On the Sigmoid Function as a Variable Permeability Model for
Brinkman Equation
D.C. ROACH
Department of Engineering
University of New Brunswick
100 Tucker Park Road, Saint John, New Brunswick, E2L 4L5
CANADA
M.H. HAMDAN
Department of Mathematics and Statistics
University of New Brunswick
100 Tucker Park Road, Saint John, New Brunswick, E2L 4L5
CANADA
Abstract: - A collection of popular variable permeability functions is presented and discussed in this work. The
functions have been used largely in Brinkman’s equation which governs the flow through a porous domain in the
presence of solid, macroscopic boundaries on which the no-slip condition is imposed, and has been used in
transition layer modelling. A convenient classification of permeability functions is also provided. The sigmoid
logistic function is presented in this work in a modified form that is suitable for variable permeability modelling,
and is used in obtaining solution to Poiseuille flow through a Brinkman porous channel.
Key-Words: - Sigmoid function, Porous media, Transition layer, Variable permeability
Received: April 19, 2021. Revised: January 20, 2022. Accepted: February 22, 2022. Published: March 23, 2022.
1 Introduction
In order to overcome some of the short-comings of
Darcy’s equation, the literature reports on a
number of its modifications and generalizations,
[1], [2], [3]. As is well known, the celebrated
Darcy’s equation is used in the study of seepage
flow through porous media and does not account
for microscopic inertial effects that arise due to
tortuosity of the flow path. It is marked by the
absence of a viscous shear term that is necessary to
account for viscous shear effects. These effects
arise when a macroscopic, solid boundary is
encountered, and are useful in treating flows with
special viscosities (cf. [1] ,[3] and the
references therein) and with flows that involve
dust particle settling (cf. [2] and the references
therein). Darcy’s equation may be valid in low
permeability, low porosity media where
variations at the microscopic, pore-level length
scale are negligible at the macroscopic length scale.
In its steady-state form,
Darcy’s equation can be written in the following form
when permeability to the fluid is a non-tensorial
quantity:
(1)
wherein is the base-fluid constant viscosity, is the
density of the fluid, is the pressure, is the
gravitational acceletarion, is the medium
permeability as a function of position , and
is the
seepage velocity vector.
Nakshatrala and Rajagopal, [3], discussed other
limitations of Darcy’s equation and its short-comings
in predicting important phenomena in flow through
porous media, and emphasized that while Darcy’s
equation merely predicts fluid flux through the
porous structure, its flux prediction is not accurate at
high pressures and pressure gradients. Consequently,
flows of fluids with variable viscosity, namely ones
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2022.17.5
D. C. Roach, M. H. Hamdan
E-ISSN: 2224-3429
29
Volume 17, 2022
in which viscosity changes with pressure, require a
generalization of Darcy’s equation in order to better
model functional dependence of viscosity on pressure
and to quantify the drag force in this type of flow and
its dependence on permeability variations.
Nakshatrala and Rajagopal, [3], provided a
generalized form of Darcy’s equation in which the
drag coefficient depended on pressure. Chang et.al.,
[4], reported Darcy’s equation in the following
generalized form:
(2)
with drag force given by:
(3)
wherein is the fluid fixed viscosity, and is the
experimentally obtained Barus coefficient.
In order to overcome the limitation imposed by
the absence of a viscous shear term in Darcy’s
equation, Brinkman, [5], modified Darcy’s equation
into the form
(4)
where is the effective viscosity of the fluid in
the porous medium and is the superficial average
velocity vector. The effective viscosity was shown
to depend on porosity of the medium and the
viscosity of the base fluid, [6].
Various authors have discussed validity and
limitations of Brinkman’s equation, [7], [8], [9],
[10]. Rudraiah, [10], suggested that
Brinkman’s equation is the most appropriate
model of flow through porous layers of finite
depth, while Nield, [8], elegantly concluded that the
use of Brinkman’s viscous shear term requires a
redefinition of the porosity near a solid boundary
due to a process referred to as channelling.
Sahraoui and Kaviany, [11], and Kaviany, [12],
studied the case of flow through variable
permeability media when using Brinkman’s
equation and emphasized the need for variable
permeability near macroscopic boundaries,
whether slip or no-slip conditions are applied.
Hamdan and Barron, [13], showed
numerically that the Laplacian in Brinkman’s
equation is significant in a thin layer near a solid
boundary, but less significant in the core of the
porous medium.
