Unsteady Stokes Flow through Porous Channel with Periodic Suction and Injection with Slip Conditions


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 4, 2018, 937-945
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Unsteady Stokes Flow through Porous Channel with
Periodic Suction and Injection with Slip Conditions

Kaleemullah Bhatti1, Zarqa Bano1,∗, Abdul Majeed Siddiqui2

1 Department of Mathematics and Social Sciences, Sukkur IBA University, Sukkur,
Pakistan
2 Department of Mathematics, Pennsylvania State University, York Campus, York,
PA 17403, U.S.A.

Abstract. This work is concerned with the influence of slip conditions on unsteady stokes flow
between parallel porous plates with periodic suction and injection. The obtained unsteady govern-
ing equations are solved analytically by similarity method. The characteristics of complex axial
velocity and complex radial velocity for different values of parameters are analyzed. Graphical
results for slip parameter reveal that it has significant influence on the axial and radial velocity
profiles. The effects of suction or injection are also observed. The problem of unsteady stokes flow
through porous plates with no slip is recovered as a special case of our problem.

2010 Mathematics Subject Classifications: 76D05, 76D07

Key Words and Phrases: Unsteady Stokes Flow, Porous Plates, Periodic Suction and Injection,
Wall Slip

1. Introduction

Unsteady Stokes flow of a Newtonian fluid in a channel is an interesting field due to its
relevance to various engineering applications. Initially Berman A.S (1953) [1] studied the
problem of unsteady laminar flow through porous walls. Ganesh [4] have recently applied
similarity transformation method to solve problem of unsteady stokes flow through parallel
porous plates when there is no slip between the fluid layer at the bottom and top and
the corresponding walls. Kirubhashankar and Ganesh [2] then modified the problem by
considering MHD fluid flow through the same channel of parallel porous plates. Unsteady
incompressible MHD fluid flow through porous walls with slip conditions is studied recently
by H. Zaman [3].

In this paper an analysis has been presented for the flow of a Newtonian fluid through
a channel having parallel porous plates with periodic outward suction at upper wall and
periodic injection of fluid through the lower wall. Also, we have considered the slippage

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v11i4.3309

Email addresses: zarqa.bano@iba-suk.edu.pk (Z. Bano), kaleem.msmts@iba-suk.edu.pk (K. Bhatti)

http://www.ejpam.com 937 c© 2018 EJPAM All rights reserved.



K.Bhatti, Z. Bano, A. M. Siddiqui / Eur. J. Pure Appl. Math, 11 (4) (2018), 937-945 938

effect of fluid layer connected to the plates. An exact solution to Navier Stokes equations,
reduced to forth order linear ordinary differential equation with the slip boundary condi-
tions is obtained. The similarity transformation method is adopted to solve the dynamic
equations for small flow through porous plates. Ganesh’s work [4] can be considered as a
special case of our work when no slip occurs.

2. Main Problem

2.1. Problem Formulation

Two infinite porous parallel plates are considered at y=0 and y=H. An incompressible
Newtonian fluid flows through these fixed flat plates. We assume the Stokes flow and
there is a periodic injection with velocity V1eiωt at lower plate and a periodic suction
with velocity V2eiωt at upper plate. In addition we consider slip between the fluid layer
in contact with the wall according to Navier slip law which states that the relative velocity
of the fluid and the wall is proportional to the shear rate at the wall.
Let u and v be the components of velocity vector in x and y direction in flow field at any
time t. We chose velocity vector ~q and pressure P in the form:

~q =
{
u(x, y)̂i+ v(x, y)ĵ

}
eiωt, (1)

P = p(x, y)eiωt, (2)

Under the above assumptions and the choice of rectangular co-ordinate system the gov-
erning equations are:

∂u

∂x
+
∂v

∂y
= 0, (3)

ρ
∂u

∂t
= −∂p

∂x
+ µ(

∂2u

∂x2
+
∂2u

∂y2
), (4)

and

ρ
∂v

∂t
= −∂p

∂y
+ µ(

∂2v

∂x2
+
∂2v

∂y2
), (5)

where µ is the co-efficient of viscosity and ρ is constant density of the fluid. The convective
terms in the governing equations are neglected due to very small Reynolds number. The
boundary conditions of the problem are:

u(x, 0) = λ

∣∣∣∣∂u∂y
∣∣∣∣
y=0

, (6)

u(x,H) = λ

∣∣∣∣∂u∂y
∣∣∣∣
y=H

, (7)

and

v(x, 0) = V1, v(x,H) = V2. (8)



K.Bhatti, Z. Bano, A. M. Siddiqui / Eur. J. Pure Appl. Math, 11 (4) (2018), 937-945 939

2.2. Analytical Solution

For a two dimensional incompressible flow a stream function ψ is introduced for which,

u(x, y) =
∂ψ

∂y
v(x, y) = −∂ψ

∂x
, (9)

such that the continuity equation (3) is automatically satisfied. Using (1), (2) and (9) in
Equations (4) and (5) we get,

