Advances in Pure Mathematics, 2011, 1, 187-192
doi:10.4236/apm.2011.14033 Published Online July 2011 (http://www.SciRP.org/journal/apm)
Unsteady Flow of a Dusty Visco-Elastic Fluid through a
Incliend Channel
Geetanjali Alle1, Aashis S. Roy2, Sangshetty Kalyane1, Ravi M. Sonth3
1Department of Physics, Singhania Un i ve rs i ty, Rajasthan, India
2Department of Materials Science, Gulbarga University, Gulbarga, India
3Department of Mat hem at i cs, K.C.T. Engineering College, Gulbarga, India
Received March 1, 2011; revised April 26, 2011; accepted May 10, 2011
Abstract
The present discussion deals with the study of an unsteady flow of a dusty fluid through an inclined channel
under the influence of pulsatile pressure gradient along with the effect of a uniform magnetic field. The ana-
lytical solutions of the problem are obtained using variable separable and Fourier transform techniques. The
graphs drawn for the velocity fields of both fluid and dust phase under the effect of Reynolds number. The
velocity profiles for the liquid and the dust particles decreases at different values of time t increases. As the
visco-elastic parameter

increases the velocity of the liquid and the dust particles deceases. When relaxa-
tion time parameter

increases, the velocity of the liquid and dust particles decreases.
Keywords: Dusty Fluid, Pulsatile Pressure Gradient, Velocities of Dust and Fluid Phase, Inclined Channel,
Reynolds Number
1. Introduction
In recent years many authors have studied the flow of
immiscible viscous electrically conducting fluids and
their different transport phenomena. These fluid also
known as non-Newtonian fluids are molten plastics.
Plups, emulsion etc., and large variety of industrial
product having visco-elastic behavior in their motion.
Such fluids are often embedded with spherical non-
conducting dust particles in the form of impurities. This
fluid also called dusty Rivlin-Ericksen second order fluid.
The influence of dust particles on visco-elastic fluid flow
has its importance in many applications such as extrusion
of plastic in the manufacture of rayon and Nylon, purifi-
cation of crude oil, pulp oil, pulp, paper industry, textile
industry and in different geophysical cases etc. In these
cases stratification effect is often observed which are
under the action of geomagnetic field.
Saffman et al., (1962) studied the stability of a laminar
flow of dusty gas with uniform distribution of dust parti-
cles. Michel (1965) considered the Kelvin-Helmholtz
instability of the dusty gas. Michael and Miller (1965)
discussed the motion of the dusty gas enclosed in the
same infinite space above a rigid plane boundary. We
have studied the unsteady dusty visco-elastic liquid in a
channel bounded by two parallel plates. The change in
velocity profiles for dust and liquid particles has been
depicted graphically.
2. Theory
Formulation and Solution of the Problem
The X-axis is taken along the plate and the Y-axis nor-
mal to it. The basic equations of hydro magnetic flow are
 

2 111 1
021
1'
'
uuu Pu
t
kN uu






 


 
(1)
 
222 12
'
'
uk
uu uu
tm




 

1
div 0u
(2)

(3)

2
div 0u

(4)

where the 1

, 2 u u

denotes the velocity vector of fluid
and dust particles respectively: the pressure: p


the density of the fluid:


t kinematic coefficient of viscosity:

the time: m, the mass o f the dust particles:
, the number density of dust particles: K, the stokes
0
N

188 G. ALLE ET AL.
resistance coefficient which for spherical particles of
radius a is



 a:

, the coefficient of viscosity of
fluid particles.
In the present analysis, the following important as-
sumptions are made:
1) The dust particles are spherical in shape are uni-
formly distributed.
2) Chemical reaction, mass transfer and radiation be-
tween the particles and fluid are not considered.
3) The temperature is uniform with in a particle.
4) Interaction between particles themselves is not con-
sidered.
5) The flow is fully developed.
6) The buoyancy force is neglected.
7) The number density of the dust particles is constant
throughout the motion.
8) The displacement current is zero, since the flow
velocity is small relative to the speed of light.
9) The Hall effects are negligible.
10) The fluid is electrically neutral, i.e., no surplus
electrical charge distribution is present in the fluid.
11) Only the electromagnetic body forces are present.
12) Fluid properties are invariable.
13) Viscous dissipation is neglected.
Maxwell’s equations, tog ether with Ohm’s law and th e
law of electromagnetic conservation, are written in the
case of zero-displacement and hall currents as:
 
B
J (5)
t

 
B
E

1E
(6)

J
VB
0
(7)
B
(8)
0
E
(9)
The usual Prandtl boundary layer assumptions along
with assumptions (5)-(9) leads to the following reduction
of the previous equations:

021
KN uu Puu
tx t y


 
 


 

 
  

2
11
2
1
 (10)

12
2
u
K
uu


12
0uu


t


 (11)
which are to be solved subject to the boundary conditions
0,t
1
0, P t
x




(c on s ta nt)C

12
,0,0u u

 
(12)
yh
Changing it into non dimension a l form by putting
2
12
22
,, ,,,
''
uh u yx tph
yxtu vp
hhh






 
 


We have

2
2
1
up ul
vu
tx t y

 


 

  
 (13)
vuv
t


 (14)
where
2
m
K
h


  Relaxation time parameter, 0
mN
l

Mass Concentration



2
h

0: =0,0tuv
Visco-elastic parameter
The boundary conditions are


0:=0, at1tu y
=0, at1uy (15)
take pC
x

0
(constant) for t.