Brinkman’s equation received considerable
attention in the literature in the analysis of flow
through porous layers with applications to heat and
mass transfer. For uni-directional flow, equation
(4) takes the following form, wherein
is the
tangential velocity component and
is the
constant driving pressure gradient:
(5)
When permeability is constant, equation (5)
admits the following solution, [14], in the flow
through a channel with solid walls at
wherein
:
(6)
While in many idealizations and analyses of uni-
directional and two-dimensional flows through
porous media, the use of constant permeability has
been the rule rather than the exception, it has long
been recognized that naturally occurring porous
media possess variable permeability. A number of
articles in the scientific literature have reported a shift
towards variable permeability modelling. A number
of reasons are behind this shift, some of which are
summarized in what follows.
1- Transition layer: In the study of flow
through channels over porous layers, the use
of a constant permeabilitty in the porous
layer results in a permeability discontinuity
at the interface. To circumvent, Nield and
Kuznetsov, [15], suggested the use of a
transition layer between a constant
permeability porous layer and free-space.
Permeability in the transition layer is
variable and ranges from the constant
permeability at the intersection with the
porous layer to its infinity value at the
interface with the free-space channel. While
Nield and Kuznetsov, [15], introduced one
model to account for variations in
permeability, various other models have
been developed.
2- Channeling effects: In dealing with the
problem of flow through a porous medium as
governed by Brinkman’s equation, Hamdan
and Kamel, [14], emphasized the need for
variable permeability modelling to be
compatible with Brinkman’s equation and to
better handle the no-slip condition on
macroscopic, solid boundaries and to
alleviate channelling effects near a solid
boundary.
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2022.17.5
D. C. Roach, M. H. Hamdan
E-ISSN: 2224-3429
30
Volume 17, 2022
3- Influence on heat and mass transfer: In
many arising applications of flow through
porous media, such as microfluidics and
flows with variable viscosity fluids, the flow
of polar fluids, [1], and the flow of dusty
gases over porous layers, [2], it is imperative
to account for permeability and porosity
variations. These variations influence heat
and mass transfer in the porous layers.
The above, and many other reasons, emphasize
the need to develop robust variable permeability
models. The main contributions in this area have
recently been reviewed by Roach and Hamdan, [16],
who also reported on the most recent models
available for the transition layer. They provided a
classification of the available models into five
categories.
In this work, further comments on the available
models and the solution to Brinkman’s equation
when Poiseuille flow is considered through a variable
permeability porous channel bounded by solid walls
have been provided. Additionally, a new variable
permeability model that is compatible with
Brinkman’s equation is developed. The model is
based on the sigmoid logistic function and can be
used in the study of flow through porous layers with
or without interfacial conditions, as well as in the
study of flow of pressure-dependent viscosity fluids
in porous channels and layers with variable
permeability. The sigmoid function avoids the
sudden variation in the permeability present in
other approaches to modeling porous flows and
can provide a more gradual, natural, variation of
the permeability.
2 Overview of Permeability Functions
2.1 Direct and Inverse Models
In three-dimensional flow through naturally
occurring porous media, permeability is a tensorial
quantity. Idealizations of flow through one and two
space dimensions, however, is important to our
understanding of the flow phenomena. This
idealization gives rise to variable permeability
models that use algebraic functions of one of the
space variables. Equation (5) describes fully-
developed, uni-directional flow and involves two
functions of the normal space variable: the
tangential velocity function
, and the medium
permeability function,
.
A determinate solution of (5) necessitates that
one of the functions must be known. If the
permeability function is specified, then solution of
equation (5) is sought for
𝑢(𝑦)
which then
describes the velocity profile associated with the
prescribed permeability distribution. This
approach is referred to as the direct method and is
more popular in the literature, where a number of
variable permeability models are available and
serve a spectrum of flow situations and flow
domains of specific industrial applications (c.f.
[14], [18], [19], [20], and the references therein).
An important set of the available variable
permeability models that received considerable
attention in the literature are classified in the
subsections to follow.
In inverse analysis, however, once can assume
the form of
𝑢(𝑦)
and solve equation (5) for
𝑘(𝑦)
to provide a permeability distribution that
produces the specified velocity profile.
