Abdul M Siddiqui1, Getinet A Gawo1 and Qurat A Azim2*
1Department of Mathematics, York Campus, Pennsylvania State University, York, PA 17403, USA
2Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Defence Road, Lahore 54000, Pakistan
*Corresponding author: Qurat A Azim, Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Defence Road, Lahore 54000, Pakistan
Submission: January 19, 2021 Published: January 28, 2021
ISSN 2637-8078Volume4 Issue3
The study of fluid flow through porous walled channels and ducts has many applications
in biomechanics and in industry. In industry, processes such as desalination, reverse osmosis
and flow through various tubular nanostructures (see, for example, [1]). However, such
fluid flow studies have gained much popularity due to their applications in biology and
biomechanics. For example, a few dozens of blood flows through capillaries and arterioles
each day. The blood pressure in arterioles is much lower in comparison with that in the
main arteries. Therefore, the pressure gradient gives rise to Poiseuille like flow in such
structures. However, the Poiseuille like flow cannot be maintained in the renal tubules of a
kidney. Kidneys are vital organs in an organism and perform the function of filtration of fluid
through the body. Kidneys balance the amount of water in the body apart from getting rid of
metabolic waste. Each day, about 200 liters of blood passes through human kidneys in order
to filter out about 2 liters of excessive water containing waste products in the form of urine
[2]. Blood enters a kidney through renal arteries for purification where the metabolic waste
in the blood enters glomerular filtrate (urine). Kidney contains more than a million small
filtration units called nephrons. The nephron can structurally be divided into two parts-the
Bowman’s capsule and the renal tubule. Absorption of useful substances like glucose, sodium,
bicarbonate, potassium, phosphate, calcium and amino acids from the filtrate takes place in
the nephron. This reabsorption takes place through small pores among the surface cells on
the tube walls. The glomerular filtrate, after the reabsorption process is completed, enters the
bladder through ureters for excretion. There have been several mathematical studies on the
analysis of fluid flow through renal tubules, both in plane channel geometry and in cylindrical
tube geometry. Researchers have assumed several variations in the type of variation that takes
place through the tubule walls. Some discussions of renal tubule models were presented by
Wesson [3] & Burgen [4]. These studies were theoretical in nature and the authors assumed
a constant rate of reabsorption. There have also been studies on a purely mathematical basis
that address the analysis of flow through porous walled channels and ducts. These works,
however, by Berman [5-9] do not include the application of flow through renal tubules. The
idea of these studies is to establish the nature of the flow as a two-dimensional flow. This is
caused by a transverse velocity component that arises due to the suction/absorption that
takes place at the surface of channel walls. Thus, the velocity pro les of such flows differs
greatly from simple Poiseuille flow.
Macey [10] presented the rest formal mathematical treatment of the flow through renal
tubules taking into account a constant or uniform reabsorption rate. Kelman [11] argued that
the reabsorption function showed an exponential decay through the proximal renal tubule in
his theoretical study. Macey [12] again incorporating the exponential decay assumption in his
work. He presented explicit solutions for axially symmetric creeping flow as well as for the
average pressure drop in the tubule. Kozinski [13] presented an extension of Macey’s work
for porous channels. Marshall [14] & Palatt et al. [15] presented their analyses of fractional
reabsorption and leakage flux in the proximal renal tubules. Radhakrishnamacharya [16]
also studied such flows in tubules of varying cross section. On similar lines, diverging and
converging tubes as well as tubes with slowly varying cross-section were also studied [17]. Ahmad [18] additional considered periodic reabsorption velocity
at tubule walls and obtained exact solutions for the flow. More
recently, Siddiqui et al. [19-24] presented various mathematical
analyses for creeping flow of Newtonian fluid through a porous slit
or tube with various functions for reabsorption at the wall, with
no-slip as well as slip effects taken into consideration. In another
advancement of the studies under consideration, it is argued that
disease in the renal tubule can make the tubule channel to act as
a porous medium. Various diseases like interstitial nephritis or
tubular proteinuria can cause excess amounts of fibers, proteins,
fatty substances and cholesterol can enter the glomerular filtrate.
These substances can then get suspended in tubule channels. These
glomerular diseases can therefore affect the permeability of the
tubular channel. Moreover, some fatty fibers may also cause full or
partial blockage of the channel. In the latter case, some material
passes through the blocked duct. This phenomenon can be modeled
effectively with the equations for flow through porous media. These
models take into account the supplementary drag forces exerted on
the flow due to the presence of solid matrix fibers [25]. Two of the
more recent studies incorporating porous medium for creeping
flow through porous walled channels are by Siddiqui et al. [26,27].
Some of the studies mentioned here use the data for rat kidneys to
study pressure differences and effects of reabsorption parameters
on the flow. Most of the works discussed and cited here are limited
to the study of Newtonian fluid flows. However, in recent years,
there have been some advancements in the field of non-Newtonian
fluids flowing through porous channels as well (See, for example,
[28-30]). There is also consideration being given to study the
flow of couple stress fluid [31] through porous channels. Couple
stress fluid has also been known to model blood flow in various
physiological situations [32]. In our opinion, the idea of using
couple stress fluid model to study the flow through porous walls its
very well in context of a diseased renal tubule. Since couple stress
fluid model is a blending model, it incorporates the presence of
particles in the fluids, and can also be useful for consideration of
polar effects. Hence with non-Newtonian fluids, the possibilities of
studying flow through renal tubules are endless.
© 2021 Qurat A Azim. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and build upon your work non-commercially.