Technical Note

Thermal dispersion effect on MHD flow of dusty gasand dust particles through hexagonal channel

Parveen Sharma, C.L. Varshney *

Department of Post-Graduate Studies and Research in Mathematics and Computer Science, S. V. College, Aligarh 202001 (UP), India

Received 29 September 2002

Abstract

In the field of dynamics for a dusty fluid, the volume of the dust particles and flow behaviour of particles in different

conditions is very important in engineering problems such as atmospheric fallout, nuclear reactor, powder technology,

performance of solid fuel rocket nozzles, air craft icing and so many others. An analysis is presented in this paper to

study the effects of thermal dispersion and Viscous dissipation on unsteady flow of a viscous incompressible dusty gas

through a hexagonal channel of uniform cross section under the influence of magnetic field and time dependent pressure

gradient. The results show the change in velocity profile of gas and particles in the presence of magnetic field with time,

thermal dispersion and volume fraction /.� 2003 Elsevier Science Ltd. All rights reserved.

1. Introduction

The laminar flow of dusty viscous fluid with different

initial and boundary conditions was studied [1–4] due to

its applications in Engineering Problems. Most of these

researches neglected the effects of thermal dispersion and

viscous dissipation on unsteady flow of viscous incom-

pressible dusty gas with volume fraction. Rudinger [5]

showed the error thus introduced from significant to

large. Nayfeh [6] developed the equation of motion of

dusty fluid taking into account the volume fraction of

dust particles.

In the Present Paper, we study the effects of thermal

dispersion and viscous dissipation on unsteady MHD

flow of a dusty gas through a hexagonal channel. Ex-

plicit expressions of velocities for both the gas and dust

particles have been obtained in exact form by using in-

tegral transform techniques. The effects of magnetic field

and thermal dispersion on the unsteady flow of dusty

gas have been shown graphically and presented in tab-

ular form.

2. Formulation

Let us consider the effects of thermal dispersion and

viscous dissipation on the motion of a dusty fluid con-

sidering volume fraction through a hexagonal channel of

uniform cross section under the influence of magnetic

field. Then, the governing equations are written in the

following form:

qð1� /Þ ouot

¼ ð1� /Þ�� op

ozþ l

o2uox2

�þ o2u

oy2

��

þ KN0ðv� uÞ � rb20uq

þ 1qGx ð1Þ

N0movot

¼ /

�� op

ozþ l

o2uox2

�þ o2u

oy2

��þ KN0ðu� vÞ

ð2Þ

where q is the density of the gas, / is the volume fractionof the dust particles, N0 is the number density of the*Corresponding author.

0017-9310/03/$ - see front matter � 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0017-9310(02)00503-3

International Journal of Heat and Mass Transfer 46 (2003) 2511–2514

www.elsevier.com/locate/ijhmt

particles, m is the mass of each dust particle, K is the

stoke�s resistance coefficient, r is the electrical conduc-tivity, Gx is the resultant body force on the gas. u and vrepresent the velocity of the gas and dust particles. In-

troducing the following non-dimensional quantities.

Introduce: x ¼ ax0, y ¼ ay0, z ¼ az0, pa2 ¼ qmp0, tm ¼a2t0, au ¼ mu0, M ¼ rb2

0

qð1�/Þ and S ¼ Gxqð1�/Þ,

av ¼ mv0

Eqs. (1) and (2) become

ouot

¼ � opoz

þ o2uox2

�þ o2u

oy2

�þ e1ðm � uÞ �Muþ S ð3Þ

ovot

¼ /0�� op

ozþ o2u

ox2

�þ o2u

oy2

��þ cðu� mÞ ð4Þ

where f ¼ mN0q

;/0 ¼ /f; c ¼ ka2

mm; e ¼ c

ð1� /Þ ; e1 ¼ f e;

ð5Þ

� opoz

¼ f ðtÞ ðan arbitrary function of timeÞ

S ¼ fGx

and c is the spin gradient viscosity. The correspondingnon-dimensional initial and boundary conditions are

