Parallel Flow of a Pressure-Dependent Viscosity Fluid
through Composite Porous Layers
Department of Mathematics and Statistics
University of New Brunswick
100 Tucker Park Road, Saint John, N.B.,
CANADA
Abstract: - Flow of a fluid with pressure-dependent viscosity through a composite of two porous layers is
considered in this work in an attempt to validate velocity and shear stress continuity conditions at the interface,
and are popular in the study of flow over porous layers and through composite layers when viscosity of the fluid
is constant. For the current problem, conditions at the interface between the porous layers reflect continuity
assumptions of velocity and shear stress, with additional continuity assumptions on pressure and viscosity.
Viscosity is assumed to vary continuously and exponentially across the layers as a function of pressure. Analytical
solutions are obtained to illustrate the effects of flow and media parameters (Darcy numbers, layer thicknesses,
angle of inclination, and viscosity adjustment parameter) on the dynamic behaviour of pressure-dependent
viscosity fluids in porous structures. All computations, simulations and graphs in this work have been carried out
and obtained using Maple 2020 software package.
Key-Words: - Composite porous layers; Pressure-Dependent Viscosity; Inclined Plane.
Received: April 8, 2021. Revised: January 5, 2022. Accepted: January 18, 2022. Published: March 1, 2022.
1 Introduction
In this work, we initiate the study of flow of fluids
with pressure-dependent viscosities through
composite porous layers. Flow of fluids with constant
viscosities through layered porous media have been
extensively studied (cf. [1, 2, 3] and the references
therein). Flow of fluids with pressure-dependent
viscosities through free-space has also received
considerable attention in the literature, (cf. [4, 5, 6, 7,
8] and the references therein). However, only
recently models of flow of fluids with pressure-
dependent viscosities through porous media have
been derived [9, 10, 11, 12].
Fluid viscosity variations include changes in
viscosity due to temperature, pressure and shear-
thinning. Many fluids, such as paint and polymers
exhibit behaviours in which a fluid becomes either
thicker, or thinner when sheared, [13, 14, 15, 16, 17].
Nakshatrala & Rajagopal [17] provided an account of
these variations. Studies of fluid viscosity variations
and viscosity dependence on pressure can be traced to
back to the nineteenth century and the work of various
authors, including Stokes [18], Barus [19, 20]. More
recently, experimental studies confirm dependence of
viscosity on pressure, [7, 8, 21].
Barus [19, 20] suggested two relationships for
dependence of viscosity on pressure, an exponential
and a linear relationship. The Barus relationships
received considerable analysis in the literatrs, (cf. [22]
and the references therein). A model describing the
dependence of viscosity on pressure, temperature
and density has been reported in Szeri [5]. Other
models of dependence have been discussed in the
literature, [23, 24].
Interest in fluids with pressure-dependent
viscosities in porous media stems from their
industrial applications in the oil industry (enhanced
oil recovery and geological carbon sequestration),
food and polymer processing, in the pharmaceutical
industry, in thin film lubrication and in filtration
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problems [6, 15, 17, 22, 25, 26]. Fusi et.al. [25]
provided elegant modelling of three important
filtration problems involving pressure-dependent
viscosity fluids in porous media, and Housiadas et.al.
[26] initiated studies of compressible fluids with
pressure-dependent viscosity, in their development of
new analytic solutions of weakly compressible
Newtonian Poiseuille flows. These and many other
applications initiated the recent interest in the
modelling and simulation of flow through porous
media with a focus on fluids with variable or
stratified viscosities, and fluids with pressure-
dependent viscosities, [12, 11, 12, 17].
Models describing the flow of fluids with pressure-
dependent viscosity through porous media have been
derived, discussed or otherwise analyzed by various
authors, including Srinivasan and Rajagopal [10],
Nakshatrala and Rajagopal [17], Kannan and
Rajagopal [13], Abu Zaytoon et.al. [11], Alharbi,
et.al. [12], Of interest to the current work is the
Brinkman-type model, [10, 13, 17], in which the flow
through a rigid porous structure is described by the
following equations of continuity and momentum,
respectively, written here for steady flow:
(1)
(2a)
(2b)
is the Cauchy stress tensor in which
(2c)
where
u
is the velocity vector field,
p
is the
pressure,
is the fluid density,
is the body force, is the
variable viscosity, and is a drag function that
has been given various forms as discussed by Kannan
and Rajagopal [13], and include exponential and
polynomial forms in terms of pressure. Governing
equations for flow in the channel are the equation of
continuity and the Navier-Stokes equations with
pressure-dependent viscosity. These equations are
similar in form to equations (1) and (2) with
in (2a).