The above ideas were implemented in devising a
permeability function for Brinkman’s equation by
Hamdan and Kamel, [14]. They determined that, in
Poiseuille flow through a porous channel, the
permeability function is of the form
(7)
and
(8)
where
; is the depth of a porous channel,
and is a dimensionless number given by
, wherein is the maximum
(constant) permeability attained in the porous
channel.
The idea above emphasizes that velocity and
permeability functions can be taken proportional to
each other. This concept was implemented by Abu
Zaytoon et al, [21], in their analysis of flow of
pressure-dependent viscosity fluid over an inclined
variable permeability porous layer whose
permeability function is of the form
(9)
where is a reference constant permeability.
Another permeability function that proved to be of
utility in the study of flow of pressure-dependent
viscosity fluids down an inclined plane was provided
by Alzahrani et.al., [22], and in which the
permeability was taken as the square of the variable
pressure function, given by:
(10)
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2022.17.5
D. C. Roach, M. H. Hamdan
E-ISSN: 2224-3429
31
Volume 17, 2022
where , is a constant, is
the density of the fluid, is the gravitational
acceleration and is the angle of inclination.
In obtaining analytic solutions to the Darcy-
Lapwood-Brinkman equation with variable
permeability, Alharbi et.al., [23], found it convenient
to introduce the following permeability function:
(11)
where , is a parameter, is Reynolds
number, defined as , wherein and
are density and viscosity of the fluid, respectively,
and and are characteristic velocity and length,
respectively.
2.2.
Exponential Models
In their study of similarity solutions for buoyancy
induced flows in a saturated porous medium adjacent
to impermeable horizontal surfaces, Chandrasekhara
et al., [24] assumed that permeability varied
exponentially from the wall and provided the model
given by
(12)
where is the value of the permeability at the edge
of the boundary layer, is a constant having
dimensions of y, and d is a constant. A similar model
was used by Hassanien et al., [25], in their study of
vortex instability of mixed convection flow.
Rees and Pop, [26], studied the free convection
in a vertical porous medium with the exponential
model of permeability
(13)
where
is the permeability at the wall,
is the
permeability of the ambient medium, and d is the
length scale over which the permeability varies.
Alloui et al., [27], analyzed convection in
binary mixtures using the exponential model of the
form
(14)
where
is a fitting parameter. When
is small,
permeability function (7) behaves like the linear
function
(15)
Abu Zaytoon et.al., [18], considered an
exponential permeability of the form
(16)
in their analysis of flow through composite porous
layers, one of which was a Darcy layer.
Pillai et.al., [28], considered the steady MHD
flow in an inclined channel over a porous layer
with a decaying exponential permeability that
depends on the depth of the porous layer, using the
following model due to Sinha and Chadda, [29]:
(17)
while Silva- Zea et.al., [30], studied MHD flow
using a model of the form
(18)
where c is a positive number, and is the average
permeability of the medium.
2.3. Polynomial Models
Other important models of one dimensional
permeability variation used in soil mechanics
include those found in Schiffman and Gibson,
[31]; Mahmoud and Deresiewicz, [32], and Jang
and Chen, [33]. Cheng, [17], used the model
(19)
where β and
are parameters of curve fittings,
and
is the characteristic permeability of the
medium. A typical value for
is 2.
In their study of the effects of variable
permeability on MHD flow in a porous channel
of depth
, Narasimha Murthy and Feyen, [34],
used the following form of equation (12):
(20)
where
is the permeability in the interior of the
porous medium, while Srivastava and Deo, [35],
employed the form:
(21)
where
, in their analysis of Couette and
Poiseuille MHD flow in a porous layer.
In their analysis of thermo-solutal convection in
a heterogeneous porous layer enclosed in a
rectangular cavity, Choukairy and Bennacer, [36],
implemented a permeability function of the
following form, wherein n is a parameter.
(22)
2.4. Periodic Models
Mathew, [37], employed the following periodic
variable peermeability model in the study of two-
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2022.17.5
D. C. Roach, M. H. Hamdan
E-ISSN: 2224-3429
32
Volume 17, 2022
dimensional MHD convective heat transfer through a
vertical porous channel:
(23)
where is a constant and is a constant reference
permeability.
2.5. Transition Layer Models
Some of the recent advances in the study of flow
through a free-space channel over porous layers
include imbedding a Brinkman transition layer of
variable permeability between the channel and a
constant permeability porous layer. This is depicted
in Fig. 1, if we take Layer 1 to be a semi-infinite
Darcy layer, Layer 2 a Brinkman variable
permeability layer and Layer 3 a free-space fluid
layer.
