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Numerical Heat Transfer, Part A: Applications
An International Journal of Computation and Methodology
ISSN: (Print) (Online) Journal homepage: www.tandfonline.com/journals/unht20
MHD peristaltic two-phase Williamson fluid flow,
heat and mass transfer through a ureteral tube
with microliths: Electromagnetic therapy simulation
P. Deepalakshmi, E. P. Siva, D. Tripathi, O. Anwar Bég & S. Kuharat
To cite this article: P. Deepalakshmi, E. P. Siva, D. Tripathi, O. Anwar Bég & S. Kuharat (01 Apr
2024): MHD peristaltic two-phase Williamson fluid flow, heat and mass transfer through a
ureteral tube with microliths: Electromagnetic therapy simulation, Numerical Heat Transfer, Part
A: Applications, DOI: 10.1080/10407782.2024.2333501
To link to this article: https://doi.org/10.1080/10407782.2024.2333501
Published online: 01 Apr 2024.
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MHD peristaltic two-phase Williamson fluid flow, heat and
mass transfer through a ureteral tube with microliths:
Electromagnetic therapy simulation
P. Deepalakshmi
a
, E. P. Siva
a
, D. Tripathi
b
, O. Anwar B�
eg
c
, and S. Kuharat
c
a
Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and
Technology, Kattankulathur, India;
b
Department of Mathematics, National Institute of Technology, Srinagar,
India;
c
Multi-Physical Engineering Sciences Group, Dept. Mechanical and Aeronautical Engineering, Corrosion/
Coatings Lab, Salford University, Manchester, UK
ABSTRACT
The ureter typically experiences a frequency of one to five peristaltic con-
tractions per minute. However, it is important to note that these contrac-
tions can be disrupted by various physical and mechanical irritants. Ionic
contents in the urine make it electrically conducting and responsive to
electromagnetic body forces. MHD can be deployed in bio magnetic ther-
apy to control or mitigate symptoms associated with peristaltic pumping
in the urinary system. This article therefore focuses on the hydromagnetic
effects on flow patterns of urine with debris (monoliths). The mechanism
of urine flow is largely coordinated by the kidneys. The flow inside the
ureter is interrupted by microliths, which are generated by the sedimenta-
tion of excretory products. To simulate this, a two-phase formulation is
adopted, comprising the electromagnetic urological viscous fluid phase
and the particulate phase for solid grains. The peristaltic propulsion of
two-phase liquid in the ureter is simulated as a sinusoidal wave propaga-
tion of incompressible non-Newtonian fluid. The Williamson viscoelastic
model is deployed for the rheology. Heat transfer is also included with
Soret thermo-diffusion and viscous heating effects. Long wave and low
Reynolds number approximations are employed based on lubrication the-
ory. The mass, momentum, energy and concentration conservation equa-
tions with associated boundary conditions are rendered non-dimensional
via appropriate scaling transformations. A numerical solution is achieved
via BVP4C MATLAB quadrature. Graphical visualizations of the velocity,
temperature and concentration (solid grains) are given for the influence of
suspension parameter (f), Hartmann number (M), Prandtl number (Pr),
Weissenburg number (We), particle volume fraction (C), Eckert number (Ec),
Soret number (Sr), Schmidt number (Sc). The novelty of the present work is
therefore the simultaneous consideration of a generalized two-phase model,
wall slip, non-Newtonian characteristics, cross diffusion, viscous dissipation,
mass diffusion, magnetic body force and curvature effects in peristaltic uro-
logical transport, which has not been undertaken previously. The detailed
simulations reveal that the flow velocity is reduced due to the presence of
solid particles and the channel curvature, in comparison to the flow in an
unobstructed channel devoid of solid particles. Enhancing the hydro-
dynamic slip parameter speeds up the movement of particles and fluid
near the channel walls, boosts wall skin friction, raises pressure difference
in the pumping area, and amplifies bolus magnitudes.The rise in peristaltic
pumping results in a reduction in solid particle concentration, which is sig-
nificant phenomena.This theoretical approach may aid in treating
ARTICLE HISTORY
Received 8 January 2024
Revised 23 February 2024
Accepted 13 March 2024
KEYWORDS
Biomagnetic therapy;
computational urological
fluid dynamics;
concentration; heat transfer;
monoliths; non-newtonian;
numerical; peristalsis; soret
thermo-diffusion; trapping
bolus; ureter flow; viscous
heating
CONTACT E. P. Siva sivae@srmist.edu.in Department of Mathematics, College of Engineering and Technology, SRM
Institute of Science and Technology, Kattankulathur – 603203, Tamil Nadu, India.
� 2024 Taylor & Francis Group, LLC
NUMERICAL HEAT TRANSFER, PART A: APPLICATIONS
https://doi.org/10.1080/10407782.2024.2333501
conditions such as Urinary Tract Infections (UTIs). The computations effect-
ively demonstrate that significant manipulation of urological pumping
characteristics can be achieved with an electromagnetic field. Some new
features of two-phase ureteral dynamics are highlighted as relevant to
magnetic therapy techniques, which may be beneficial to clinicians.
CULMINATIONS
A detailed new formulation is given for magnetohydrodynamic (MHD) two-
phase Williamson non-Newtonian ureteral transport with mass diffusion,
Soret cross diffusion, viscous heating and peristaltic wave propulsion.
We compute numerically the pressure gradient, skin friction, Nusselt num-
ber, and wall shear stress in a planar channel with flexible walls, which
serves as a model for the ureter under a transverse magnetic field.
We also investigate the impact of Hartmann number and Weissenberg
non-Newtonian number on fluid and particle phase velocities.
The present work reveals some interesting insights into electromagnetic
ureteral peristaltic multi-phase non-Newtonian thermo-solutal transport
phenomena via extensive visualization.
There is a strong suppression in the ureteral fluid phase velocity for
greater magnetic field confirming the excellent flow control abilities of
Magnetic Ureteral Therapy (MUT).