When the drag function is expressed as the ratio
between viscosity of the fluid and permeability of the
porous medium, namely, , equation (2)
reduces to a model derived using intrinsic volume
averaging, [11, 12]. This form of drag function
facilitates studies of flow through constant and
variable permeability porous media [27, 28]. We
implement this form in the current study where we
consider the parallel flow of a fluid with pressure-
dependent viscosity through two porous layers
inclined to the horizontal at an angle.
The flow domain of a porous channel inclined to
the horizontal has been a model configuration for
many problems, including thin film lubrication and
wave, [29], and, we believe, it facilitates the
introduction of a continuous pressure function on
which viscosity depends. Conditions at the interface
between the porous layer that emphasize continuity
of pressure, viscosity, shear stress and velocity.
2 Problem Formulation
The steady flow of an incompressible fluid with
pressure-dependent viscosity through a porous
medium is governed by continuity and momentum
equations (1) and (2), above, in which we take
.
Now, consider the flow through the configuration
shown in Fig.1, where a fluid flows through two
composite porous layers each with different
permeability. We set layer 1 to and layer
2 to . Layers 1 and 2 are assumed to be
parallel and meet at a sharp interface with
angle of inclination ϑ. The layers are bounded by
solid, impermeable walls at
Fig. 1. Representative sketch
For the unidirectional flow at hand, the flow is in
the x-direction, taken along the inclined wall, with
. Continuity equation (1) implies that
Equation (2) reduces to
, where is the body force component in
the y-direction, namely, Flow in the x-
direction is under the effect of body force (gravity)
whose component in the x-direction is .
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Pressure gradient in the x-direction is zero, ,
which emphasizes that
We now define the following dimensionless
quantities and Darcy numbers with respect to total
thickness of the flow domain and with respect to a
characteristic velocity, :
(3)
Dropping the asterisk (*), the governing equations
for the ith layer, where refer to the lower and
upper layer, respectively, take the following forms:
(4)
(5)
At the solid boundaries, , the
velocity vanishes. We also assume that the value of
pressure at y=1 is given as . Boundary conditions
are thus as follows:
(6)
0 (7)
(8)
(9)
(10)
(11)
(12)
In order to find the pressure distribution, we integrate
equation (5) and use conditions (8) and (12) to obtain
(13)
(14)
At the interface between the two layers, the
pressure is given by
(15)
In order to solve (16), we first need to specify
viscosities as functions of pressure distributions
(13) and (14). In this work we assume that viscosities
vary exponentially according to Barus’ relationship
of the form:
(16)
where
A
and
are positive constants, referred to as
viscosity control parameters. Their role is to keep
values of viscosities within realistic limits. We note
that and are equal at the interface,
, hence condition (11) is satisfied. Equations (4) thus
take the following form:
(17)
where
(18)
(19)
3 Problem Solution
3.1. Velocity Profiles
General solutions to equations (17) are given by:
(20)
;
(21)
where are arbitrary constants and
,
,
,
(22)
Velocity at the interface is given by either (20) or
(21), evaluated at , namely
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(23)
. (24)
Now, to find the values of arbitrary constants
appearing in (20) and (21), we apply
equation (6), (7), (9) and (10) to obtain the following
system of linear equations:
(25)
(26)
(27)
(28)
Equations (25)-(28) are written in the matrix form
, which can be solved numerically using
Maple, where
(29)
3.2. Vorticity and Shear Stress
Vorticities, in the lower layer and in the upper
layer, take the following forms with the help of (20)
and (21), respectively:
(30)
(31)
with values at the interface given by
(32)
.
(33)
We note that as a consequence of
conditions (10) and (11).
The shear stresses, in the lower layer in the
upper layer, are written as follows with the help of
(20) and (21), respectively:
(34)
(35)
The shear stress at the interface () is given by:
(36)
(37)
We note that due to (10).
4. Results and Discussion
Results have been obtained for the following ranges
that are representative of the media and flow
parameters:
Ranges of Darcy Numbers:
, and
. We point out that when then
the two layers behave like one layer.
