Journal o]’Applied Mathematics and Stochastic Analysis 5, Number 2, Summer 1992, 131.138
ON THE UNSTEADY FLOW OF TWO VISCO-ELASTIC FLUIDS
BETWEEN TWO INCLINED POROUS PLATES
P.P. SENGUPTA
University
of Kalyani
Department of Mathematics
Kalyani, West Bengal, INDIA
T.K. RAY
River Research Institute
Mohanpur, West Bengal, INDIA
L. DEBNATH
University
of Central Florida
Department of Mathematics
Orlando, Florida 32816, USA
ABSTlCT
This study s concerned wth boh hydrodynamic nd
hydromgnetc unsteady slow flows of wo mmisdble visco-elastc fluids
of lvln-Eficksen ype between two porous prllel nonconducting plates
certain ngle to the horizontal. The exact solutions for the
nclned
velocity fields, skin frictions, and the interface velocity distributions are
found for both fluid models. Numerical results are presented in graphs.
A comparison is made between the hydrodynamic and hydromagnetic
velocity profiles. It is shown that the velocity is diminished due to the
presence of a transverse magnetic field.
Key words: Visco-elastic hydromagnetic and hydrodynamic
fluids, skin frictions and interface velocity fields.
AMS (MOS) subject classifications:
76W05.
1. INTRODUCTION
The study of fluid flows in a porous medium plays an important role in the recovery of
crude oil from the pores of reservoir rocks by displacement with immiscible water and forming
polymetric adhesive joints between the solids. Various hydrodynamic and hydromagnetic flows
in different fluid configurations have received considerable attention in recent years by several
researchers including Kapur
[1], Kaput
[2], Bhattacharya [3], Gupta and Goyal [4],
associates [7-9]. In spite of this progress, some
and Sukhla
Gupta and Singh [5-6], and Sengupta and his
problems remained unsolved. Two such problems are considered in this paper.
1Received: September, 1991.
Revised: January, 1992.
Printed in the U.S.A. (C) 1992 The Society of Applied Mathematics, Modeling and Simulation
131
P.R. Sengupta, T.K. Ray and L. Debnatla
132
This study is concerned wih both hydrodynamic and hydromagnetic unsteady slow
flows of two immiscible visco-elastic fluids of Rivlin-Ericksen type between two porous parallel
nonconducting plates inclined at a certain angle to the horizontal. The exact solutions for the
velocity fields, skin frictions, and the interface velocity distributions are found for both fluid
models.
Numerical results are presented in graphs.
The hydrodynamic and hydromagnetic
velocity profiles are compared. It is shown that velocity is diminished in the latter case due to
the presence of a transverse magnetic field.
2. FORMULATION OF TIlE PROBLEM
We consider the unsteady flow of two incompressible, immiscible Rivlin-Ericksen fluids,
each occupying a certain height between two porous parallel stationary plates inclined at an
a
We set a Cartesian coordinate system with the z-axis along the
interface of the two fluids and parallel to the direction of the flow while the z-axis is chosen
upward. Assuming that uj = uj(z, z, t), vj = O, wj = 0 and 0 = 0 where j 1, 2, the equation
angle
to the horizontal.
of continuity
Ouj
= 0 leads to uj = uj(z, t).
The unsteady equation of motion for the incompressible visco-elastic fluids in a porous
medium is
duj
d2uj
1 i)p
aj
where #j, v,j, j, kj and r/j are densities, coefficients of kinematic viscosigy, kinematic viscoelasticity coefficients, permeabilities, and coefficients of viscosity of ghe fluids respectively, j = 1
refers
o he first fluid (0 _< z <_ L), and j = 2 refers to he second fluid (- L _< z <_ 0).
The required boundary conditions are
u 1 -0
at z
= L,
u1
= u0
at z
0
u2
=0
at z
u2
= uo
at z
J
L,
=0
J
for the first fluid
(2.lab)
for the second fluid.
(2.2ab)
3. SOLUTIONS OF TIIE llYDRODYNAMIC PROBLEM
With the usual initial conditions, it is convenient to introduce the Laplace transform
with respect to time
oo
j=
/
o
e-Stujdt, Re(s)> 0
(3.1)
On the Unsteady Flow of Two Visco-Elastic Fhdds
Application of this transform reduces equation
d2uj
where
to the form
Rj( u + pj) = o,
z
aj
(2.1)
--j
Rj=s+---kj, Nj=j+st3j and
-pj=
-
10p
--j-b-+gsinO with j= 1,2.