1. Introduction
Two-phase flows arise in numerous applications in medicine and technology. These flows are
characterized by particles suspended in a viscous medium. Examples of applications include ven-
tricular assist devices (VADs), also called blood pumps [1], magnetic pharmacology [2], lymphe-
dema and transport in the swollen lymphatic nodes [3], biomicrofluidic separation devices [4],
orthopedic biofluid dynamics [5], tissue transdermal transport [6], capillary filtration [7], dialysis
treatments [8], interstitial flows [9], nanoparticle transport in asthmatic therapy [10], hazardous
biowaste conveyance [11], interfacial hydrodynamics in blood flows [12] and synovial hydro-
dynamic lubrication [13]. As noted, two-phase flow applications are growing in twenty first cen-
tury microfluidics, which is the study of fluid flow in small channels and devices, wherein it may
be deployed for manipulating and analyzing cells and particles. During MRI, the injection of a
contrast agent into the blood stream also creates a two-phase system. Another complex and sig-
nificant application of two-phase flows in medical fluid dynamics is the propulsion of urine in
the ureter, i.e. urodynamic transport. When contaminants and pathogens infect the ureter, sedi-
mentation of particles in urological fluids can precipitate serious disorders, including alluviation
calculi in the ureter [14]. While in the majority of patients with normal urinary tract and kidney
function and no predisposing co-morbidities, urinary tract infections (UTIs) can often be self-
limiting or readily cleared with a short course of antibiotics. However, approximately 25% [15,16]
of UTI patients experience persistent or recurrent infection, treatment failure and complicated
UTIs carry an increased likelihood of such outcomes. These complications have motivated scien-
tists and engineers to utilize simulation models to investigate the mechanics of ureteral infection
and how to combat the associated negative effects. The essential mechanism underlying urological
transport is peristalsis. This is a complex rhythmic motion produced by successive waves of con-
traction in elastic, tubular structures that push their fluid-like contents forward. Peristalsis
achieves exceptional efficiency in the urinary system by creating an involuntary sinusoidal muscu-
lar shrinkage of the uterine wall, which pushes the urine from the kidneys to the urinary bladder
via synergized wavy wall motions of the ureter controlled by electrical impulses. This mechanism
is one of the most effective in nature for internal propulsion and also features in embryonic heart
2 P. DEEPALAKSHMI ET AL.
development, pulmonary circulation, swallowing, digestive mixing, lymphatic dynamics, etc. It
also features prominently in external locomotion in snakes, eels, earthworms, etc. Although many
extensive investigations of peristalsis have been conducted by biologists for over a century, it was
only in the late 1960s that engineers began to develop mathematical hydrodynamic models and
experimental simulations of peristaltic propulsion. The seminal contributions of Fung and cow-
orkers at UC San Diego [17] and Shapiro and coworkers at MIT [18] introduced the lubrication
approximation for formulating peristaltic flow problems. This approach transforms the transient
fixed frame scenario to a laboratory (moving) frame and assumes very low Reynolds numbers
and high wave lengths for the peristaltic motion. It therefore dramatically simplifies the Navier-
Stokes 3-D viscous flow model to axisymmetric flow in an infinitely long distensible tube. These
studies produced comprehensive solutions for ranges of peristaltic motion depending on pressure
difference and plotted streamlines, velocity distributions and bolus characteristics. This approach
was adopted subsequently by Lykoudis [19] to analyze the ureteral pumping. Boyarsky and
Weinberg [20] extended the work in [19], examining in detail the hydrodynamics of the uretero-
pelvic junction and conus (specialised components of the ureter) and observing that in peristaltic
pumping, the bolus at this point does not influence the pressures, flows, or volume above it sig-
nificantly. Most of the subsequent work in this area followed a similar methodology until the
twenty first century, when CFD (computational fluid dynamics) emerged as a feasible tool.
Lozano [21] explored both numerically and experimentally the dynamics of ureteral peristaltic
motion. He assumed a Newtonian model and using small wavenumbers, described regular per-
turbation expansions and identified local and global bifurcations in streamline patterns. He also
computed particle paths and addressed local bifurcations and their topological modifications with
dynamical system techniques. This study identified a triplet of unique hydrodynamic phenomena,
backward (reflux) pumping, trapping and augmented flow and furthermore noted the presence of
a number of stagnation points. Kiil [22] studied the response of dilatation in the ureteral wall
during catheter insertion. Vahidi et al., [23] deployed ADINA FSI finite element software to com-
pute the two-way fluid-structure interaction between the compliant ureteral wall and the sur-
rounding urological Newtonian fluid.
Nomenclature
~
Uf,~
Vf Axial and Transverse fluid phase velocity
components
~
a Amplitude
~
K Channel’s length dependent constant
f Suspension parameter
~
P Pressure
t Time
l Dynamic viscosity
Re Reynolds number
k Wavelength
~
b Channel’s half-width at the inlet
~
c Constant velocity
C Partial volume fraction
qf Williamson urological fluid density
~s~
X~
X&~s~
Y~
Y Shear stress tensor components
We Weissenburg Number
Sr Soret number
~
H~
X,~
t
Channel wall
X Non-dimensional axial coordinate
~
Up,~
VpAxial and Transverse particle phase vel-
ocity components
s Stoke’s number
r Electrical conductivity of the Williamson
urological fluid
B0 Transverse magnetic field intensity
ðqcpÞf Specific heat capacity of the Williamson
urological fluid
~
Tf Fluid phase temperature
K Thermal conductivity of the Williamson
fluid (isotropic)
qp Particulate density
x~
T Particle phase Saffman wall slip coefficient
~
Tp Particle phase temperature
~
Kf Particulate concentration
Dm Molecular diffusivity of the particles
M Hartmann number
Pr Prandtl number
Ec Eckert number
Sc Schmidt number
h Non-dimensional channel wall
Y Dimensionless transverse coordinate
NUMERICAL HEAT TRANSFER, PART A: APPLICATIONS 3
The wall stimulation was accommodated as a nonlinear contact mechanics problem, and time-
dependent alterations in the ureteral wall intraluminal shear stress during peristalsis were com-
puted. They gave a comprehensive visualization of key characteristics, including the influence of
ureteral wall compliance on contraction wave velocity, the quantity of contraction waves on the
ureteral outlet flow, the pressure difference between the ureteral inlet and outlet and also the
peak amplitude of the contraction wave. They observed that much greater shear stress arises in
the proximal part of the ureter relative to the distal or central locations, middle and distal parts.
Furthermore, they demonstrated that greater hydromechanical efficiency is achieved for larger
amplitudes of the contraction wave and that sub-optimal performance of the ureteropelvic junc-
tion precipitates significant urological reflux even when there is a very gradual initiation stage for
the peristaltic contraction wave. G�
omez-Blanco et al., [24] used a fluid-structure interaction simu-
lation to study the optimal design of stents for ureteral peristaltic pumping. They scrutinized
closely the interaction between urine flow and a double-J stented ureter and deployed a variety of
nonlinear viscoelastic models for the quasi-incompressible and isotropic ureteral wall deform-
ation. They noted that wall compliance has a key effect on the peak amplitude of the contraction
wave and also the tensile stress in the ureteric wall. The above simulations were confined to sin-
gle-phase Newtonian peristaltic pumping. However, the presence of contaminants in urological flu-
ids can produce multi-phase flows. Two-phase phenomena are common since sedimentary
deposits in the form of particles may flow with the viscous urological base liquid. The resulting
hydrodynamic problem therefore requires an appropriate fluid-particle suspension model. A very
popular methodology was introduced by Caltech applied mathematician Saffman [25] in the early
1960s, which he termed a “dusty fluid”. In this approach, suspended particles are studied as
spherical rigid bodes and interfacial momentum transfer is possible. Marble [26] generalized the
Saffman model to consider thermal effects as well as other forms of slip. These models have been
extensively deployed in biological fluid dynamics in addition to other areas of complex industrial
suspension rheology. Srivastava et al., [27] studied peristaltic two-phase flow with the fluid-par-
ticle suspension model. They evaluated the results of velocity using momentum equations in both
phases and made comparisons between them. Their computations showed that there is a reduc-
tion of fluid velocity in the axial direction when there is an increase in the solute concentration.