ðiÞ uðx; y; tÞ ¼ 0 ¼ mðx; y; tÞ for t6 0 ð6Þ

ðiiÞ uðx; y; tÞ ¼ 0

¼ mðx; y; tÞ on the boundary for tP 0

ð7Þ

Let

x1 ¼ y; x2 ¼ y �p3x and x3 ¼ y þp

3x ð8Þ

Under the transformation of Eq. (8), Eqs. (3) and (4)

become

ouot

¼ f ðtÞ þ o2

ox21

�þ 4o

2

ox22þ 4o

2

ox23þ 2 o2

ox1 ox2þ 2 o2

ox1 ox3

� 4 o2

ox2 ox3

�uþ e1ðm � uÞ �Muþ S ð9Þ

omot

¼ /0 f ðtÞ�

þ o2

ox21

�þ 4o

2

ox22þ 4o

2

ox23þ 2 o2

ox1 ox2

þ 2 o2

ox1 ox3� 4 o2

ox2 ox3

�u�þ cðu� vÞ ð10Þ

Subject to the initial and boundary conditions.

ðiÞ uðx1; x2; x3; tÞ ¼ 0

¼ mðx1; x2; x3; tÞ for t6 0; tP 0 at

x1 ¼ffiffiffi3

p

2; x2 ¼ �

ffiffiffi3

p¼ x3: ð11Þ

ðiiÞ ouox1

¼ 0 ¼ omox1

;ouox2

¼ 0 ¼ omox2

;ouox3

¼ 0 ¼ omox3

ð12Þ

We use the technique of integral transforms to solve

the problem as follows:

On account of u and v being an even function of x1 x2x3 the finite Fourier sine transforms vanish. MultiplyingEqs. (9) and (10) by cosðPpx1Þ cosðQqx2Þ cosðRrx3Þ, andthen integrate it with the limit 0 to

ffiffi3

p

2, 0 to

p3, and 0 top

3 and using conditions. (11) and (12), we get.

oUot

¼ apqrf ðtÞ � bpqrU þ e1ðV � UÞ �MU þ S; ð13Þ

oVot

¼ /1 apqrf ðtÞ�

� bpqrUþ cðu� vÞ; ð14Þ

where apqr ¼ð�1Þpþqþr

PpQqRrand bpqr ¼ P 2p þ Q2

q þ R2r ð15Þ

and

u ¼ v

¼Z ffiffiffiffiffi

3=2p

0

Z ffiffi3

p

0

Z ffiffi3

p

0

uðx1; x2; x3; tÞ cosðPpx1Þ

cosðQqx2Þ cosðRrx3ÞdX1 dX2 dX3 ð16Þ

Solving Eqs. (13) and (14) by using Laplace transforms

under the transformed initial condition U ¼ 0 ¼ V and

t ¼ 0, we get

SU ¼ apqr�ff ðSÞ � bpqrU ¼ e1ðV � UÞ �MU þ S ð17Þ

SV ¼ /0½apqr�ff ðSÞ � bpqrU þ cðU � V Þ ð18Þ

where U , V , �ff ðsÞ are the Laplace transform of respective

quantities.

U ¼ apqrf 0ðSÞðS1 � S2Þ

S1 þ eS � S1

�� S2 þ eS � S2

�ð19Þ

V ¼ apqr�ff ðSÞðS1 � S2Þ

S1/0 þ e

S � S1

"� S2/

0 þ eS � S2

#ð20Þ

S1 and S2 are two roots of quadratic equation

S2 þ ðbpqr þ e1 þ c þM � SÞ þ ebpqr ¼ 0: ð21Þ

and hence,

S1 ¼ � 12

ðbpqr�

þ e1 þ c þ M � SÞ

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðbpqr þ e1 þ c þM � SÞ2 � 4ebpqr

n or �ð22Þ

2512 P. Sharma, C.L. Varshney / International Journal of Heat and Mass Transfer 46 (2003) 2511–2514

S2 ¼ � 12

ðbpqr�

þ e1 þ c þ M � SÞ

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðbpqr þ e1 þ c þM � SÞ2 � 4ebpqr

n or �

ð23Þ

By applying convolution theorem and put f ðtÞ ¼ C,and C is an absolute constant, we get