At the interface with the channel, permeability of
the transition layer approaches infinity, while at the
interface with the porous layer, it assumes the value
of the constant permeability of the porous layer.
Nield and Kuznetsov, [15], introduced a permeability
function in their analysis of the transition layer, in
such a way that the reciprocal of the permeability
varies linearly across the layer, according to
(24)
where H is the overall porous layer thickness, is
related to the thickness of the transition layer, is the
transition layer permeability, and is the constant
permeability of the underlying porous layer. This
choice of permeability distribution reduces
Brinkman’s equation in the transition layer to an
Airy’s inhomogeneous ordinary differential
equation.
The Nield and Kuznetsov approach has been
successfully implemented in obtaining solutions to
flow through composite porous layers of variable
permeability by Abu Zaytoon et.al., [19]. They, [19]
also provided analysis suggesting that the
permeability distribution in the transition layer can be
modelled in a way that reduces the governing
Brinkman’s equation to the generalized
inhomogeneous Airy’s equation, of which the Nield
and Kuznetsov, [15], model is a special case. This is
depicted in Fig. 1, if Layer 1 is taken to be a free-
space fluid layer, Layer 2 a Brinkman variable
permeability layer and Layer 3 a Brinkman constant
permeability layer. Permeability function in this case
has been described as:
(25)
Although Airy’s and generalized Airy’s equations
are useful in the analysis of the transition layer, there
are situations that lead to other special differential
equations. One such flow configuration is where a
porous layer of variable permeability is immersed in
a free-space channel, [38].
If, in Fig. 1, Layer 1 is a free-space fluid layer,
Layer 2 is a Brinkman variable permeability layer
and Layer 3 is a free-space fluid layer, then
permeability distribution across the Brinkman porous
layer, with interfaces at and , can be
modelled using the following function:
(26)
where is a reference constant permeability, and
is an adjustable parameter such that
, or . The resulting porous
layer thickness is . Equation (26) produces
both symmetric and non-symmetric regions and can
be used in a number of ways, one of which is
selecting the widths of the lower and upper fluid layer
then determining . It should be noted that
this choice of permeability function reduces
Brinkman’s equation to a Weber inhomogeneous
differential equation whose solution has been
discussed in details by Abu Zaytoon and Hamdan,
[38].
Fig. 1. Representative Sketch
3 Sigmoid Function
A function that has received considerable attention in
neural networks and deep learning is the
sigmoid function, [39], [40], [41], [42]. This
function possesses a characteristic S-shaped curve
and maps the number line onto a finite-length
subinterval, such as (0,1).
Three popular sigmoid functions are the logistic
function, the arctangent and hyperbolic tangent. In
its
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2022.17.5
D. C. Roach, M. H. Hamdan
E-ISSN: 2224-3429
33
Volume 17, 2022
most common form, the sigmoid logistic function
function is given by:
(27)
and has horizontal asymptotes at with
and
(28)
Its first two derivatives are given by
(29)
(30)
is increasing on its domain and has a point
of inflection at . It represents the solution to the
initial value problem
(31)
and its integral given by
C (32)
where C is a constant.
The sigmoid function enjoys various applications
in deep learning and neural networks as it serves as
an activation function (cf. [40], [41] and the
references therein). Since this smoothly-increasing
function has not made it to the porous media
literature, the intention here is to present it as a
candidate for modelling variations in permeability
across a porous layer. The S-shaped graph of the
sigmoid function, [42], makes it appealing in the
study of transition layers. . In what follows, a
modification of is provided and used in the
Poiseuille flow through a Brinkman porous layer of
variable permeability. The sigmoid function is used
here to create continuously varying permeability
between relatively constant permeability regions.
This approach, which treats the flow domain as one
region with variable permeability, seems to replicate
the flow in layered media.
Consider the unidirectional, fully-developed flow
through the porous domain described by
. The porous
layer is bounded by solid, impermeable walls at
and , and the flow is governed by Brinkman’s
equation (5). On the macroscopic solid walls, the no-
slip condition is used. It is
assumed that the permeability in the layer varies in
the lateral direction, , and permeability variations
are governed by the following modified form of the
sigmoid function:
(33)
where , , and is adjusted to
change the rate of transition between and .