Kamal et al., [28] studied the hydrodynamic wall slip effect on fluid-particle suspension peristaltic
pumping in a distensible two-dimensional channel under sinusoidal waves. They showed that the
critical value of the pressure gradient is comparatively lower when particles are present relative to
a purely single-phase viscous Newtonian fluid. Many other studies have been communicated in
peristaltic two-phase fluid dynamics addressing a variety of multi-physics effects, including
coupled thermal solutal transport [29], wall damping [30], viscoplastic fluidity [31], nanofluids
and tapered conduit geometries [32], curved tubes [33], couple stress and thermal slip effects [34]
and mass diffusion [35]. All these studies have revealed the significant influence of particle sus-
pensions on peristaltic flow characteristics.
In recent times, a new therapy for ureteral infections has emerged. This exploits the electric-
ally-conducting nature of urological liquids, which is due to the presence of ions and other chem-
icals that respond to external electrical and magnetic fields. Magnetohydrodynamic ureteral
therapy (MUT) has been shown to be very effective in the treatment of monolith obstructions in
both ureteral and gynaecological performance [36]. Impulse magnetic field (IMF) control can be
strategically deployed to activate the impulse activity of ureteral smooth muscles in over 60% of
patients and thereby optimally manage ureterolith fragments. More recently, pioneering work in
biomagnetic therapy at Stanford University [37] has led to the creation of a new medical device,
MagSToNE (Magnetic System for Total Nephrolith Extraction), which utilizes magnetization to
remove kidney stones during ureteroscopy to maximize the stone-free rate and minimize operat-
ing time. This technique has proven to be far superior to existing approaches such as uretero-
scopy, in which a ureteroscope is passed endoscopically up to the stone and a laser fiber is used
4 P. DEEPALAKSHMI ET AL.
to fragment or dust the stone. MagStoNE has achieved comparable success rates but with much
shorter operating times as fragments. It is much less intrusive and portable and exploits a differ-
ent approach to urological hydrodynamics, namely magnetohydrodynamics (MHD) technology.
The invention consists of two components: a small-diameter flexible magnetic wire (MagWIRE)
and superparamagnetic particles with surface chemistries that bind to kidney stones. After a kid-
ney stone is fragmented, a superparamagnetic particle solution is instilled through the ureteral
access sheath and coats the fragments, rendering them magnetizable. The MagWIRE has been
tested rigorously and can enter through the ureteroscope or the access sheath and uses a unique
magnetic configuration to generate a strong magnetic field to optimize the capture of stone frag-
ments along the entire length of the wire. Furthermore, this device can be deployed multiple
times in succession to extract all fragments of monoliths not retrievable with conventional thera-
pies. It is compatible with existing ureteroscopy setups and is set to revolutionize urological dis-
order treatments around the world.
Motivated by the afore-said magnetohydrodynamic ureteral therapy devices, in the present
study we generalize previous investigations to consider peristaltic pumping in ureteral two-phase
(fluid-particle suspension) with non-Newtonian and magnetohydrodynamic body force effects [38].
Lorentz body force is included to simulate magnetohydrodynamic drag. The Saffman fluid-par-
ticle suspension model is adopted, and heat transfer is also considered, as are Soret thermal diffu-
sion and viscous dissipation effects [39]. The robust Williamson rheological model [40–42] is
utilized to accommodate the shear-thinning (pseudoplastic) and shear-thickening (dilatant) behav-
ior of the electrically conducting urological fluid-particle suspension. It is important to note that
a number of other non-Newtonian studies have been reported in peristaltic pumping of relevance
to ureteral dynamics with other rheological models. Ajithkumar et al., [43] explored the flexibility
of walls by peristaltic pumping a bi-viscous Bingham nanofluid through a porous medium under
convective boundary conditions by employing the R-K-based shooting technique. Ajithkumar
et al., [44,45] addressed the movement of a magnetohydrodynamic Ree-Eyring nanofluid through
a porous conduit, along with considering activation energy and thermal radiation. Vajravelu and
Ajithkumar et al., [46,47] examined the peristaltic pumping of different nanofluids through a
lubrication approach. Jagadesh [48] studied convective peristaltic pumping of a Casson fluid
across an inclined porous wavy channel by employing a regular perturbation technique.
Ajithkumar et al., [49,50] studied the bioconvective peristaltic transport of a non-Newtonian
nanofluid similar to Sutterby and Jeffrey across a porous symmetric channel with compliant walls.
Hina and Kayani et al., [51,52] analyzed the peristaltic motion of a non-Newtonian nanofluid fol-
lowing the Carreau-Yasuda (CY) model in a compliant walled channel using Buongiorno’s model
without assuming constant diffusion coefficients. Yasin et al., [53] primarily address the peristaltic
flow of a hybrid nanofluid including copper (Cu) and silver (Ag) nanoparticles with blood as the
carrier fluid in a symmetrical channel. Yasin et al., [54] scrutinized a viscous-elastic (Maxwell)
fluid with a slip condition to analyze hemodynamics in arteries and capillaries, taking into
account Hall current features. Long-wave and low Reynolds number approximations are
employed based on lubrication theory. The mass, momentum, energy, and concentration conser-
vation equations with associated boundary conditions are rendered non-dimensional via appro-
priate scaling transformations. A numerical solution is achieved via BVP4C MATLAB
quadrature. Graphical visualizations of the velocity, temperature and concentration (solid grains)
are given for the influence of the suspension parameter (f), Hartmann number (M), Prandtl
number (Pr), Weissenburg number (We), particle volume fraction (C), Eckert number (Ec), Soret
number (Sr), and Schmidt number (Sc). The simulations demonstrate that significant manipula-
tion of urological pumping characteristics can be achieved with an electromagnetic field. Some
new features of two-phase ureteral dynamics are highlighted as relevant to magnetic therapy tech-
niques, which may be beneficial to clinicians.
NUMERICAL HEAT TRANSFER, PART A: APPLICATIONS 5
2. Mathematical model for magnetized ureteral two-phase rheological flow
The physical model to be investigated is depicted in Figure 1, with a rendition of the feed mech-
anism via the kidneys to the ureter. Two-dimensional viscous fluid-particle suspension hydromag-
netic peristaltic pumping with rheological, thermal and viscous heating effects is considered in a
planar channel with distensible walls, as a model of the ureter under a transverse magnetic field.