Range of inclination angle 30, 60, and 75 degrees.
Range of 2, 3, and 5.
Range of 0.25, 0.5 and 0.75.
Range of 0.01, 0.1 and 1.
We can take and as is Kannan and
Rajagopal [13]. We also take A=U=1 as
representative values.
4.1. Velocity Profiles
Velocities in the porous layers are given by equations
(20) and (21), respectively. Effects of the various
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flow and medium parameters on the velocity profiles
are discussed in what follows.
4.1.1. Effects of Darcy Numbers
Fig. 2 illustrates the effects of Darcy numbers on the
velocity profiles across the two layers, when they are
of the same thickness. As is well known, velocity
increases with increasing Darcy number. When
, permeabilities are the same and the two
layers behave like a single layer. The velocity profile
shown for the case of is parabolic
and symmetric about y=0.5.
In the lower layer, y < 0.5, Fig. 2 shows the
increase in velocity with increasing . By keeping
fixed at 0.1, Fig. 2 shows a greater influence on
the upper layer by the lower layer with increasing
. This influence results in distorting the parabolic
shape of the velocity profile in the upper layer.
Fig. 2 Velocity across the two layers for various
values of .
, , , , 30,
.
4.1.2. Effects of
The value of represents the fraction of the total
thickness of the flow domain that the lower layer
occupies. When , both layers are of the same
thickness.
In Fig. 3, we illustrate the effects of when the
lower layer has a higher Darcy number than the upper
layer. This translates into the lower layer having a
higher influence on the upper layer by virtue of the
higher velocity associated with higher Da. With
increasing , velocity at the interface increases, as per
equations (23) and (24).
4.1.3. Effects of
Fig. 4 illustrates the effects of on the velocity
profile. It shows that with decreasing velocity
increases due to the decrease in viscosity that is
associated with decreasing . Decreasing viscosity
translates into a lesser resistance to the flow, hence a
greater velocity.
Fig. 3 Velocity across the two layers various values
of
, , 30, , and
.
4.1.4. Effects of
Associated with increasing angle of inclination is
a decrease in the pressure distribution, hence a
decrease in viscosity, hence an increase in velocity.
In addition, with increasing inclination angle there is
a greater gravitational force driving the flow. The net
effect across the configuration is a velocity increases
with increasing angle of inclination, as shown in Fig.
5.
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4.1.5. Effects of
For a given pressure distribution, when increases
then viscosity increases, which results in a decrease
in velocity across the layers. This is illustrated in Fig.
6 which shows the slowing down of the flow as
increases.
Fig. 4 Velocity across the two layers for various
values of .
, , , 30, , and
.
Figure 5 Velocity across the two layers for various
values of .
, , , , , and
Fig. 6 Velocity across the two layers for various
values of
, , , 30, , and
4.2. Velocity, Vorticity and Shear Stress at the
Interface
Velocity, vorticity and shear stress at the interface
are tabulated, Tables 1 through 5, below for
different parameters using expressions (23), (24),
(32), (33), (36) and (37).
Table 1 illustrates the effects of Darcy number on
interfacial quantities and shows the expected increase
of velocity at the interface with increasing for a
fixed value of . This behaviour is in agreement
with the effects of Darcy number on the velocity
profile, shown in Fig. 2. Vorticity decreases,
numerically, with decreasing , and shear stress at
the interface increases, numerically with decreasing
This is also in line with the velocity profiles
depicted in Fig. 2 if one considers the behaviour of
slopes of the velocity curves at the interface .
Also in line with the velocity profile figures, Fig.
2 to Fig. 6, are the effects of the parameters , ,
and on the velocity at the interface, illustrated in
Tables 2 to 5 in which we document the values of
vorticity and shear stress at the interface. We
summarize their effects as follows. Increasing
results in a decrease in the interfacial velocity and a
decrease in the vorticity, and a numerical increase in
shear stress, as shown in Table 2.
Increasing , results in an increase in the
interfacial velocity, and a corresponding increase in
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the vorticity, and a numerical decrease in the shear
stress at the interface, as shown in Table 3.
Increasing results in a decrease in the interfacial
velocity, and a decrease in the vorticity. The shear
stress at the interface remains unaffected by changes
in , as shown in Table 4.