In this work, varying widths of permeability
transition are simulated and the different parameters
in (33) investigated. Graphs of are shown in
Fig. 2 for h=0.1m, , ,
and various values of n. With increasing n and the
creation of very narrow variable permeability region,
a stepwise change in permeability is observed. For
n=0 and n=-1, permeability varies approximately
linearly, with a minimum amount of change when n
< 0. This is quite significant and implies that the
thickness of the variable permeability layer (region)
can be controlled as it varies smoothly between two
constant permeability layers (regions) without having
to resort to interfacial conditions.
Fig. 2. Variable Permeability as a Function of
Parameter n
It should be noted that the permeability on the
solid bounding walls is zero. This creates a jump in
the permeability as we move away from the solid
walls into the porous layer. However, this situation is
no different than the usual practice of analyzing the
flow through a constant permeability layer bounded
by solid walls, where there is a jump in permeability
as it starts with a value of zero on the boundary and
takes on a non-zero constant value in the porous
layer.
Equation (5), with given by (33), was solved
numerically using fifth order algorithm in MATLAB
with relative error tolerance of and absolute
error tolerance of . All quantities and parameters
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2022.17.5
D. C. Roach, M. H. Hamdan
E-ISSN: 2224-3429
34
Volume 17, 2022
have been left in dimensional form for ease of
physical interpretation of results. Results were
generated for the following dimensional quantities:
(water)
Velocity profiles for various values of n are
illustrated in Fig. 3, which shows the faster flow
regiments in the upper parts of the channel with
increasing n. In these regions, permeability is higher,
thus resulting in faster flow. For the cases of n=0 and
n=-1, the velocity profiles resemble those of
Brinkman flow through constant permeability layers.
This is an expected behaviour as the permeability
variations across the layer are linear with small
variations when n=0 and n=-1. Furthermore,
inflection in the velocity profiles is found to increase
in strength as the transition region narrows with
increasing .
Fig. 3. Velocity Profiles for Various Values of n
Darcy friction, or Darcy resistance in Brinkman’s
equation, is given by the quantity
. This has
been calculated for various values of n and illustrated
in Fig. 4, which shows the effects of variable
permeability on resistance offered to the flow across
the layer, and the thickness of variable permeability
region that is impacted (in particular, for n = 1,2,3).
As the transition layer thickness decreases with
increasing n, the flow approaches a step-change in
permeability and the peak of the Darcy friction
increases dramatically. For n=0 and n=-1, the near
constant permeability across the channel results in a
parabolic Darcy resistance profile, as there are no
spikes or sudden changes in permeability.
Fig. 4. Darcy Friction for Various Values of n
Viscous shear stress, across the porous layer
is given by the expression:
. Fig. 5
illustrates the effects of the employed permeability
model and the values of n in the Sigmoid function on
shear stress across the channel. For a given effective
viscosity coefficient, , the term
gives the
instantaneous change in velocity with respect to the
lateral variable. In regions where the transition across
the variable permeability layer is rapid (i.e. a thin
transition layer), these changes are abrupt, as shown
for the cases of n=2, 3.
For the case of n=1, changes are smooth, and for
n=0, -1, the shear stress profiles are closer to
linearity. Since the corresponding velocity profiles
are almost parabolic, the viscous shear profiles
should be almost linear. The reason that they are not
linear could be ascribed to the fact that a no-slip
condition was used on the solid walls, which resulted
in a jump in the permeability as one moves into the
porous channel. It is expected that the viscous shear
profiles would be different in cases of a slip condition
associated with a non-zero permeability on a porous
boundary.
The above behaviour also influences, and
explains, the behaviour of the net viscous shear
profiles, given in Fig. 6 for various n. The net viscous
shear stress across a fluid element is given by the net
instantaneous change of the shear stress term with
respect to the lateral direction, namely,
(34)
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2022.17.5
D. C. Roach, M. H. Hamdan
E-ISSN: 2224-3429
35
Volume 17, 2022
As the value of n increases and the permeability
approaches a step-change, the net viscous shear
across a fluid element must also adjust abruptly to
maintain the force balance modelled by equation (5).
As can be seen in Fig. 6 when comparing the net
viscous shear profiles of rapidly varying permeability
to those of near constant permeability, there are
substantial deviations to the net viscous shear force.
For example in the cases of n=0, -1, the net viscous
shear is always negative while for the positive values
of n there are regions of positive net viscous shear.