The following are considered as assumptions
Two dimensional
Incompressible laminar flow
Cartesian co-ordinate system ~
X,~
Y
non-Newtonian fluid (Williamson fluid)
Sinusoidal wave along tapered channel with particles
A sinusoidal wave of amplitude ~
a, ~
K represents the channel length-dependent constant, wave-
length k, channel’s half-width at the inlet is denoted by ~
b and constant velocity ~
c propagates
along the channel wall with height ~
H~
X,~
t
defined as:
~
H~
X,~
t
¼~
asin 2p
k~
X−~
c~
t
ð Þ
þ~
bþ~
K~
X(1)
Magnetic induction, thermal dispersion, Dufour (diffuso-thermal) and thermal stratification
effects are neglected. The magnetic field is sufficiently weak to negate hall current effects. The
governing conservation equations, i.e. fluid phase and particulate phase momentum, temperature,
and concentration equations in a coordinate system, amalgamating the models in [32,33] and
extending to include hydromagnetic effects, can be shown to take the form:
Fluid-Phase:
qf~
Uf
@~
Uf
@~
Xþ~
Vf
@~
Uf
@~
Yþ@~
Uf
@~
t
" #¼−
@~
P
@~
Xþ@~s~
X~
X
@~
Xþ@~s~
X~
Y
@~
Y
þSC
1−C~
Up−~
Uf
−rB2
0~
Uf, (2)
qf~
Uf
@~
Vf
@~
Xþ~
Vf
@~
Vf
@~
Yþ@~
Vf
@~
t
" #¼−
@~
P
@~
Yþ@~s~
X~
Y
@~
Xþ@~s~
Y~
Y
@~
Y
þSC
1−C~
Vp−~
Vf
, (3)
ð1−CÞðqcpÞf~
Uf
@~
Tf
@~
Xþ~
Vf
@~
Tf
@~
Yþ@~
Tf
@~
t
" #¼ ð1−CÞK@2~
Tf
@~
X2þ@2~
Tf
@~
Y2
!þCSð~
Uf−~
UpÞ2
þqpcpC
x~
Tð~
Tp−~
TfÞ, (4)
Figure 1. Schematic diagram of peristaltic ureteral MHD two-phase flow.
6 P. DEEPALAKSHMI ET AL.
ð1−CÞ~
Uf
@~
Kf
@~
Xþ~
Vf
@~
Kf
@~
Yþ@~
Kf
@~
t
" #¼ ð1−CÞDm
@2~
Kf
@~
X2þ@2~
Kf
@~
Y2
!þqpC
qfx~
Kð~
Kp−~
KfÞ
þDm
~
T0ð1−CÞK~
T
@2~
Tf
@~
X2þ@2~
Tf
@~
Y2
!:(5)
In the aforementioned equations [40],
s~
X~
X¼2 1 þWe_
cð Þ@U
@X,
s~
X~
Y¼1þWe_
cð Þ @U
@Yþd@V
@X
,
s~
Y~
Y¼2d1þWe_
cð Þ@V
@Y,
_
c¼2@U
@X
2
þ@U
@Yþd2@V
@X
2
þ2@V
@Y
2
!1=2
:
Particle Phase:
qpC~
Up
@~
Up
@~
Xþ~
Vp
@~
Up
@~
Yþ@~
Up
@~
t
" #¼−C@~
P
@~
XþCS ~
Uf−~
Up
, (6)
qpC~
Up
@~
Vp
@~
Xþ~
Vp
@~
Vp
@~
Yþ@~
Vp
@~
t
" #¼−C@~
P
@~
YþCS ~
Vf−~
Vp
, (7)
qpCcp~
Up
@~
Tp
@~
Xþ~
Vp
@~
Tp
@~
Yþ@~
Tp
@~
t
" #¼Cqpcp
x~
Tð~
Tf−~
TpÞ, (8)
~
Up
@~
Kp
@~
Xþ~
Vp
@~
Kp
@~
Yþ@~
Kp
@~
t
" #¼1
x~
Kð~
Kf−~
KpÞ:(9)
Where the subscripts f and p denote fluid phase and solid (particulate granules) phase, respect-
ively. In Eqs. (2)–(9) the following notation applies: C is the partial volume fraction parameter,
~
X,~
Y,~
t
are axial and transverse coordinates and time, qf is Williamson urological fluid density,
~
Uf,~
Vf are axial and transverse fluid phase velocity components, ~
P is pressure, ~s~
X~
X and ~s~
Y~
Y are
the shear stress tensor components, ~
Up,~
Vp are axial and transverse particle phase velocity compo-
nents, S is Stokes number, r is electrical conductivity of the Williamson urological fluid and B0is
transverse magnetic field intensity, ðqcpÞf is specific heat capacity of the Williamson urological fluid,
~
Tf is fluid phase temperature, K is thermal conductivity of the Williamson fluid (isotropic), qp is
particulate density, x~
T is particle phase Saffman wall slip coefficient, ~
Tp is particle phase tempera-
ture, ~
Kf is particulate concentration, Dm is molecular diffusivity of the particles in urological
Williamson fluid. The following dimensional boundary conditions [39] are prescribed at the chan-
nel walls ( ~
Y¼0:~
Y¼~
H).