u ¼ 16C

3ffiffiffi3

pX1

p¼q¼r¼0

apqrbpqr

1

�� 1

S1 � S2fðS1 þ bpqrÞeS2t

� ðS2 þ bpqrÞeS1tg�cosðPpX1Þ cosðQqX2Þ cosðRrX3Þ

ð24Þ

v ¼ 16C

3ffiffiffi3

pX1

p¼q¼r¼0

apqrbpqr

1

�� 1

S1 � S2fðS1 þ /0bpqrÞeS2t

� ðS2 þ /0bpqrÞcS1tg�cosðPpX1Þ cosðQqX2Þ cosðRrX3Þ

ð25Þ

3. Discussion

Table 1 and Figs. 1–4 are self explanatory. Here, the

variations of velocity of dust and gas particles for vari-

ous values of the parameters such as the thermal dis-

persion parameter S, the magnetic parameter M , and thevolume fraction / are shown. These figures and the tableclearly show that the gas particles move faster than the

Table 1

Variation of U=C and V =C for various values of U, X ,M , and S

S X M U U=C V =C

0 0 5 0 0.1586 0.1499

0.04 0.1595 0.1518

0.08 0.1604 0.1536

10 0 0.1498 0.1393

0.04 0.1511 0.1417

0.08 0.1524 0.144

0.2 5 0 0.1474 0.1394

0.04 0.1482 0.1412

0.08 0.1491 0.1429

10 0 0.1394 0.1298

0.04 0.1406 0.132

0.08 0.1418 0.1341

1 0 5 0 0.1604 0.1522

0.04 0.1612 0.154

0.08 0.162 0.1557

10 0 0.1517 0.1416

0.04 0.153 0.1438

0.08 0.1542 0.1461

0.2 5 0 0.1491 0.1415

0.04 0.1498 0.1432

0.08 0.1505 0.1447

10 0 0.1412 0.1319

0.04 0.1423 0.134

0.08 0.1434 0.136

2 0 5 0 0.1621 0.1545

0.04 0.1629 0.1561

0.08 0.1636 0.1577

10 0 0.1537 0.1438

0.04 0.1548 0.146

0.08 0.156 0.1482

0.2 5 0 0.1506 0.1436

0.04 0.1513 0.1452

0.08 0.1519 0.1466

10 0 0.1429 0.1339

0.04 0.144 0.136

0.08 0.145 0.1379Fig. 1. Velocity profile for f ¼ 1:0, g ¼ 5:66, S ¼ 1 at

M ¼ 0; 5; 10.

P. Sharma, C.L. Varshney / International Journal of Heat and Mass Transfer 46 (2003) 2511–2514 2513

dust particles, whereas the increasing values of the

magnetic parameter M decelerate the velocities of both

the gas and the dust particles, on the other hand in-

creasing values of thermal dispersion parameter S in-

crease the velocities of both the gas and the dust

particles. Also the similar effect on the velocities of both

the gas and dust particles is observed when the volume

fraction / increases.

References

[1] Mitra, Acta Cinecia Indica VI(m) (3) (1980) 144.

[2] D.H. Michael, P.W. Norey, Quart. J. Mech. Appl. Maths 21

(1968) 375.

[3] P.G. Saffman, J. Fluid Mech. 13 (1962) 120.

[4] M. Gupta, O.P. Varshney, J. Engng. Appl. Sci. 2 (1983)

277.

[5] G. Rudinger, A. J. A. A. J. 3 (1965) 1217.

[6] A.H. Nayfeh, A. J. A. A. J. 4 (1966) 1868.

Fig. 2. Velocity profile for f ¼ 1:0, g ¼ 5:66, S ¼ 1 at

M ¼ 0; 5; 10.

Fig. 3. Velocity profile for f ¼ 1:0, g ¼ 5:66, M ¼ 0 at

S ¼ 0; 1; 2.

Fig. 4. Velocity profile for f ¼ 1:0, g ¼ 5:66, M ¼ 0 at

S ¼ 0; 1; 2.

2514 P. Sharma, C.L. Varshney / International Journal of Heat and Mass Transfer 46 (2003) 2511–2514