This signifies a profound physical change in the
physics governing the flow across the layer as the
direction of the net viscous shear force changes from
impeding the flow (negative net viscous shear) to
enhancing the velocity (positive net viscous shear).
The direction change of the net viscous shear
force acting on a fluid element that can be seen for
the cases of n=1, 2, 3 results from the development
of the inflection in the velocity profiles, which
themselves are produced by the rapidly varying
permeability. In these regions of inflection, regions
of faster flow attempt to accelerate the adjacent
slower regions in the direction of flow. Furthermore,
in the cases of n=2, 3 it is seen that there is abrupt
sign change in the net viscous shear, the location of
which correlates to the region of rapid variation of the
permeability.
Fig. 5. Viscous Shear Stress for Various Values of n
Fig. 6. Net Viscous Shear Stress for Various n
4 Conclusion
In this work, a listing and a classification of variable
permeability models that have been reported and
used frequently in the literature on porous media,
have been provided. The classification provided
groups the models into five different sub-classes for
ease of reference.
A variable permeability model that is based on
the sigmoid logistic function, which was modified to
describe smooth transition between two regions of
constant permeability in a single domain without the
need for interfacial conditions, has been
implemented. The sigmoid function approach is
believed to be promising in the analysis of flow
through variable permeability layers and in the
simulation of flow in the transition layer.
Furthermore, the effect of imposing step-wise
changes to the permeability on the various terms
governing flow in porous media has been examined.
Future aspects of this work is to compare the
different classes using a number of flow situations of
interest and of practical significance, and to extend
the models to the study of pressure-dependent
viscosity fluid flows.
References:
[1] Kamel, M.T., Roach, D.C., and Hamdan, On the
Micropolar Fluid Flow through Porous Media.
Mathematical Methods, System Theory and
Control, ISBN 978-960-474-094-9, WSEAS
Press, 2009, pp. 190-197.
[2] Hamdan, M.H., Roach, D.C., Silva-Zea,R.,
Erazo-Bone, R., Chuchuca-Aguilar, F. and
Escobar-Segovia, K., On the Flow of a Dusty
Gas through a Channel over a Porous Layer
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2022.17.5
D. C. Roach, M. H. Hamdan
E-ISSN: 2224-3429
36
Volume 17, 2022
Saturated with a Dusty Gas. IOSR Journal of
Engineering, Vol. 10(9), 2020, pp. 35-44.
[3] Nakshatrala, K.B. and Rajagopal, K.R., A
Numerical Study of Fluids with Pressure-
dependent Viscosity Flowing through a Rigid
Porous Medium. Int. J. Numer. Meth. Fluids,
Vol. 67, 2011, pp. 342-368.
[4] Chang, J., Nakashatrala, K.B. and Reddy, J.N.,
Modification to Darcy-Forchheimer Model Due
to Pressure-dependent Viscosity: Consequences
and Numerical Solutions. J. Porous Media, Vol.
20(3), 2017, 263-285.
[5] Brinkman, H.C., A Calculation of the Viscous
Force Exerted by a Flowing Fluid on a Dense
Swarm of Particles. Applied Scientific Research,
A1, 1947, pp. 27-34.
[6] Lundgren, T. S., Slow Flow through Stationary
Random Beds and Suspensions of Spheres. J.
Fluid Mechanics, Vol. 51, 1972, pp. 273–299.
[7] Auriault, J.-L., On the Domain of Validity of
Brinkman’s Equation. Transport in Porous
Media, Vol. 79(2), 2009, pp. 215-223.
[8] Nield, D. A., The Limitations of the Brinkman-
Forchheimer Equation in Modeling Flow in a
Saturated Porous Medium and at an Interface.
Int. J. Heat and Fluid Flow, Vol. 12(3), 1991, pp.
269-272.
[9] Nield, D. A., and Bejan, A., Mechanics of Fluid
Flow through Porous Media. In: D.A. Nield and
A. Bejan, Convection in Porous Media, 3rd ed.,
Springer, USA, 2006, pp. 14-16.
[10] Rudraiah, N., Flow Past Porous Layers and
their Stability. In N. P. Cheremisinoff (Ed.),
Encyclopedia of Fluid Mechanics, Slurry Flow
Technology, Gulf Publishing, Houston, Texas,
USA, Vol. 8, 1986, pp. 567-647.
