Thermal Characteristics of Bioconvective Flow of a Shear-thinning
Fluid Conveying Nanoparticles and Gyrotactic Cells within a Stratified
Region
TOSIN OREYENI1, ANSELM O. OYEM2,3, BASMA SOUAYEH4, FELIX O. OKUNLOLA5
1Department of Physical Sciences,
Precious Cornerstone University, Ibadan,
NIGERIA
2Department of Mathematics,
Federal University Lokoja,
NIGERIA
3Department of Mathematics,
Busitema University,
UGANDA
4Department of Physics,
King Faisal University,
SAUDI ARABIA
5Department of Natural Sciences,
Precious Cornerstone University, Ibadan,
NIGERIA
*Corresponding Author
Abstract: - Thermal stratification in solar thermal systems is important for energy extraction and storage, as
well as for improving the efficiency and utilization of the trapped heat energy, leading to better economic
feasibility for renewable energy sources. The significance of triple stratification with the Cattaneo-Christov
model in the bio-convective nanoparticles flow of thixotropic fluid coexisting with gyrotactic microorganisms
is presented in this study. The incorporation of the Cattaneo-Christov heat and mass flux into the fluid model
allows for a more precise prediction of heat and mass phenomena in the fluid system. The governing partial
differential equations describing fluid flow are parametrized to produce a system of ordinary differential
equations. Using the Optimal Homotopy Analysis Method (OHAM), the series solutions are obtained. The
effects of selected pertinent parameters on the various profiles are revealed and properly reported. It is
envisioned that larger values of thermal stratification result in a decrease in temperature and concentration
distribution when, and .
Key-Words: - Cattaneo-Christov model, Stratification, Optimal Homotopy Analysis Method, Bioconvection,
Nanofluid, Renewable energy systems.
Received: June 6, 2023. Revised: November 7, 2023. Accepted: December 20, 2023. Published: December 31, 2023.
1 Introduction
Thixotropy is one of the most important phenomena
in science and engineering. Some materials exhibit
viscosity when their shear stress or shear rate varies
with the time. When exposed to mechanical
agitation, for instance, shaking, these materials are
rendered less viscid. When unused, they are solid-
like and thus, stand for their initial viscosity. It
represents a reversible process in which the
damaged structure destroys and reconstructs itself at
rest. This affects industrialization and engineering
applications very much. It can be noticed that this
kind of viscoelastic fluid takes part in the production
of paint, ink, and glue in the specified industries.
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The viscous viscosity for these products is low and
therefore they could easily be applied on surfaces.
However, as it is sheared, it ends up as a paste that
does not simply run or leak as [1] puts it. This has
been very important in making paints and inks that
are homogeneous throughout their surface; hence,
they don’t get pooled or run out. Thixotropic fluids
are often employed when creating drilling muds and
concrete by engineers. Drill mud is a thixotropic
material, which is mainly applied to the
reinforcement of the borehole against its collapse.
They have the ability to retain low viscosities when
pumped and circulated, meaning they flow
effectively through the wellbore. However, when
left standing they thicken to provide the necessary
support, [2]. Understanding thixotropic behavior is
important to the biomedical and pharmaceutical
industries. Certain pharmaceutical formulations find
it beneficial to have thixotropic properties as they
help in the ease of dispensing or application during
use as well as improve product stability during shelf
time. This includes creams, gel, and ointment. Some
thixotropic materials are used in drug delivery
systems whose purpose is a slow release of active
ingredients.
Thixotropy may be advantageous for some
uses such as coatings, adhesives, and food
processors; but it could be counterproductive for
medical-related applications and structure
constructs. It can pose problems in the design of
structural elements, and this is exemplified by the
case where a structural material exhibiting the
thixotropy character, suffers a reduction in apparent
viscosity with time while undergoing static loads
and cyclic loads, [3]. Thixotropy can have negative
effects on drug delivery systems in the biomedical
industry especially in pharmaceutical formulations.
Even though it is used in some formulations to
improve their usability, it can also have an impact
on the consistency and stability of pharmaceutical
products. The viscosity and flow characteristics of
hydrogels, ointments, suspensions, and emulsions,
for instance, may change as a result of thixotropic
behavior, which may have an impact on their
functionality and therapeutic efficacy, [4]. In the
biomedical sector, thorough evaluation and
characterization of pharmaceutical formulations can
aid in determining how thixotropy affects product
performance and allow for the necessary
adjustments to ensure consistent and reliable drug
delivery. Analysis of boundary layer flow of
thixotropic fluid under various effects has been
considered in the literature. Recently, [5], presented
the retardation effects of the Lorentz and Darcy-
Forcheimmer forces on the boundary layer flow of a
thixotropic fluid containing nanoparticles. It was
observed in their findings that thixotropy
characteristics reinforce the motion of nanofluid.