ρiω
∂ψ

∂y
= −∂p

∂x
+ µ

∂

∂y

(
∇2ψ

)
, (10)

and

ρiω
∂ψ

∂x
=
∂p

∂y
+ µ

∂

∂x

(
∇2ψ

)
. (11)

For the constant wall velocity vw = 0 and given BC’s, a suitable choice of stream
function is,

ψ(x, y) =
(
H
u0
a
− v2x

)
f (η) , (12)

where u0 is the average entrance velocity. a = 1 − v1/v2 , with 1 / |v1| / |v2| and
η = y/h is called dimensionless distance. Using equation (12) in equations (10) and (11)
we get,

−1

ρ

∂p

∂x
=
(u0
a
− v2x

H

) [
iωf ′ (η)− ν

H2
f ′′′ (η)

]
, (13)

and

− 1

ρH

∂p

∂η
= iωv2f (η)− ν

H2
v2f
′′ (η) . (14)

Differentiating (13) with respect to η and (14) with respect to x we get,

−1

ρ

∂2p

∂η∂x
=
(u0
a
− v2x

H

) [
iωf ′′ (η)− ν

H2
f (iv) (η)

]
, (15)

and
∂2p

∂x∂η
= 0. (16)

Comparing (15) and (16) we get,

f (iv) (η)− iωH2ρ

µ
f ′′ (η) = 0. (17)

Assuming α2 = iωρ
µ , (17) reduces to the form,

f (iv) (η)− α2H2f ′′ (η) = 0. (18)



K.Bhatti, Z. Bano, A. M. Siddiqui / Eur. J. Pure Appl. Math, 11 (4) (2018), 937-945 940

Also using stream function defined by (9) and (12) and the slip boundary conditions
(6)to (8), we get following new boundary conditions on f (η),

hf ′(0) = λf ′′(0), (19)

hf ′(1) = λf ′′(1), (20)

f(0) = 1− a, (21)

and
f(1) = 1. (22)

Solving (18) subjected to the above boundary conditions to get f(η) and then using this
accompanied with (1) and (9) we get,

u = u(x, y)eiωt,

u =
∂ψ

∂y
eiωt,

u =
(u0
a
− v2x

H

)
f ′(η)eiωt,

u = Hαa
(u0
a
− v2x

H

)
[
αλ(αλ sinhαH + coshαy − coshα(H − y))

α3Hλ2 sinhαH − αH sinhαH + 2 coshαH − 2

+
sinhα(H − y) + sinhαy − sinhαH

α3Hλ2 sinhαH − αH sinhαH + 2 coshαH − 2
]eiωt, (23)

and
v = v(x, y)eiωt,

v = −∂ψ
∂x

eiωt,

v = v2[
aα(λ sinhαy + λ sinhα(H − y)− y sinhαH +H sinhαH − λ coshαH)

α3Hλ2 sinhαH − αH sinhαH + 2 coshαH − 2

+
a(coshαy − coshαH − coshα(H − y)−H sinhαH + 1)

α3Hλ2 sinhαH − αH sinhαH + 2 coshαH − 2

+
α3λ2(H − aH + ay) sinhαH + 2(coshαH − 1)

α3Hλ2 sinhαH − αH sinhαH + 2 coshαH − 2
]eiωt (24)

As a special case we present here the expression for velocity profile obtained by for
λ=0.

u = Hαa
(u0
a
− v2x

H

)
[
sinhα(H − y) + sinhαy − sinhαH

−αH sinhαH + 2 coshαH − 2
]eiωt, (25)

and

v = v2
α(aH − ay −H) sinhαH + (2− a) coshαH + a(coshαy − coshα(H − y))− 2

−αH sinhαH + 2 coshαH − 2
eiωt

(26)
Equation (25) and (26) match with the axial and radial velocity profiles in [4].



K.Bhatti, Z. Bano, A. M. Siddiqui / Eur. J. Pure Appl. Math, 11 (4) (2018), 937-945 941

2.3. Pressure Distribution

The pressure distribution can be obtained by extracting the pressure gradients from
(13) and (14) and then by integrating with respect to x and η respectively. Thus∫ x

0

∂p

∂x
dx = p(x, η)− p(0, η) (27)

∫ η

0

∂p

∂η
dη = p(x, η)− p(x, 0) (28)

Now it follows from (27) and (28) that∫ x

0

∂p

∂x
dx+

(∫ η

0

∂p

∂η
dη

)
x=0

= p(x, η)− p(0, 0) (29)

Using (13) and (14) we get,

p(x, η) = p(0, 0) +
ρv2
2H

(
iwf(η)− νf ′′′(η)

H2

)
x2 − ρu0

a

(
iwf(η)− νf ′′′(η)

H2

)
x

−ρv2
H

(∫ η

0
(H2iwf(η)− νf ′′(η))dη

)
(30)

Where p(0,0) is pressure at the entrance of the channel.