Uo + Pl
] SRsnhLv/Ri.]N.,1.
The inverse Laplace transform gives the velocity in 0
2pL2kUlr In
Z0o= (- )n {sin-[(Ln( L2
The transformed velocity
2)n=lZ (L
27rUoL2(k2
.
2P2L2k2
In the limit
.=
(- )
- + z)2r k1)4-
(3.3)
_< z < L,
sir" }
Vl(r2n2kl q- L2)
h(,::Z x + z)
t
(3.4)
32 in -L _< z _< 0 is
oo
-b
..
1 for the first fluid is
)sinh(L- z)v/Ri/N 1 Pl
The solution of the transformed velocity
l (z, s) =
133
l )nsin(L +
n(
2
z\
q_ r2.22)(L2 -t- 7rzr’:k2)
zzd"(
n{siny(L + z)- sin}
(+)
-(r. + )
t
(3.6)
(3.4) and (3.6) tend to zero and hence the
steady state solutions are attained and are given by the first two terms in (3.4) for ul(z,t), and
by the first two terms in (3.6) for u2(z, t).
as too, the exponential terms in
-
Interface Velocity: The tangential stress is continuous at the interface of the liquids.
Thus we get
Oul
rh--
z=O
Ou21
= r/:-
(3.7)
z=O
P.R. Sengupta, T.K. Ray and L. Debnatta
134
The interface velocity can be written as
2V%(
On putting qj =
in the steady-state solutions, we obtain
(3.9)
(sinhz/22 )
sinh(L+z)/
u-’ = (1 2q2k2) ’sinhL/2 + 2q2k2 SinhL] +
u2
(3.10)
1
4. SOLUTIONS OF THE IIYDRODYNAMIC PROBLEM
We consider the visco-elastic fluids electrically conducting in the presence of a constant
transverse magnetic field B0 along the z-axis. Thus, the equation of motion is
Ouj
02uj
10p
vjuj
<rjB
where o’j is the electrical conductivity of the fluids. This equation is to be solved by the same
initial and boundary conditions as for the hydrodynamic problem. The Laplace transformed
equation of (4.2) is
az
+sM
N--y
= o,
where
j 2o
:i
Mj = s +-j+
..pj and j = 1,2.
v
<r
B
(4.4)
The solution of (4.3) with the transformed boundary condition is
l(Z, s)=
+
z)v/Mi/N I + Pl sinhzv/Mi/Ni
SM i SinhLv/M’;/i
S}a ) sinh(L’s’inhv/ml/N
1
The inverse transform gives
uo
+
Pl
SM;
(4.5)
On the Unsteady Flow of Two V’tsco.Elastic Fluids
c
-I"
271"0n=lE
(- I]hn
135
nL2(klPlPl 1Pl/21 (rlBlkl)Sin(L Z)-(L 2 +r127rl)(r22.lPl...+.L2.pll..+.o.1Boi2], )
gl(n2rr2kl + L2)pl + a’lB20L2kl
klPl(L 2 + n2’21 )
ex
t
)
(4.6)
Similarly, we find
2(z, s) =
-
+ S-M2).h( + )v//i
p2
::SinhL"
f sinhzv/M2/N2 +
)
(4.7)
The inverse Laplace transform gives
+
r2B2k2)(sin(L + z)--). 1 )n
2
2
’n;22 f2)(’N 22"
L2
r"’k2’p"V2 ;’ 2 P;v L2’,;2"+"a’B2
i"
n(k2u2p2 u2p2
2ruLEn=, (L +
2
+
v(n2r2k + L2)p2 + o’2B2oL2k2
(L 2 + n2rr2fl2)k2P2
ex
2p2L2k2
,,__,
(- 1) n
t
)
p2(sin-(L + z)- sin-q-)
(4.8)
(L: + n2;’2f3:)’k:p:
In the limit
t, the transient terms involved in (4.6) and (4.8) decay and the
steady state solutions are attained and given by the first two terms in each of the results (4.6)
and (4.8). These steady state solutions can be written
u,(z,t)=
(sinhz
( Uo+v,rPlsihL-z’
) sinhL +lTi ksinhL
Pl
1
)
(4.9)
P.R. Sengupta, T.K. Ray and L. Debnath
136
and
p
sinh(L +z)x2
(
sinhz
where
’J Bo2
Ti = --jj +
1
1
Hj2
(4.11)
and H.. are the Hartmann numbers.