This made [6] to examine nasal cavity mucus
velocity variations using two distinct power laws
and thixotropic mucus layers. The effects of
thixotropy and shear-thinning through consideration
of the pipe flow of organic kerosene gel for different
pumping conditions were looked into by [7].
Bioconvection is the term used to describe the
mass movement of microorganisms in response to
environmental cues such as light, temperature
gradients, or chemical gradients. It is a common
phenomenon in biofilms or microbial suspensions.
On the other hand, nanofluids are fluids with
suspended nanoparticles that have unusual thermal
and flow characteristics causing these nanofluid
properties to impact greatly on the behaviour of
bioconvective microorganisms thereby, resulting in
a change of its thermophysical properties.
Researchers have looked into the modulation of bio-
convection by nanofluids which may allow for
better management and manipulation of microbial
populations. Fields such as nutrient cycling,
distribution and transportation of microorganisms
around natural water bodies, ecology, and
ecosystems, and treatment of wastewater, among
others have attracted many scientific researchers to
bioconvection processes enhancing the efficiency of
bioremediation techniques, and facilitating the
removal of pollutants from water systems. In the oil
and gas industry, bio-convection studies are relevant
to microbial-enhanced oil recovery (MEOR)
techniques by employing microorganisms to
increase oil recovery from reservoirs. In so doing,
optimizing MEOR strategies and predicting the
movement of injected microbes within the reservoir
can help, ultimately increasing oil production
through an understanding of bio-convection patterns
and microbial behavior.
Many authors have considered the concept of
bio-convection flow together with the suspension of
nanoparticles. [8], considered the implication of bio-
convection on the thermal effect of cross nanofluid
while, [9], analyzed the bioconvective assessment
for rate type nanofluid by numerical technique and,
[10], presented a numerical approach to bio-
convection caused by hydromagnetic flow with
nanoparticles. Furthermore, [11], examined the
impact of magnetic field and activation energy on
immiscible steady nanofluid moving over an elastic
stretched surface containing motile gyrotactic
microorganisms as, [12], discussed active and
passive controls of a shear-thinning fluid containing
nanoparticles and gyrotactic microorganisms while
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the characteristics of bioconvective flow of Casson
fluid with nanoparticles on surface with non-
uniform thickness was presented by [13]. Recently,
[14], presented the vertical bioconvective flow of
nanofluid containing microorganisms as, [15],
considered magnetohydrodynamic bioconvective
Eyring-Powell fluid flow on a permeable cone and
plate with activation energy and viscous dissipation
of a non-Newtonian fluid.
Thermal stratification is a process of layering
or separating fluids or gases based on their
temperatures in a medium such as air or water. It is
important in solar water heaters and thermal energy
storage systems, among others. When a fluid or gas
is heated, it loses density and rises whereas when a
fluid or gas is cooled it sinks due to increased
density. Within the medium, this creates distinct
layers or strata of varying temperatures. Thermal
stratification is important in natural bodies of water
such as lakes and oceans because it determines the
distribution of aquatic organisms and influences
ecosystem dynamics. Similarly, stratification can be
used in thermal energy storage systems to store and
extract heat energy such as in concentrated solar
power plants or district heating systems. Managing
thermal stratification can lead to more efficient and
cost-effective systems and processes which may
result in energy savings, reduced environmental
impact, and improved economic performance.
Likewise, [16], investigated the significance of
triple stratifications in the dynamics of a micropolar
fluid using nanoparticles and exponential heat
production. Their result showed that as stratification
increases the temperature gradient between the
surface and the free stream is reduced, which lowers
both fluid velocity and temperature. By using the
hottest water first, the system's efficiency is
improved, [17], and similar researches on
stratification were done by [18], [19], [20], while an
analytical approach to examine the effects of double
stratification and variable fluid properties on an
upper convected Maxwell fluid chemical reaction
was done by [21] and [22], discussed the impact of
bioconvection flow of nanofluid over a medium,
among others, [23], [24], [25], [26], [27].
From the aforementioned literature, numerous
researches on the concept of bioconvection flow
with various fluid models have been made however,
the bioconvective flow of a thixotropic fluid with
triple stratification using a Cattaneo-Christov model,
and the suspension of nanoparticles have not been
considered. Hence, this research looks into the flow
of a thixotropic fluid under the unique effects of
nanoparticles, stratifications, Lorentz force, and
Cattaneo-Christov heat and mass flux. This research
will support microbial enhanced oil recovery
(MEOR), thermal energy storage systems, and the
biomedical industry.