2.4. Results and Discussion

The numerical values of axial and radial velocity components have been calculated for
different values x, ωt and slip parameter λ.

Figure 1: Axial velocity profile for λ=0.1, v1 = 1, v2 = 2 x = 0, H = 10, u0 = 0.5 and α=1 for different
values of ωt

Figures 1 to 7 are axial velocity profiles at different cross sections of the channel namely
at x = 0, x = 2 and x = 4 and for different values of slip parameter namely for λ = 0,



K.Bhatti, Z. Bano, A. M. Siddiqui / Eur. J. Pure Appl. Math, 11 (4) (2018), 937-945 942

Figure 2: Axial velocity profile for λ=0.1, v1 = 1, v2 = 2 x = 2, H = 10, u0 = 0.5 and α=1 for different
values of ωt

Figure 3: Axial velocity profile for λ=0.1, v1 = 1, v2 = 2 x = 4, H = 10, u0 = 0.5 and α=1 for different
values of ωt

Figure 4: Axial velocity profile for λ=0.2, v1 = 1, v2 = 2 x = 0, H = 10, u0 = 0.5 and α=1 for different
values of ωt



K.Bhatti, Z. Bano, A. M. Siddiqui / Eur. J. Pure Appl. Math, 11 (4) (2018), 937-945 943

Figure 5: Axial velocity profile for λ=0.3, v1 = 1, v2 = 2 x = 0, H = 10, u0 = 0.5 and α=1 for different
values of ωt

Figure 6: Axial velocity profile for λ=0.4, v1 = 1, v2 = 2 x = 0, H = 10, u0 = 0.5 and α=1 for different
values of ωt

Figure 7: Axial velocity profile for λ=0, v1 = 1, v2 = 2 x = 0, H = 10, u0 = 0.5 and α=1 for different values
of ωt



K.Bhatti, Z. Bano, A. M. Siddiqui / Eur. J. Pure Appl. Math, 11 (4) (2018), 937-945 944

Figure 8: Radial velocity profile for λ=0.1, v1 = 1, v2 = 2, H = 10 and α=1 for different values of ωt

Figure 9: Radial velocity profile for λ=0.5, v1 = 1, v2 = 2, H = 10 and α=1 for different values of ωt

Figure 10: Radial velocity profile for λ=1, v1 = 1, v2 = 2, H = 10 and α=1 for different values of ωt

λ = 0.1, λ = 0.2 and λ = 0.3 and λ = 0.4, when average entrance velocity is u0 = 0.5.
It is observed from Figure 1, 2 and 3 that the magnitude of axial velocity decreases as



REFERENCES 945

x increases from 0 to 4. The same result is obtained by Ganesh [4] for no slip problem.
Figures 1, 4, 5, 6 and 7 show the effect of slip parameter λ. It is clearly seen that as λ
increases from 0 to 0.4 the magnitude of axial velocity near the plates at y = 0 and y = 10
also increases. Figure 8, 9 and 10 depict the radial velocity profile for v1 = 1, v2 = 2
and α = 1, for different values of ωt. It is very clear that the radial velocity completely
vanishes for ωt = π

2 and it is non-linear for other values of ωt. It is also seen that when
slip parameter increases from λ = 0 to λ = 1, the magnitude of radial velocity decreases
and behaves nearly like a constant.

2.5. Conclusion

In the above analysis, an exact solution for velocity field of unsteady stokes flow of
viscous fluid between two parallel plates in the presence of porous walls and slippage
effects is constructed. The graphical analysis has been made for different values of the slip
parameter at various cross sections of the channel. The expressions of axial and radial
velocities which are presented here reduce to the Ganesh’s results for the case when there
is no slip at wall. We presume that the results obtained here add one more class of exact
solution to that of a few presently available in the literature.

References

[1] Abraham S. Berman. Laminar flow in channels with porous walls. Journal of Applied
Physics, 24(9):1232–1235, 1953.

[2] Dr. S. Ganesh C. K. Kirubhashankar and A. Mohamed Ismail. An Exact Solution of
The Problem Of Unsteady MHD Flow Through Parralel Porous Plates. Int’l Journal
of Advances in Mechanical Automobile Engg., 1:39–42, 2014.

[3] H.Zaman. Hall Effects on The Unsteady Incompressible MHD Fluid Flow With Slip
Conditions And Porous Walls. Applied Mathematics and Physics, 1(2):31–38, 2013.

[4] Ganesh S and Krishnambal S. Unsteady Stokes Flow Of Viscous Fluid Between Two
Parallel Porous Plates. Journal on Information Sciences and Computing, 1(1):63–66,
2007.