@~
Uf
@~
Y¼0, @~
Tf
@~
Y¼0, @~
Kf
@~
Y¼0, @~
Up
@~
Y¼0, @~
Tp
@~
Y¼0, @~
Kp
@~
Y¼0at Y ¼10, (10)
NUMERICAL HEAT TRANSFER, PART A: APPLICATIONS 7
~
Uf¼0, ~
Tf¼~
Th,~
Kf¼~
Khat ~
Y¼~
H:(11)
The nonlinear boundary value problem defined by Eqs. (10)–(11) in primitive form is feasible
to solve even with modern numerical methods. Also, this requires explicit data for each fluid and
particle property. To facilitate a solution, scaling transformations are therefore introduced. These
dramatically simplify the problem at hand and simultaneously enable scaling of different thermal,
magentic and hydrodynamic effects via appropriate dimensionless numbers, a very powerful tool
in fluid dynamics. Proceeding with the analysis, the following non-dimensional variables are
invoked [32–34, 40]:
X¼~
X
k,Y¼~
Y
~
a,Uf¼~
Uf
~
c,Up¼~
Up
~
c,Vf¼~
Vfk
~
a~
c,Vp¼~
Vpk
~
a~
c,P¼~
a2~
P
l~
ck,
Re ¼qf~
a~
c
l,hf¼~
Tf−~
T0
~
Th−~
T0
,hp¼~
Tp−~
T0
~
Th−~
T0
h¼~
H
~
a,af¼~
Kf−~
K0
~
Kh−~
K0
,ap¼~
Kp−~
K0
~
Kh−~
K0
,
t¼~
c~
t
k,d¼~
a
k,sXX ¼k~s~
X~
X
l~
c,sXY ¼~
a~s~
X~
Y
l~
c,sYY ¼~
a~s~
Y~
Y
l~
c,We ¼C~
c
~
a,_
c¼~
_
c~
a
~
c,S¼lf
~
a2,
Ha ¼ffiffiffi
r
l
rB0~
a,Ec ¼~
c2
~
Th−~
T0
cp
, Pr ¼lcp
K,Sr ¼DmqfK~
T
l~
T0
~
Th−~
T0
~
Kh−~
K0
!,Sc ¼Dmqf
l:
(12)
These parameters denote respectively the dimensionless ~
X,~
Y
axial and transverse coordi-
nates, non-dimensional axial fluid phase velocity, non-dimensional axial particulate phase vel-
ocity, dimensionless transverse particle phase velocity, dimensionless pressure, Reynolds number
(based on wave amplitude and peristaltic wave speed), dimensionless fluid phase temperature
function, dimensionless particulate phase temperature function, dimensionless height of channel
(ureteral conduit), fluid concentration parameter, particulate concentration parameter, dimension-
less time, amplitude to wave length ratio, non-dimensional shear stress components, Weissenberg
number, shear rate, momentum Stokes number, Hartmann magnetic number, Eckert (viscous
heating) number, Prandtl number, Soret number and Schmidt number. Saffman suspension par-
ameter is denoted ðfÞ:Implementing the transformations (12) in Eqs. (2)–(11), the transformed
non-dimensional boundary value problem emerges as:
Dimensionless Fluid phase:
@P
@X¼@2Uf
@Y21þ2We @Uf
@Y
þfC
1−CUp−Uf
ð Þ −ðMÞ2Uf, (13)
@P
@Y¼0, (14)
@2hf
@Y2
þCPrEc
1−C
1
f
dP
dX
2
¼0, (15)
@2af
@Y2
þSrSc @2hf
@Y2
¼0:(16)
Dimensionless Particle phase:
@P
@X¼−fUp−Uf
ð Þ, (17)
hf¼hp, (18)
8 P. DEEPALAKSHMI ET AL.
af¼ap:(19)
Dimensionless Boundary conditions
@Uf
@Y¼0, @hf
@Y¼0, @af
@Y¼0, @Up
@Y¼0, @hp
@Y¼0, @ap
@Y¼0 at Y¼0, (20)
Uf¼0, hf¼1, af¼1 at Y¼h(21)
Here the transformed wall equation is mentioned as,
h¼1þgþdsin 2pX−t
½ ,
Where g¼~
K~
X
~
a,d¼~
b
~
a<1:
The skin-friction coefficient, also known as the wall shear stress, is determined by applying Eq.
(22). Similarly, the thermal flux, which represents the rate of thermal exchange, is determined by
applying Eq. (23).
The skin friction parameter ðCfÞon the wall is precisely described as
Cf¼−l@U
@Y
at Y ¼h, (22)
The local Nusselt number parameter (Nu)is precisely defined as
Nu ¼−
@h
@Y
at Y ¼h, (23)
The equation governing the rate of mass transfer through the wall is expressed as Sherwood
number (Sh)
Sh ¼−
@a
@Y
at Y ¼h:(24)
3. Methodology
The above mentioned non-linear ordinary differential Eqs. (13)–(19) were solved numerically uti-
lizing MATLAB’s bvp4c function, together with Eqs. (20) and (21) associated with the boundary
conditions.
To aid in the achievement of this resolution, the following procedures were implemented:
System Reduction: The introduction of new variables led to the transformation of a complex
set of higher-order partial differential equations into a simpler set of first-order ordinary dif-
ferential equations.
Generation of Boundary Conditions: To ensure adherence to the problem’s constraints,
boundary conditions were established for the variables that were recently incorporated.
The suitable initial approximations for these newly introduced variables were ascertained,
thereby providing a basis for the numerical resolution.
By meticulously adhering to these procedures, the intended solution was achieved utilizing the
bvp4c function, thereby successfully resolving the resulting system.
The second-order governing equations, including both fluid and particle phases, were trans-
formed into a set of first-order ordinary differential equations through the inclusion of new varia-
bles. The variables and were defined as f1 and f2, respectively. and were defined as f3 and f4,
while and were defined as f5 and f6, respectively. The procedure described the transformation of
the original set of linked higher-order differential equations and their corresponding boundary
NUMERICAL HEAT TRANSFER, PART A: APPLICATIONS 9
conditions into a system of three first-order differential equations. The boundary conditions were
adjusted to match the converted equations. The system of first-order linear ordinary differential
equations generated is as follows:
U0¼f2, (25)
U00 ¼f20¼1
1þ2We f 2
ð Þ½ Ha
ð Þ2f1þP1
1−C
, (26)
h0¼f4, (27)
h00 ¼f40¼CPrEc
1−C
1
fP2, (28)
r0¼f6, (29)
r00 ¼f60¼−SrSc f 40
:(30)
The associated boundary conditions are
fa2
ð Þ ¼0, fa4
ð Þ ¼0, fa6
ð Þ ¼0, (31)
fb1
ð Þ ¼0, fb3
ð Þ ¼1, fb5
ð Þ ¼1:(32)
A numerical solution is achieved via BVP4C MATLAB quadrature. This technique has been
applied to many multi-physical fluid dynamics problems, e.g. triple diffusive convection duct
flows [55], robotic smart lubrication films [56], bio magnetic hypodermic coating flows [57], car-
bon wall nanotube bio-coatings [58], ternary hybrid nanofluid magnetic functional coatings [59]
and electromagnetic squeezing tribological applications [60,61]. It can accommodate any order of
derivatives. In MATLAB, this quadrature is used to obtain solutions for the fluid and particulate
velocity functions, temperature functions and concentration functions. BVP4C uses stepping for-
mulae, which are summarized in [57]. Further details are given in Kattan [62]. The algorithm
relies on an iteration structure. BVP4C is a numerical platform that implements the Lobato IIIa
three-stage formula. This is a collocation formula that is formed by a polynomial collocation. It
provides a C1-continuous solution that is fourth-order accurate uniformly in x2a,b
½ :
The fourth-order formulae are given below:
k1¼L f xn,yn
ð Þ, (33)
k2¼L f xnþL
2,ynþk1
2
, (34)
k3¼L f xnþL
2,ynþk2
2
, (35)
k4¼L f xnþL,ynþk3
ð Þ, (36)
ynþ1¼ynþk1
6þk2
3þk3
3þk4
6þO L5
ð Þ:(37)
where L¼ ðxiþ1−xiÞrepresents the size of each subinterval. The crucial part in utilizing bvp4c is
the variation step and early guessing of the mesh point. Besides, efficiency will eventually depend
on the programmer’s ability to provide the algorithm with an initial guess for the solution. Two
folders can be created, for example, “code a” and “code b,” for the trial-and-error initial guess
and continuous iterations that approximate closely to the initial guess, respectively. The above-
described computing approach cannot be used without transforming the higher-order differential
equations to differential equations of order one. Some commands in handling the function, such
10 P. DEEPALAKSHMI ET AL.
as “@odeBVP” and “@odeBC”, are from the syntax of the solver “sol ¼bvp4c” (@OdeBVP,
@OdeBC, solinit, options). The iterative process is carried out until an accuracy of 10−6 is
achieved which is obtained for the values of the boundary conditions and step size. The numer-
ical results obtained from the solver are then plotted as graphs.