[11] Sahraoui, M., and Kaviany, M., Slip and No-
slip Velocity Boundary Conditions at Interface of
Porous, Plain Media. Int. J. Heat and Mass
Transfer, Vol. 35(4), 1992, pp. 927-943.
[12] Kaviany, M., Part I: Single Phase Flow. In
M. Kaviany, Principles of Heat Transfer in
Porous Media, 2nd ed., Spinger, Mechanical
Engineering Series, 1995, pp. 95-100.
[13] Hamdan, M.H. and Barron, R.M., Analysis
of the Darcy-Lapwood and the Darcy-Lapwood-
Brinkman Models: Significance of the Laplacian.
Applied Math. Comput., Vol. 44(2), 1991, pp.
121-141.
[14] Hamdan, M.H. and Kamel, M.T., Flow
through Variable Permeability Porous Layers.
Adv. Theor. Appl. Mech., Vol. 4(3), 2011,
pp.135-145.
[15] Nield, D.A. and Kuznetsov, A.V., The Effect
of a Transition Layer between a Fluid and a
Porous Medium: Shear Flow in a Channel.
Transport in Porous Media, 78, 2009, pp. 477-
487.
[16] Roach, D.C. and Hamdan, M.H., Variable
Permeability and Transition Layer Models for
Brinkman Equation. International Khazar
Scientific Researches Conference-III, January 7-
9, 2022, Khazar University, Baku, Azerbaijan,
Proceedings, ISBN: 978-625-8423-84-6, IKSAD
Publishing House. 2022, pp. 184-191.
[17] Cheng, A. H.-D., Darcy's Flow with Variable
Permeability: A Boundary Integral Solution.
Water Resources Research, Vol. 20(7), 1984, pp.
980-984.
[18] Abu Zaytoon, M.S., Alderson, T.L. and
Hamdan, M. H., Flow over a Darcy Porous Layer
of Variable Permeability. J. Applied
Mathematics and Physics, Vol. 4, 2016, pp. 86-
99.
[19] Abu Zaytoon, M.S., Alderson, T.L. and Hamdan,
M. H., Flow through a Variable Permeability
Brinkman Porous Core. J. Applied Mathematics
and Physics, Vol. 4, 2016, pp. 766-778.
[20] Hamdan, M. H., and Abu Zaytoon, M. S., Flow
Over a Finite Forchheimer Porous Layer with
Variable Permeability. IOSR Journal of
Mechanical and Civil Engineering, Vol. 14(3),
2017, pp. 15-22.
[21] Abu Zaytoon, M.S., Xiao, Y. and Hamdan,
M. H., Flow of a Fluid with Pressure-dependent
Viscosity through Variable Permeability Porous
Layers. WSEAS Transactions on Applied and
Theoretical Mechanics, Vol. 16, 2021, pp. 204-
212.
[22] Alzahrani, S. M., Gadoura, I. and Hamdan,
M. H., A Note on the Flow of a Fluid with
Pressure-dependent Viscosity through a Porous
Medium with Variable Permeability. J. Modern
Technology and Engineering, Vol. 2(1), 2017,
pp. 21-33.
[23] Alharbi, S.O., Alderson, T.L. and Hamdan,
M.H., Analytic Solutions to the Darcy-Lapwood-
Brinkman Equation with Variable Permeability.
Int. J. Eng. Res. Applic., Vol. 6(4), 2016, pp. 42-
48.
[24] Chandrasekhara, B., and Namboodiri, P.,
Influence of Variable Permeability on Combined
Free and Forced Convection About Inclined
Surfaces in Porous Media. Int. J. Heat and Mass
Transfer, Vol. 28(1), 1985, pp. 199-206.
[25] Hassanien, I., Salama, A., and Elaiw, A.,
Variable Permeability Effect on Vortex
Instability of Mixed Convection Flow in a Semi-
infinite Porous Medium Bounded by a
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2022.17.5
D. C. Roach, M. H. Hamdan
E-ISSN: 2224-3429
37
Volume 17, 2022
Horizontal Surface. Applied Math. and Comput.,
Vol. 146(2-3), 2003, pp. 829-847.
[26] Rees, D. A., and Pop, I., Vertical Free
Convection in a Porous Medium with Variable
Permeability Effects. Int. J. Heat and Mass
Transfer, Vol. 43, 2000, pp. 2565-2571.