Focusing on the analysis of the hydromagnetic
bio-convective flow of shear-thinning fluid with
nanoparticles, triple stratification, and Cattaneo-
Christov heat and mass flux through a mathematical
model framed using partial differential equations is
presented and with the application of similarity
transformations, these partial differential equations
are converted and parameterized into the system of
ordinary differential equations and the approximate
analytic solution is obtained using Homotopy
Analysis Method (HAM). This study is useful in the
field of microbial-enhanced oil recovery (MEOR).
This process is an economical approach for the
recovery of unrecovered oil which is based on
injecting live microorganisms containing essential
nutrients into oil reservoirs through injection wells.
Hence, this research provides answers to the
following questions during the investigation;
i. What effect does the magnetic parameter have
on various distributions at the lowest layer of
stratifications?
ii. What effect does thermal stratification pose on
various profiles when Brownian motion and
thermophoretic parameters are raised?
iii. What is the impact of gyrotactic parameters on
the density of motile microorganism’s profile?
iv. What is the significance of the study to
industries and contributions to SDG 7?
2 Mathematical Formulation
Taking into account the Cattaneo-Christov heat and
mass flux, stratifications, fluid properties, and
bioconvective hydromagnetic flow of thixotropic
fluid with nanoparticles, the motile gyrotactic
microorganisms in the thixotropic fluid, swims in
the direction of the concentration gradient thus,
forming a bioconvection (Figure 1, Appendix). This
movement of gyrotactic microorganisms within the
thixotropic fluid containing nanoparticles is
intended to stabilize the nanoparticles' distribution
and prevent the appliance from corrosion and
sedimentation. The uniform surface temperature,
concentration, and gyrotactic microorganisms are
introduced, and based on the assumptions
mentioned above, the governing partial differential
equations of the bio-convective flow, [12], [20],
[28], [29], [30], [31], [32], [33], [34], [35], [36],
[37], [38], is given as Eq. (1) – (5) subject to
boundary condition (6) – (7), where Eq. 5 is the
density of gyrotactic microorganisms:
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subjected to the boundary conditions;
Where, and are velocity components along
x and y directions respectively,
is the ratio
of heat capacity of nanoparticle material to heat
capacity of base fluid, is dynamic viscosity, is
density of base fluid, is motile microorganism
concentration, is thermal diffusivity, is specific
heat capacity, is electrical conductivity, is
strength of uniform magnetic field, are
temperature, nanoparticle concentration,
microorganism concentration within the boundary
layer, , , are temperature, nanoparticle
concentration, microorganism concentration of
ambient fluid, is acceleration due to gravity, ,
, are thermal, solutal and motile microorganism
expansion coefficient, is thermal conductivity of
base fluid, is Brownian diffusion coefficient,
is thermophoresis diffusion coefficient, is
microorganism diffusion coefficient, is
volumetric heat generation coefficient, is
chemotaxis constant, and are the relaxation
time for heat and mass flux, respectively.
The suitable similarity transformation used is,
The model of temperature dependent viscosity
obtained from Batchelor’s experimental data is of
the form
To make the inclusion of stratification in the
fluid model clear, the thermal stratification at the
wall , solutal stratification at the wall ,
motile microorganisms at the wall and the free
stream., , are defined as:
where, and are the reference
temperature, nanoparticles concentration, and motile
micro-organisms’ concentration respectively, and
, and are the
temperature, nanoparticles concentration, and motile
microorganisms’ concentration coefficients
respectively.
Introducing the stream function and
other similarity variables, Eq. is satisfied
automatically and Eqs. with the boundary
conditions becomes;
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subject to the dimensionless boundary conditions
Where the governing dimensionless parameters;
are the magnetic parameter, modified local Grashof
number, Buoyancy ratio parameter, Prandtl number,
bio-convection Rayleigh number, micropolar
parameter, thermophoresis parameter, Brownian
motion parameter, Lewis number, Schmidt number
for diffusing motile microorganisms, Peclet number,
space-dependent internal heat source parameter,
thixotropic parameters, relaxation time parameter
due to heat flux, relaxation time parameter due to
mass flux, gyrotactic microorganisms concentration
difference parameter, thermal stratification
parameter, solutal stratification parameter,
gyrotactic microorganism density stratification
parameter respectively.
3 Method of Solution
Without solving the given nonlinear partial
differential problem Eq. (1) – (5) subject to
boundary conditions (6) – (7), it is required to
determine what types of base functions are
appropriate to represent the solution; first by
examining the physical context and the
initial/boundary conditions of the nonlinear
differential problem. Given the dimensionless
boundary conditions (17) and (18), , ,
and can be expressed by setting the base
functions to be:
The solutions for and can be represented
in series form as:
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where
,
,
and
are the coefficients.