4. Results and discussion
Extensive computations have been conducted in MATLAB to determine the influence of all key
control parameters on the peristaltic multi-physical transport problem. Specifically these parame-
ters are Saffman suspension parameter (f), Hartmann number (M), Prandtl Number (Pr),
Weissenburg number (We), particle volume fraction (C), Eckert number (Ec), Soret number (Sr)
and Schmidt number (Sc). The effects of these parameters on fluid/particle velocity functions,
fluid/particulate temperature function and particlulate concentration function are depicted in
Figures 2–13. The pressure gradient distribution is sketched in Figures 14–16. All data utilized is
extracted from clinically viable sources [63,64] and earlier studies [65]. Velocity, temperature and
concentration are considered in turn.
4.1. Velocity characteristics
Figure 2 shows that a significant decrement in both fluid and particle phase velocity is induced
with elevation in the Hartmann magnetic number (M). The major reason for the reduction in
fluid phase velocity is due to the action of the Lorentz force on the electrically conducting non-
Newtonian urological liquid. This arises in the linear retarding body force term in Eq. (13),
−M
ð Þ2Uf:The influence of magnetic field generates a strong resistance to the axial flow which
induces deceleration. The urological volumetric flow rate which is proportional to velocity will
also be suppressed. Higher Hartmann number therefore achieves excellent flow control via a
non-intrusive means which has been shown to be beneficial in urological disorders [37,38]. By
virtue of the definition of M ¼ffiffir
l
qB0, expresses the relative influence of the Lorentzian magnetic
force to the viscous force in the regime. When M¼1 both forces contribute equally, and
Hartmann-Stokes boundary layers may arise at the interior of the ureteral duct. When M¼0
magnetic effects vanish, and the urological liquid is electrically non-conducting. When M>1 the
magnetic force dominates the viscous force, and this effectively stifles momentum development in
the duct. The range of magnetic field intensities studied here is representative of the actual mag-
nitudes explored in [35]. In all the profiles peak velocity magnitude is computed at the duct cen-
terline (Y¼0) since only the upper duct half space region is plotted due to symmetry (the actual
Figure 2. Impact Of hartmann number (M) on fluid and particle phase velocity ðUf,pÞ:
NUMERICAL HEAT TRANSFER, PART A: APPLICATIONS 11
duct depth extremities extend over the range, (−1<Y<1). The opposing force augmented the
non-Newtonian potency of the reactive Williamson fluid and diminished the heat generation
within the system. Consequently, the viscosity of the fluid is enhanced as the molecular bond is
stimulated throughout the flow regime. It is discovered that a greater transverse magnetic field,
Figure 3. Impact Of saffman supension parameter (f) on fluid and particle phase velocity ðUf,pÞ:
Figure 4. Impact Of weissenberg number (We) on fluid and particle phase velocity ðUf,pÞ:
Figure 5. Impact Of prandtl number (Pr) on fluid and particle phase velocity temperature h
ð Þ
f, p
.
12 P. DEEPALAKSHMI ET AL.
represented by an increasing Hartmann number, similar retardation may be found for both fluid
and particle.
Figure 3 reveals the response in velocity with an alteration in Saffman suspension parameter,
f:This parameter features in the momentum Stokes number, S ¼lf
~
a2, and arises in the fluid
Figure 6. Impact Of particle volume fraction (C) on fluid and particle phase temperature h
ð Þ
f, p
.
Figure 7. Impact Of eckert number (Ec) on fluid and particle phase temperature h
ð Þ
f, p
.
Figure 8. Impact Of saffman suspension parameter ðfÞon fluid and particle phase temperature h
ð Þ
f, p
.
NUMERICAL HEAT TRANSFER, PART A: APPLICATIONS 13
phase-particulate phase coupling term, þfC
1-CUp−Uf
ð Þ in the fluid phase momentum Eq. (13).
Contrary to the Lorentzian drag term, the coupling term is an assistive body force and accentu-
ation in the parameter, f implies greater slip between the fluid and particles leading to an acceler-
ation in the fluid phase. The suspension of granules accelerates with enhancement of average
Figure 9. Impact of saffman suspension parameter ðfÞon concentration a
ð Þ
f, p
.
Figure 10. Impact of schmidt number (Sc) on concentration a
ð Þ
f, p
.
Figure 11. Impact of soret number (Sr) on concentration a
ð Þ
f, p
.
14 P. DEEPALAKSHMI ET AL.
velocity of particles. A significant decrease in axial particle phase velocity is observed. Both phases
are presented here with opposite results.
Figure 4 Illustrates the evolution in fluid and particle phase velocity with increasing
Weissenberg number (We). This parameter arises uniquely in the modified shear term,
þ2We @Uf
@Y
@2Uf
@Y2
in Eq. (13) which is a mixed derivative term. The Weissenberg number (We)
embodies the relative contribution of the elastic forces to the viscous forces in the rheological
urinary fluid. It also expresses the ratio of stress relaxation time of the fluid to the specific pro-
cess time. It is most deployed for simple shear flows where it also describes the product of the
shear rate and the relaxation time. When We ¼0 non-Newtonian effects vanish since elastic
effects swamp out viscous effects. Since all rheological liquids including magnetic urological fluids
are elastico-viscous, they combine elastic and viscous properties. For situations wherein the relax-
ation time of a flow is much less than the timescale of an elastic-viscous fluid, then viscous effects
dominate. Conversely when relaxation time of the flow exceeds the timescale, elastic effects will
be amplified over viscous effects. The Weissenberg number can also be regarded as the inverse
Deborah number (ratio of the time scale of the flow and the stress relaxation time).
It is evident that as the Weissenberg number increases, the viscous forces increase relative to
the elastic forces, which induces strong deceleration in the urological fluid (tensile stresses related
to the first normal stress difference are reduced, which contributes significantly to this process).
It also slows down the movement of particles. The Weissenberg number in rheology also
Figure 12. Impact of eckert number (Ec) on concentration a
ð Þ
f, p
.
Figure 13. Impact of particle volume fraction (C) on concentration a
ð Þ
f, p
.