Finally, the interface velocity is
In terms of the notation q j =
the steady state solution can be written in the
form
u" =
1-’x ] sin’hL’
Tx\sinhL-
(4.13)
1
and
(
u2 = 1u"
2q2sinh(L + z)x2 + 2q2 sinhzv + 1
-2 ] sinhLy2 T2sinhL2
)
(4.14)
5. DISCUSSIONS AND CONCLUSIONS
It follows from the numerical calculation that, for a conducting fluid in the presence of a
transverse magnetic field, the velocity on the interface is at its maximum and then gradually
decreases with continuous increase of distance in the upward or downward direction according to
whether the upper or lower fluid is involved.
For the upper fluid, the velocity attains the
minimum value zero at the upper boundary plate, while for the lower fluid, the velocity reaches
the minimum value zero at the lower boundary plate. In both cases, the velocity profile is very
smooth and continuous. For nonconducting fluids
(or in the absence of a magnetic fields), the
maximum velocity does not occur at the interface; rather, it is attained for the upper fluid at a
point a bit higher than the interface, while for the lower fluid it is attained at a point a bit
lower than the interface.
This shows a striking difference in the flow pattern between a
conducting and a nonconducting fluid mode. However, the nature of the variation of the flow
patterns of the conducting and nonconducting fluids remains very similar.
ql
In both fluid models that Ul/Uo and u2/u o increase with the increase in ql and q2 where
and q2 represents the pressure gradients. On the other hand, these velocity ratios increase
On the Unsteady Flow o]’ Two V’tsco-Elastic Fhdds
137
with the increase in permeabilities k 1 and k 2 of the fluids. It is important to observe that when
u1
u1
=
=
= 0.785
2 = 0,
2 = 1, then n’ = 0.68 and when
for the same value of z/L. This means that the velocity decreases by 13% in the presence of a
rlB/PlU
o’2B/P2U
O’lB2o/PlUl o’2B/P2U
transverse magnetic field. Thus the decrease of the velocity for the hydromagnetic case is worth
noting.
The velocity profiles are drawn in Figure 1 and Figure 2 for the hydrodynamic and
hydromagnetic fluid models.
ACKNOWLEDGMENT
This work was partially supported by the University of Central Florida.
Authors
express grateful thanks to the referee and the Principal Editor for several suggestions.
REFERENCES
[i]
J.N. Kaput, Applied Sci. Res., Section 10B (1968), pp. 183-193.
J.N. Kapur and J.B. Sukhla, ZAMM, 44 (1964), pp. 268-269.
[3]
R.N. Bhattacharya, Bull. Cal. Math. Soc., 60 (1970), pp. 127-136.
[4]
M.C. Gupta and C.M. Goyal, Indian J. Pure and AppI. Math., 4 (1973), pp. 1250-1260.
[5]
M. Gupta, Agra Univ. Journal of Research, (1978).
[61
M. Gupta and D.P. Singh, Indian J. Theoret. Phys., 32 (1984), pp. 275-283.
[r]
P.R. Sengupta and S.K. Bhattacharyaa, Rev. Roum. Sci. Tech. Mech. Appl., Tome 25
(1980),
[s]
pp. 171-181.
P.R. Sengupta and J. Ray Mohapatra, Rev. Roum. Sci. Tech. Mech. Appl., 16 (1971),
pp. 1023-1041.
P.R. Sengupta, T.K. Ray and L. Debnath
138
U2
Uo
U1
Uo
I- NON MHD CASE q=q=2
II- MHD CASE q= q= 2
III- MHD CASE q=q= 1.5
k1
1
k-2
2-/// !11
0.8
0.6
",\\
jB o2
Pj)j
(
"’\
0.4
0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8
-z/L
FIGURE 1
z/L
U2
/’/
////
1
0.8 0.6 0.4
-z/L
III
U1
1
\\
0.8
...1
0.2 0
"’\
I-NON MHD CASE
!!-MHD CASE
JB
II- MHD CASE
IB
,
0.2 0.4 0.6 0.8 1
FIGURE 2
z/L
q=q=2
k.-I
k:z = 2
PIll
jj
=2