present analyticalresults will be of use in validating computer routines for numerical solutions of morecomplex two-phase particulate suspension flows in channels and in stimulating neededexperimental work in this area.

It should be pointed out that the particle-phase pressure is assumed negligibleand that the particles are being dragged along with the fluid phase.

The initial and boundary conditions of the problem are given as

Vðr; 0Þ ¼ 0; Vpðr; 0Þ ¼ 0; ð8aÞ@Vð0; tÞ@r

¼ 0;@Vpð0; tÞ

@r¼ 0; Vðd; tÞ ¼ 0; Vpðd; tÞ ¼ 0 ð8bÞ

where d is the pipe radius.Equations (7)–(8) constitute an initial-value problem, which can be made dimen-

sionless by introducing the following dimensionless variables and parameters:

�rr ¼ r

d; �tt ¼ tl

qd2; Go ¼ �

@P

@z; k ¼

qp/

qð1� /Þ ;

Vðr; tÞ ¼ lVðr; tÞGod2

; V pðr; tÞ ¼lVpðr; tÞ

God2;

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and a ¼ Nd2q=l is the inverse Stokes’ number, B ¼ qlp=lqp is the viscosity ratio,and Ha ¼ Bod

ffiffiffiffiffiffiffiffir=l

pis the Hartmann number (Sutton and Sherman, 1965).

By introducing the above dimensionless variables and parameters as well as theexpression of the fluid viscosity defined above, Equations (7)–(8) can be written as(the bars are dropped):

@V

@t¼ 1þ @

2V

@r2þ 1

r

@V

@rþ kaðVp � VÞ � Ha2Bi

Bi2 þ Be2V ð9Þ

@Vp

@t¼ B

@2Vp

@r2þ 1

r

@Vp

@r

� �þ aðV � VpÞ ð10Þ

Vðr; 0Þ ¼ 0; Vpðr; 0Þ ¼ 0; ð11aÞ

@Vð0; tÞ@r

¼ 0;@Vpð0; tÞ

@r¼ 0; Vð1; tÞ ¼ 0; Vpð1; tÞ ¼ 0 ð11bÞ

The volumetric flow rates and skin-friction coefficients for both the fluid andparticle phases are defined, respectively, as (Chamkha, 1994):

Q¼ 2pZ 1

0

rVðr; tÞdr; Qp ¼ 2pZ 1

0

rVpðr; tÞdr; C ¼�@Vð1; tÞ@r

; Cp ¼�BK@Vpð1; tÞ

@r

ð12Þ

Results and Discussion

The solutions for V and Vp that satisfy the initial-value problem represented byEquations (9)–(11) can be assumed to take the forms

Vðr; tÞ ¼X1n¼1

HnðtÞJoðknrÞ; ð13aÞ

Vpðr; tÞ ¼X1n¼1

HpnðtÞJoðknrÞ; ð13bÞ

where kn are the roots of the equation JoðknÞ ¼ 0 (Jo being the zero-order Besselfunction of the first kind (Selvadurai, 2000)). Substituting Equations (13a) and(13b) and their derivatives into Equations (9) and (10) (with the inhomogeneous partof these equations represented by a series in JoðknrÞ, multiplying by rJoðkmrÞ, to takeadvantage of the orthogonality property of Bessel functions) and then integratingwith respect to r from 0 to 1 yield

_HHn þ ðk2n þ kaþMÞHn � kaHpn ¼ bn ð14Þ

_HHpn þ ðBk2n þ aÞHpn � aHn ¼ 0 ð15Þ

where a dot denotes ordinary differentiation with respect to t, M ¼ Ha2Bi=ðBi2 þ Be2Þ, and bn ¼ 2=ðknJ1ðknÞÞ (J1 being the first-order Bessel function of thefirst kind). These equations are combined to give

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€HHn þ ðk2nð1þ BÞ þ ð1þ kÞaþMÞ _HHn þ ðaðk2

n þMÞþ Bk2

nðk2n þ kaþMÞÞHn ¼ ðaþ Bk2

nÞbn ð16Þ

The solution of the above linear ordinary inhomogeneous differential equationsubject to