NUMERICAL HEAT TRANSFER, PART A: APPLICATIONS 15
Figure 14. Impact of weissenberg (We) on pressure gradient ðdP=dXÞ:
Figure 15. Impact of suspension parameter (f) on pressure gradient dP=dXð Þ:
Figure 16. Impact of hartmann number (M) on pressure gradient dP=dXð Þ:
16 P. DEEPALAKSHMI ET AL.
computes the degree of anisotropy or orientation generated by the fluid deformation and is rele-
vant for constant stretch history flows, as studied here. In cases where non-constant stretch his-
tory arises, e.g. complex polymers in bionics, it is more appropriate to deploy the Deborah
number, which better expresses physically the rate at which elastic energy is stored or released in
the fluid.
4.2. Temperature characteristics
Figure 5 visualizes the impact of Prandtl number (Pr) on fluid phase temperature. Prandtl num-
ber expresses the relation between momentum diffusivity and thermal diffusivity. It is also an
inverse function of thermal conductivity of the urological fluid. A strong decrement is induced in
temperature in the fluid phase. Higher Prandtl number fluids diffuse thermal energy (heat) much
less efficiently than lower Prandtl number fluids. The values for rheological urological fluid are
closer to water (Pr 7 at room temperature). Prandtl number also can be used to quantify the
relative thickness of the momentum and thermal boundary layers developing at the ureteral tube
internal walls. When Prandtl number equals unity the momentum and energy diffusion rates are
equivalent in the peristaltic regime. However, for Prandtl number in excess of unity the momen-
tum diffusion rate greatly exceeds the thermal diffusion rate. This certainly applies in the present
case of ureteral hydrodynamics [63,64]. The influence of particle volume fraction (C) on fluid
phase temperature is depicted in Figure 6. Peak temperature always arises at the ureteral tube
center line. It vanishes at the upper wall (Y¼1). As the percentage of monolith particles
increases, the thermal convection process is inhibited. The parameter C features in both the fluid
phase momentum Eq. (13) as þfC
1−CUp−Uf
ð Þ and also in the fluid phase temperature Eq. (15) as
the term, þCPrEc
1−C
1
fdP
dX
2:It clearly has a detrimental effect on temperature as it appears in the
denominator (1-C) and therefore elevation in C reduces the thermal diffusion in the fluid phase
and manifests in a depletion in temperatures across the upper half space of the duct. Even a
minor enhancement in volume fraction from 10% (C¼0.1) to 12% (C¼0.12) has a dramatic
effect on thermal distributions in the urological fluid. The volumetric heat capacity of the mono-
lith granule particles is enhanced with volume fraction which boosts thermal conduction in the
suspended particles but counteracts heat diffusion in the urological base fluid [63]. Figure 7 illus-
trates the influence of Eckert number (Ec) on temperature parameter. Eckert number character-
izes the viscous heating in the urological fluid which is attributable to internal friction at the
molecular level. The friction induced generates thermal energy which energizes the urological
fluid and boosts temperatures. Internal friction alters the viscosity of the urological fluid which
also contributes to a temperature change since via the Prandtl number there will be a modifica-
tion in momentum diffusion rate relative to thermal diffusion rate due to viscous heating. The
rate of change of temperature with height of the duct is also elevated as Eckert number increases
(i.e. a steeper gradient is produced). Figure 8 visualizes the impact of the Saffman suspension par-
ameter (f) on temperature evolution in the ureteral duct. The slip effect between particles and
the base urological fluid is encouraged with higher values of this parameter. This intensifies the
micro-convection round the suspended particles which contributes to enhanced thermal diffusion
in the base fluid and the resulting enhancement in temperature. Monotonic decays are computed
consistently from the duct centerline to the upper wall, and no cross-over in profiles is witnessed.
4.3. Suspended particle concentration characteristics and pressure gradient
In this final section of the discussion, we address the impact of selected control parameters on
the particulate concentration. This is distinct from the volume fraction (C) which relates to the
percentage volume of suspended particles in the urological fluid. The interconnection between
NUMERICAL HEAT TRANSFER, PART A: APPLICATIONS 17
suspension parameter (f) and concentration in peristaltic flow exhibits a high degree of complex-
ity and is subject to the influence of multiple factors which is depicted in Figure 9. The dimen-
sions and morphology of suspended particles exert a significant influence. Particles that are larger
in size or have irregular shapes may exhibit distinct interactions with the fluid and the ureteral
tube walls in comparison to smaller, spherical particles, resulting in differences in the flow and
mass transfer characteristics. The findings indicate that there is an inverse relationship between
the distribution of the suspension parameter and concentration, i.e. that larger values of the
Saffman suspension parameter (f) produce a depletion in concentration magnitudes. Figure 10
visualizes the influence of the Schmidt number (Sc) on the concentration of the particles sus-
pended in the urological fluid. The Schmidt number represents the relative rates of mass diffusion
and momentum diffusion inside the peristaltic regime. A high Schmidt number signifies that the
rate of momentum diffusion is greater than that of mass diffusion. Hence, the concentration is
suppressed with an increment in the Schmidt number. The sub-unity values of Sc in Figure 10
are representative of actual urological contamination with monoliths [see, for example, 64, 66]. A
similar effect on the concentration profile (Figure 11) is noticed with variation in the thermo-dif-
fusive parameter, i.e. Soret number Sr. This parameter also features in the same term as the
Schmidt number, viz, þSrSc @2hf
@Y2
in the fluid phase temperature Eq. (16). When the Soret num-
ber increases from zero, this indicates that thermal diffusion is the dominant mechanism.
Concentration magnitudes clearly diminish with magnifying the values of Soret number (Sr). The
term “concentration” in the context of peristaltic flow often pertains to the existence of solutes or
particles inside the fluid. The distribution of concentration within the tube may exhibit variations
due to the influence of other mixing and transport phenomena. This effect is observed in Figure
12 as a decrease in the fluid concentration profile is computed with increasing values of viscous
dissipation parameter, Eckert number (Ec). This parameter features, as noted earlier, in the fluid
phase temperature Eq. (15), specifically the term, þCPrEc
1−C
1
fdP
dX
2:The correlation between particle
concentration and particle volume fraction is contingent upon the characteristics of both the par-
ticles and the fluid in which they are immersed. Figure 13 shows that with all other variables
being constant, the increase in the number of particles in the urological fluid i.e. volume fraction,
C, substantially enhances the concentration in the fluid. Figure 14 gives various values of the
Weissenberg number (We) with a¼0.5, b¼0.5, d¼1, and C¼0.6. The study reveals that the
rate of change of pressure with respect to distance ðdP=dXÞfalls as the Weissenberg number
(We) increases. The pressure gradient is significantly reduced in Figure 15 and significantly
increased in Figure 16. In these figures, positive values of dP=dXð Þ are maintained for all x values,
indicating the absence of a reverse pressure gradient. This is a key feature of peristaltic pumping
and is highly beneficial for enhancing the efficiency of EMHD [71,72] micropumps in medical
applications. The magnetic field boosts the peristaltic flow, leading to significant changes in pres-
sure distributions due to the inverse relationship between velocity and pressure.