Increasing the angle of inclination, , results in an
increase in the interfacial velocity, as per Fig. 5, and
a corresponding increase in the vorticity, while the
shear stress at the interface decreases with increasing
inclination angle, as shown in Table 5.
Table 1 Velocity at the interface for various
,,, , 30,
1
1()= 2()
1()= 2()
1()= 2()
1
0.0034873
0.00976403
-0.08958714
0.1
0.0026393
0.00401309
-0.03682098
0.01
0.0009683
-0.0031813
0.0292059
0.001
0.000283
-0.0056960
0.0522625
0.0001
0.000086
-0.0064038
0.0587570
Table 2 Velocity, Vorticity and Shear stress at the
interface for different . , , 30,
, and ,
1()= 2()
1()= 2()
1()= 2()
1
0.00096826
0.00476337
-0.05426985
0.1
0.00892301
0.05001707
-0.06379441
0.01
0.01113969
0.06319418
-0.06475056
Table 3 Velocity, Vorticity and Shear stress at the
interface for different
, , 30, , and
1()= 2()
1()= 2()
1()= 2()
0.25
0.00052199
0.00131358
-0.01858354
0.5
0.00096826
0.00476337
-0.05426985
0.75
0.00132741
0.00812923
-0.07458747
Table 4 Velocity, Vorticity and Shear stress at the
interface for different
, , , 30, , and
0
1()= 2()
1()= 2()
1()= 2()
2
0.00096826
0.00476337
-0.05426985
3
0.00035620
0.00175235
-0.05426985
5
0.00004821
0.00023715
-0.05426985
Table 5 Velocity, Vorticity and Shear stress at the
interface for different . , , ,
, , and .
1()= 2()
1()= 2()
1()= 2()
30
0.00096826
0.00476337
-0.05426985
60
0.00204452
0.01072180
-0.10172564
75
0.00259720
0.01416932
-0.11916257
5 Conclusion
In this work, we modelled, and provided a solution
to, parallel flow of a fluid with pressure-dependent
viscosity through a composite of two inclined porous
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layers. It was assumed that permeabilities are
constant and the flow in each layer is governed by
Brinkman’s equation for flow of fluids with variable
viscosity. The layers were saturated by the same
fluid. Pressure variations through the flow domain
were described by a continuous linear function.
Viscosity was then expressed as a continuous,
exponential function of pressure. These choices of
viscosity and pressure distributions helped maintain
the continuity conditions at the interface between
layers. Model equations were solved subject to
continuity of velocity, pressure, viscosity and shear
stress at the interface, and the no-slip condition on
solid boundaries. Results obtained support the
following conclusions:
1) Parameters that influence the flow are the
angle of inclination, , pressure condition
at the upper channel wall, thickness of the
lower porous layer, viscosity adjustment
parameter, , and the Darcy number of each
layer.
2) For a given pressure distribution, viscosity
distribution is the most sensitive parameter
as it controls the behaviour and values of
viscosity. High and low values of could
result in unrealistic viscosity values.
3) Effects of the parameters on velocity at the
interface are as follows:
i) Increasing , all other parameters fixed, increases
the velocity and vorticity, but decreases shear stress
at the interface.
ii) Increasing results in a decrease in the interfacial
velocity, and a decrease in the vorticity. The shear
stress at the interface remains unaffected by changes
in .
iii) Increasing , results in an increase in the
interfacial velocity, and a corresponding increase in
the vorticity, and a numerical decrease in the shear
stress at the interface.
iv) Increasing results in a decrease in the interfacial
velocity and a decrease in the vorticity, and a
numerical increase in shear stress.
v) Velocity at the interface increases with increasing
for a fixed value of . Vorticity decreases,
numerically, with decreasing , and shear stress at
the interface increases, numerically with decreasing
.
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Contribution of individual authors
Both authors participated in pertinent literature
review.
Both authors independently obtained the solutions to
governing equations.
M.S. outlined the steps to take in conducting this
research.
M.S. provided calculations and graphing of results
using Maple, and provided initial analysis of results.
M.H. identified what quantities to calculate and
values of parameters to be used.
M.H. analysed the results wrote the manuscript.
Sources of funding
This research did not receive any specific grant
from funding agencies in the public, commercial,
or not-for-profit sectors.
Creative Commons Attribution License 4.0
(Attribution 4.0 International , CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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