Then the equation (2.2.13) becomes

2
2
1
uul
Cvu
tt y


 (16)

 

 

Appling the Laplace Transform, we have from (14) and
(16)
 
2
2
1
Cul
SuSv u
Sy

 (17)

 

Svuv

 (18)
where
00
ed,ed
st st
uutvvt




The boundary conditions (15) are transformed to
0, 01uvaty

 (19)
Solving equations (17) and (18) subject to the bound-
ary conditions (19) we have

22
2
d
1 d
uC
uSS
y



 (20)
where



21
11
SSls
SS





 (21)
Finally

2
cosh
1cosh
1
Cy
uSS









(22)

G. ALLE ET AL.
189

211
vSS S
cosh
1cosh
Cy


 




(23)
Applying Laplace Inversion formula
1ed
2
iSt
i
uut
i





 (24)
Here

is greatest then the real part of all the Singu-
larities of u

2
1
2c 1
i
i
C
u
iSS







cosh
1ed
osh
St
y
t






(25)
Taking Inversion Laplace Transform and with the help
of calculus of residues the above equations (22) and (23)
yields.
 


 


1
2
11
22
1 e
1 e
St
St
S
S






2
2
133
2
133
22 cosh
1cosh
21
1cos 1
82
21
21
1cos 1
82
21
r
r
r
r
uQy
CQ Q
ryS
rA
ryS
rA
























(26)
and
 


 


1
2
11
22
1 e
1 e
St
St
S
S







2
133
133
22 cosh
1cosh
21
1cos 1
82
21
21
1cos 1
82
21
r
r
r
r
vQy
CQ Q
ryS
rA
ryS
rA
























(2.2.25)
where
22 2
111
2
111
S Sl
S
23312
01
lS
QatS ASS







 

 

22 2
222
2
222
S Sl
S
33 12
1
lS
ASS




 
 
 
11
22
21
22
XX
rr










12
22
,
21
21 21
XX
SS
 




 





where
22
22
121 21
122
rr
Xl 

 

 
 
22
22 2
121 21
41 22
rr
XX



 
 

 
 


(27)
3. Results and Discussion
The unsteady flow of a dusty visco-elastic fluid through
a channel is studied. From Figures 1 and 2 it shows that
the velocity profiles for the liquid and the dust particles
decreases at different values of time t increases. As the
visco-elastic parameter

increases the velocity of the
liquid and the dust particles deceases as shown in Figure
3 and 4. When relaxation time parameter

increases
the velocity of the liquid and dust particles decreases as
shown in Figure 5 and 6. F rom Figure 7 and 8 it can be
observed that as mass concentration increases the veloc-
ity of the liquid and the dust particles deceases. In case
when gravity or inclination angle 0


0 and visco-
elastic parameter

 and adding the magnetic field
term then the present model becomes that of Singh and
Ram.
Figure 1. Show the variation of velocity pr ofile of liquid for
different value of time at fixed (σ = 0.8, λ = 0.5, I = 0.5).

G. ALLE ET AL.
190
Figure 2. Show the variation of velocity profile of dust for
different values of time at fixed (σ = 0.8, λ = 0.5, I = 0.5).
Figure 3. Show the variation of velocity pr ofile of liquid for
different values of time at fixed (σ = 0.8, λ = 0.5, I = 0.5).
Figure 4. Show the variation of velocity profile of dust for
different values of time at fixed (σ = 0.8, λ = 0.5, I = 0.5).
Figure 5. Show the variation of velocity profile of liquid or
different values of relaxation and at fixed (λ = 0.8, t = 0.5, I
= 0.5).

G. ALLE ET AL.
191
Figure 6. Show the variation of velocity profile of dust or
different values of relaxation and at fixed (λ = 0.8, t = 0.5, I
= 0.5).
Figure 7. Show the variation of velocity pr ofile of liquid for
different values of I mass concentration and at fixed (σ =
0.8, t = 0.5, λ = 0.5).
Figure 8. Show the variation of velocity profile of dust for
different values of I mass concentration and at fixed (σ =
0.8, t = 0.5, λ = 0.5).
4. References
[1] D. H. Michael, “Kelvin-Helmholtz Instability of A Dusty
Gas,” Proceeding Cambridge Philosophical Sciety, Vol.
61, No. 2, 1965, pp. 569-571.
doi:10.1017/S030500410000414X
[2] D. H. Michael and D. A. Millar, “Plane Parallel Flow of a
Dusty Gas,” Mathematika, Vol. 13, No. 1, 1966, pp. 97-
109. doi:10.1112/S0025579300004289
[3] P. G. Saffman, “On stability of a Laminar Flow of a
Dusty Gas,” Journal of Fluid Mechanics, Vol. 13, No. 1,
1962, pp. 120-128.
doi:10.1017/S0022112062000555
[4] C. B. Singh and P. C Ram, “Unsteady Flow of an Elec-
trically Conducting Dusty Viscous Liquid Through a
Channel,” Indian Journal of Pure and Applied Mathe-
matics, Vol. 8, No. 9, 1977, pp. 1022-1028.
[5] G. C. Sharma, “Unsteady Flow of an Electrically Con-
ducting Dusty Viscous Liquid between Two Parallel
Plates,” International Journal of Pure & Applied Mathe-
matical, Vol. 18, No. 12, 1987, p. 1131.
[6] K. K. Singh, “Unsteady Flow of Conducting Dusty Vis-
cous Liquid in an Annulas,” Acta Ciecia Indica, Vol. 3,
No. 3, 1977, p. 264.
[7] O. P. Varshney, “Flow of a Dusty Rivlin-Ericksen Fluid
through a Channel. Ph.D Thesis,” Agra University, Agra,
1983.

G. ALLE ET AL.
192
[8] M. L. Sharma, “MHD Flow of a Conducting Dusty Vis-
cous Liquid through a Long Elliptic Duct with Pressure Gradient as Function of a Time,” Ph.D Thesis Agra Uni- versity, Agra, 1980.