[27] Alloui, Z., Bennacer, R. and Vasseur, P.,
Variable Permeability Effect on Convection in
Binary Mixtures Saturating a Porous Layer. Heat
and Mass Transfer, Vol. 45(8), 2009, pp. 1117-
1127.
[28] Pillai, K., Varma, S., and Babu, M. S.,
Aligned Magnetic Effects through Varying
Permeable Bed. Proc. Indian Acad. Sci. (Math.
Sci.), Vol. 96(1), 1987, pp. 61-69.
[29] Sinha, A., & Chadda, G., Steady Laminar
Viscous Flow Down an Open Inclined Channel
with a Bed of Varying Permeability. Indian J.
Pure Appl. Math., Vol. 15(9), 1984, pp. 1004-
1013.
[30] Silva-Zea, R., Hamdan, M.H., Erazo-Bone,
R., Chuchuca-Aguilar, F. and Escobar-Segovia,
K., MHPM Solution to MHD Fluid Flow through
Porous Medium with an Exponentially Varying
Permeability. ACI Advances in Science and
Engineering, Vol. 13(2), 2021, pp. 1-24.
[31] Schiffman, R., and Gibson, R.,
Consolidation of Nonhomogeneous Clay Layers.
J. Soil Mechanics and Foundations Division,
Vol. 90(5), 1964, pp. 1-30.
[32] Mahmoud, M. S. and Deresiewicz, H.,
Settlement of Inhomogeneous Consolidating
Soils—I: The Single‐drained Layer under
Confined Compression. Int. J. for Numerical and
Analytical Methods in Geomechanics, Vol. 4(1),
1980, pp. 57-72.
[33] Jang, J.-Y., & Chen, J.-L. (1993). Variable
Porosity Effect on Vortex Instability of a
Horizontal Mixed Convection Flow in a
Saturated Porous Medium. Int. J. Heat and Mass
Transfer, Vol. 36(6), 1993, pp. 1573-1582.
[34] Narasimha Murthy, S. and Feyen, J.,
Influence of Variable Permeability on the
Dispersion of a Chemically Reacting Solute in
Porous Media. Int. J. Engineering Science, Vol.
27(12), 1989, pp. 1661-1671.
[35] Srivastava, B., and Deo, S., Effect of
Magnetic Field on the Viscous Fluid Flow in a
Channel Filled with Porous Medium of Variable
Permeability. Applied Mathematics and
Computation, Vol. 219(17), 2013, pp. 8959-
8964.
[36] Choukairy, K., & Bennacer, R., Numerical
and Analytical Analysis of the Thermosolutal
Convection in an Heterogeneous Porous Cavity.
FDMP-Fluid Dynamics & Materials Processing,
Vol. 8(2), 2012, pp. 155-172.
[37] Mathew, S., Mathematical Analysis. In: S.
Mathew, MHD Convective Heat Transfer
through a Porous Medium in a Vertical Channel
with Periodic Permeability. Sri Krishnadevaraya
University, Anantapur, India. Retrieved from
http://hdl.handle.net/10603/64940. 2005, pp. 7-
10.
[38] Abu Zaytoon, M.S. and Hamdan, M. H.,
Weber Equation Model of Flow through a
Variable-Permeability Porous Core Bounded by
Fluid Layers. Trans. AMSE. J. Fluids Eng. Apr.
2022, 144(4): 041302 (9 pp.). Paper No: FE-21-
1303 https://doi.org/10.1115/1.4052927
[39] Wood, T., Sigmoid Function. Retrieved
March 8, 2022. https://deepai.org/machine-
learning-glossary-and-terms/sigmoid-function.
[40] Skapura, D.M., Building Neural Networks.
ACM Press, New York, NY, 1996.
[41] Haykin, S., Neural Networks: A
Comprehensive Foundation, 2/e. Prentice-Hall,
Inc., 1999
[42] von Seggern, D. CRC Standard Curves and
Surfaces with Mathematics, 2nd ed. Boca Raton,
FL: CRC Press, 2007.
Contribution of individual authors
Both authors carried out the literature survey,
problem formulation, result analysis, and manuscript
preparation.
D.C. Roach formulated the sigmoid model and
carried out the simulation.
Sources of funding: No funds were received
from any source in support of this work.
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
_US
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2022.17.5
D. C. Roach, M. H. Hamdan
E-ISSN: 2224-3429
38
Volume 17, 2022
Conflicts of Interest
The authors have no conflicts of interest to
declare .