Hnð0Þ ¼ 0 and Hpnð0Þ ¼ 0 ð17Þ

can be obtained by the usual method of solving such equations to yield

Hn ¼ c1 expðs1tÞ þ c2 expðs2tÞ þ bnðaþ Bk2nÞ

ðaðk2n þMÞ þ Bk2

nðk2n þ kaþMÞÞ

ð18Þ

where s1 and s2 are the roots of the quadratic auxiliary equation:

s2 þ ðk2nð1þ BÞ þ ð1þ kÞaþMÞsþ aðk2

n þMÞ þ Bk2nðk

2n þ kaþMÞ ¼ 0 ð19Þ

and are given by

s1 ¼ð�Aþ ðA2 � 4DÞ1=2Þ

2ð20aÞ

s2 ¼ð�A� ðA2 � 4DÞ1=2Þ

2ð20bÞ

A ¼ k2nð1þ BÞ þ ð1þ kÞaþM ð21aÞ

D ¼ aðk2n þMÞ þ Bk2

nðk2n þ kaþMÞ ð21bÞ

c1 ¼ �bnð1þ s2ðaþ Bk2nÞ=ðaðk

2n þMÞ þ Bk2

nðk2n þ kaþMÞÞÞ=ðs2 � s1Þ ð22aÞ

c2 ¼ bnð1þ s1ðaþ Bk2nÞ=ðaðk2

n þMÞ þ Bk2nðk2

n þ kaþMÞÞÞ=ðs2 � s1Þ ð22bÞ

The corresponding solution for Hpn can be shown to be

Hpn¼1=ðkaÞðc1ðs1þk2nþkaþMÞexpðs1tÞþc2ðs2þk2

nþkaþMÞexpðs2tÞÞþbn=ðkaÞððk2

nþkaþMÞðaþBk2nÞ=ðaðk

2nþMÞþBk2

nðk2nþkaþMÞÞ�1Þ ð23Þ

With the solutions for Hn and Hpn known, then V and Vp can be determinedfrom Equation (13). The solutions for Q, Qp, C, and Cp can be calculated from

Q ¼ 4pX1n¼1

Hn=ðbnk2n=ðkaÞÞ; Qp ¼ 4p

X1n¼1

Hpn=ðbnk2n=ðkaÞÞ; ð24aÞ

C ¼ 2X1n¼1

Hn=ðbn=ðkaÞÞ; Cp ¼ 2BkX1n¼1

Hpn=ðbn=ðkaÞÞ; ð24bÞ

It should be noted that Equation (23) is obtained by substituting Equations (13a)and (13b) and their derivatives into Equation (12).

The series solutions reported earlier are numerically evaluated and graphicallyplotted to elucidate the effects of the various physical parameters on the solutions.

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Computations have been made for a ¼ 1 and k ¼ 10. It should be mentioned thatthe convergence of the series solutions, given by Equation (13), and the corre-sponding results obtained herein is checked by comparison with other work. Itis found that the unsteady results reduce to those reported by Dube and Sharma(1975) and Ritter and Peddieson (1977) for the cases of nonmagnetic and inviscidparticle phase and Chamkha (1994) for the case where the Hall current and ionslip are neglected. To further check the exact solutions presented earlier, the gov-erning equations are solved numerically using a finite-difference scheme. Thenumerical results are found to be in excellent agreement with the exact solutions.These comparisons lend confidence in the accuracy and correctness ofthe solutions and, in turn, in the convergence of the two series defining the exactsolution.

Figures 1(a) and (b) present the time evolution of the velocity of the fluid V anddust particles Vp at the center of the pipe, respectively, for various values of theion slip parameter Bi and the Hall parameter Be and for Ha ¼ 3 and B ¼ 0.5.

Figure 1. Effect of the parameters Be and Bi on the time evolution of: (a) V at r ¼ 0; (b) Vp

at r ¼ 0.

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Both V and Vp increase with time and V reaches the steady state faster than Vp for allvalues of Be and Bi. It is clear from the figures that increasing Be or Bi increases bothV and Vp while its effect on their steady-state times can be neglected. This is due tothe decrease in the effective conductivity, which reduces the damping magnetic forceon V.

Figures 2(a) and 2(b) present the time evolution of the velocity of the fluid Vand dust particles Vp at the center of the pipe, respectively, for various values ofthe ion slip parameter Bi and the Hartmann number Ha and for Be ¼ 1 andB ¼ 0.5. It is clear that increasing Ha decreases V and Vp and their steady-statetimes for all values of Bi due to the increase in the damping magnetic force. Thefigures indicate also that the effect of Bi on V and Vp becomes more pronouncedfor higher values of Ha.

Figures 3(a) and (b) present the time evolution of the velocity of the fluid V anddust particles Vp at the center of the pipe, respectively, for various values of the ionslip parameter Bi and the viscosity ratio B and for Be ¼ 1 and Ha ¼ 1. The figures

Figure 2. Effect of the parameters Ha and Bi on the time evolution of: (a) V at r ¼ 0; (b) Vp

at r ¼ 0.

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indicate that increasing B decreases both V and Vp and their steady-state times for allvalues of Bi. The effect of the parameter Bi on V and Vp is more apparent for highervalues of the parameter B.