4.4. Local skin friction, local nusselt number and local sherwood number
Tables 1 and 2display the physical meaning of local skin friction, local nusselt number, and local
sherwood number in both the X and Y axes. The Table 1 displays the local axial and transverse
skin frictions caused by the parameters f, Ha, C, and We. The williamson parameter has an
inverse relationship with the yield stress, resulting in increased viscosity at the surface and height-
ened axial skin friction. After examining Table 1, it is clear that the skin-friction coefficient shows
a positive link with the increasing values of f, C, Ha, and We. Table 2 exposed to view the local
nusselt number for various combinations of C and Pr. The table includes four alternative values
of the prandtl number, specifically Pr ¼2, 5, 6, and 7. Increasing the prandtl number or thermal
radiation leads to an increase in the rate of heat transfer, hence enhancing the local nusselt
18 P. DEEPALAKSHMI ET AL.
number. Similar observations are made across the entire table for varying values of the remaining
parameter. The williamson parameters primarily affect the velocity profile, but have minimal
impact on the local nusselt number. An opposite phenomenon is observed for the schmidt and
soret number in Table 2, which decrease the local sherwood number. The velocity slip plays a
crucial role in determining the local sherwood number, and it experiences a substantial drop.
5. Validation of the results
Table 3:
Table 1. The fluctuation of local skin friction number for specific values f, Ha, C and We.
Local Skin friction
fHa C We Skin friction
0.01 2.23 0.4 0.2 0.045
0.02 0.057
0.03 0.0675
0.04 0.0944
0.02 1 0.4 0.2 0.055
2 0.0892
3 0.0939
4 0.1024
0.02 1 0 0.3 0.0357
0.1 0.0578
0.2 0.0827
0.3 0.1265
0.02 1 0.3 1 0.0364
3 0.693
5 0.0875
7 0.112
Table 2. The fluctuations of local nusselt and sherwood numbers for specific values C, Pr, Sc and Sr.
Local Nusselt Number and Sherwood number
C Pr Sc Sr Nusselt Number Sherwood number
0.1 0.2 0.0768
0.2 0.096
0.3 0.1152
0.3 2 0.0461
5 0.0691
7 0.1613
0.1 0.7199
0.2 0.6746
0.3 0.5178
0.4 0.6126
0.45 0.6092
0.5 0.5264
Table 3. The acquired results are compared with the available literature [32], which serves as a limiting case for the topic
being studied on velocity profile.
We C fM Maraj et al., [32] Present Study
0 0.3 0.2 0 0.1376 0.1399
0 0.4 0.2 0 0.1365 0.1388
0 0.5 0.2 0 0.1328 0.1331
0 0.5 0.3 0 0.1386 0.1429
0 0.5 0.6 0 0.1374 0.1418
0 0.5 0.9 0 0.1339 0.1385
NUMERICAL HEAT TRANSFER, PART A: APPLICATIONS 19
The acquired results are compared with the literature of Maraj et al., [32] as a limiting case of
the reported issue, and are found to be in excellent agreement, as shown in Table 3.
i. This investigation may become a single-phase problem when volume fraction C equals zero.
ii. The values almost coincide with existing literature.
The velocity of a varying magnetic field is plotted in Figure 17 and validated with previous
results. Ongoing problems imply the flow of Williamson fluid in the presence of a magnetic field.
Maraj et al., [32] solved and experienced different non-Newtonian fluids in the absence of a mag-
netic field. Based on this comparison, it is evident that the findings of this inquiry provide robust
and accurate simultaneous results.
6. Conclusions
Motivated by emerging applications in electromagnetic ureteral flow therapy for combating
monoliths [67], a novel mathematical model in two-phase flow [42] has been developed for simu-
lating the two-dimensional, two-phase magnetohydrodynamic (MHD) [68] urologically incom-
pressible non-Newtonian peristaltic propulsion in the ureter under sinusoidal waves along the
boundaries. Saffman’s fluid-particle suspension model has been utilized. The Williamson visco-
elastic model has been deployed for rheology, and heat transfer has also been incorporated into
the model with Soret thermo-diffusion and viscous heating effects. Using long wave and low
Reynolds number approximations (lubrication theory), the mass, momentum, energy, and con-
centration conservation equations with associated boundary conditions have been transformed
into a non-dimensional boundary value problem. A numerical solution has been obtained via
BVP4C MATLAB quadrature. Graphical visualizations of the velocity, temperature, and concen-
tration (solid grains) have been presented for the influence of Saffman suspension parameter (1),
Hartmann number (M), Prandtl Number (Pr), Weissenburg number (We), particle volume frac-
tion (C), Eckert number (Ec), Soret number (Sr), Schmidt number (Sc). The simulations have
shown that:
i. With greater magnetic field intensity, i.e., greater Hartmann number, there is a strong sup-
pression in the ureteral fluid phase velocity confirming the excellent flow control abilities of
Magnetic Ureteral Therapy (MUT).
ii. With an elevation in Saffman suspension parameter ðfÞ, greater slip between the fluid and
particles is produced which elevates fluid phase velocity magnitudes.
iii. With increasing Weissenberg number (We) i.e., stronger elastic force relative to viscous
force in the rheological urodynamic fluid, a strong deceleration in the flow is computed.
iv. A strong depletion in temperature in the fluid phase accompanies an increase in Prandtl
number.
Figure 17. Validation of velocity parameters with maraj et al., [32].
20 P. DEEPALAKSHMI ET AL.
v. Increasing particle volume fraction (C) significantly reduces the fluid phase temperature
whereas an increase in Eckert number and the Saffman suspension parameter (f) both gen-
erates temperature elevation in the peristaltic regime.
vi. Increasing values of Saffman suspension parameter (f), Schmidt number (Sc), Soret number
(Sr) and Eckert number (Ec) all consistently induce a reduction in concentration
magnitudes.
The present work has revealed some interesting insights into electromagnetic ureteral peristal-
tic multi-phase non-Newtonian thermo-solutal transport phenomena. Future studies may also
consider visualization of the bolus dynamics and the deployment of electrical fields. Efforts in
these directions are currently underway and will be reported imminently. It is also noteworthy
that the renal pelvis plays a reservoir function role in urodynamics, and improved simulations of
electromagnetic therapy may require computational fluid dynamics (CFD) software to capture all
the complex characteristics, including 3-D fluid-structure interaction with the viscoelastic ureteral
wall, mural tension and mixing of contents as controlled by neurological functions. Excellent
commercial software such as FREEFEMþþ [69] and ANSYS FLUENT [70] is available for
achieving this level of sophistication and are also currently being explored.
Disclosure statement
No potential conflict of interest was reported by the author(s).
ORCID
E. P. Siva http://orcid.org/0000-0001-5663-1002
D. Tripathi http://orcid.org/0000-0003-4798-1021
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