Tables I and II present the steady-state values of the fluid-phase volumetricflow rate Q, the particle-phase volumetric flow rate Qp, the fluid-phase skin fric-tion coefficient C, and the particle-phase skin friction coefficient Cp for variousvalues of the parameters Bi and B and for Ha ¼ 1 and Ha ¼ 5, respectively.In these tables Be ¼ 1. It is shown that increasing Bi increases Q, Qp, C, andCp for all values of B and Ha. Also, increasing B decreases Q, Qp, and C, butincreases Cp for all values of Bi and Ha. This is due to the explicit dependenceof Cp on B as indicated in Equation (7). It is clear also from the tables thatthe effect of B on the quantities Q, Qp, C, and Cp is more pronounced for highervalues of Ha, while the effect of Bi on these quantities is more apparent forsmaller values of Ha.

Figure 3. Effect of the parameters B and Bi on the time evolution of: (a) V at r ¼ 0; (b) Vp atr ¼ 0.

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Conclusions

The transient MHD flow of a particulate suspension in an electrically conductingfluid through a circular pipe is studied considering ion slip. A series solution forthe governing partial differential equations is obtained for the whole range of the

Table II. Steady-state values of Q, Qp, C, Cp for various values ofBi and B and for Ha ¼ 5, Be ¼ 1

Bi ¼ 0 Bi ¼ 1 Bi ¼ 3

B ¼ 0Q 0.0929 0.1057 0.1358Qp 0.0742 0.0835 0.1051C 0.2246 0.2415 0.2797Cp 0 0 0

B ¼ 0.5Q 0.0736 0.0816 0.0995Qp 0.0193 0.0214 0.0260C 0.1944 0.2056 0.2294Cp 0.1159 0.1274 0.1525

B ¼ 1Q 0.0705 0.0779 0.0939Qp 0.0106 0.0117 0.0142C 0.1909 0.2015 0.2234Cp 0.1264 0.1384 0.1645

Table I. Steady-state values of Q, Qp, C, Cp for various values of Biand B and for Ha ¼ 1, Be ¼ 1

Bi ¼ 0 Bi ¼ 1 Bi ¼ 3

B ¼ 0Q 0.2120 0.2142 0.2178Qp 0.1569 0.1583 0.1607C 0.3697 0.3721 0.3763Cp 0 0 0

B ¼ 0.5Q 0.1391 0.1401 0.1419Qp 0.0363 0.0366 0.0370C 0.2786 0.2798 0.2819Cp 0.2070 0.2085 0.2108

B ¼ 1Q 0.1286 0.1295 0.1310Qp 0.0193 0.0195 0.0197C 0.2675 0.2685 0.2704Cp 0.2196 0.2210 0.2234

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magnetic field parameter Ha, the Hall parameter Be, the ion slip parameter Bi, andthe particle-phase viscosity B. It is shown that increasing the magnetic field or theviscosity ratio decreases the fluid and particle velocities, while increasing the Hallparameter or the ion slip parameter increases both velocities. It is found that theeffect of ion slip on the fluid and particle velocities is more apparent for higher valuesof the magnetic field or the viscosity ratio.

References

Attia, H. A. (2003). Can. J. Phys., 81(3).Chamkha, A. J. (1994). Mech. Res. Commun., 21(3), 281.Crammer, K. R. and Pai, S.-I. (1973). Magnetofluid Dynamics for Engineers and Applied

Physicists, McGraw-Hill, New York.Dube, S. N. and Sharma, C. L. (1975). J. Phys. Soc. Jpn., 38, 298.Evans, G. A., Blackledge, J. M., and Yardley, P. D. (2000). Numerical Methods for Partial

Differential Equations, Springer Verlag, New York.Gadiraju, M., Peddieson, J., and Munukutla, S. (1992). Mech. Ress Commun., 19(1), 7.Gidaspow, D., Tsuo, Y. P., and Luo, K. M. (1989). Paper presented at Fluidization IV,

International Fluidization Conference, Banff, Alberta, Canada, May.Grace, J. R. (1982). Fluidized-bed hydrodynamics, in: Handbook of Multiphase Systems,

ed. G. Hetsroni, ch. 8.1, McGraw-Hill, New York.Ritter, J. M. and Peddieson, J. (1977). Paper presented at the Sixth Canadian Congress

of Applied Mechanics.Selvadurai, A. P. S. (2000). Partial Differential Equations in Mechanics, vol. 2, Springer

Verlag, New York.Sinclair, J. L. and Jackson, R. (1989). AIChE J., 35, 1473.Soo, S. L. (1969). Appl. Sci. Res., 21, 68.Sutton, G. W. and Sherman, A. (1965). Engineering Magnetohydrodynamics, McGraw-Hill,

New York.

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