As long as such a set of base functions are
determined, the auxiliary function the initial
approximations ,, and and
the auxiliary linear operators , , and are
selected in such a way that all solutions exist and are
expressed by these sets of base functions. Therefore,
in framing the Optimal Homotopy Analysis Method
(OHAM), the rule of solution expressions in
choosing the auxiliary function , the initial
approximation ,, and is
applied to Eqs. (14) – (16) together with the
dimensionless conditions (17) and (18) and
expressed as:
Let the Linear operators , and are:
The operators ,, and have the following
properties;
where, ,,,,,,,, are constants
and the linear operators are solved using Wolfram
Mathematica.
The simulations for different values of the
controlling parameters were conducted using the
approximate analytical method described in the
preceding sections. The effects of various embedded
physical factors on the flow will be covered in the
following section. In the analysis, the chosen values
for the selected parameters are;
.
Table 1 (Appendix) displays the numerical
values of skin friction coefficients and
reduced Nusselt number for various values
of . It is observed that coefficient of skin friction
is raised for the first two entries of and later
diminishes for the last two entries of , and
reduced Nusselt number declines for all the entries
of Table 2 (Appendix) shows the numerical
values of the skin friction coefficients and
reduced Nusselt number for various values
of . It is observed that skin friction coefficient
increases for the first two and the last two entries of
while reduced Nusselt number increases for all
values of .
Figure 2 (Appendix) reveals the impact of non-
Newtonian parameters and on the velocity
distribution when and
. It is noticed that incremental values of
and results in a rise in the velocity profiles. At
the lowest layer of fluid stratification (hypolimnion)
and at a high value Brownian motion, there is
sufficient energy to boost the velocity of the fluid
and associated boundary layer thickness. While on
the contrary, temperature and concentration of
nanoparticles decreases with increasing values of
and in Figure 3 and Figure 4 in Appendix.
Figure 5 (Appendix) displays the impact of
magnetic parameter on the velocity distribution
with and . For
higher values of , it is observed that velocity
profiles decline and this trend is due to the fact that
Lorentz force arises from the interaction between a
charged particle and, both electric and magnetic
field which poses influence on the velocity profiles
of the fluid flows. When magnetic fields are present,
it has the power to change the velocity profiles of
fluids hence, incremental values of correspond to
significant increase in the profiles of temperature,
concentration of nanoparticles and density of motile
gyrotactic micro-organisms respectively as
displayed in Figure 6, Figure 7 and Figure 8 in
Appendix. This observation is as result of the fact
that when an electrically conducting fluid flows
through a magnetic field, the Lorentz force plays a
role of a drag-like force, generating electrical
currents within the fluid. The electric currents
generated, leads to dissipation of energy in the form
of heat and thereafter, result in localized
temperature increase at the boundaries of fluid.
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The effect of thermal stratification is revealed
in Figure 9 and Figure 10 in Appendix. It is obvious
from these figures that increasing values of lead
to diminution of temperature and concentration of
nanoparticles profiles when and
. This reaction is attributed to the fact
that influences the heat transfer characteristics of
the fluid. In Figure 11 and Figure 12 in Appendix,
the influence of solutal stratification parameter
on temperature and concentration of nanoparticles
profiles are displayed. It is observed that is a
decreasing function of both temperature and
concentration of nanoparticles profiles. That is,
when there is strong solutal stratification, there is a
corresponding reduction in the convective heat
transfer between different fluid layers which leads
to decrease in temperature profiles. Likewise, when
solutal stratification is strong, diffusion process
becomes more prominent and intensifies as solute
particles tend to redistribute themselves hence,
reducing the concentration difference between
regions.
In Figure 13 and Figure 14 in Appendix, the
contribution of thermal relaxation parameter on
various profiles is presented. From Figure 13
(Appendix), it is noticed that temperature profile
diminishes with augmenting values of the thermal
relaxation parameter this is due to the fact that a
larger relaxation parameter implies that the system
can reach a state of equilibrium more quickly. An
enhancement in the concentration of nanoparticles
profiles for larger values of is envisioned in
Figure 14 (Appendix). The escalation of solutal
relaxation parameter leads to diminution of the
temperature profile as observed in Figure 15
(Appendix) and incremental values of solutal
relaxation parameter causes a noticeable decline
in concentration of nanoparticles profiles when
in Figure 16 (Appendix) while,
Figure 17 (Appendix) depicts that a conspicuous
decline is noticed in density of the motile
microorganism profile when gyrotactic
microorganism’s parameter is raised.
3.1 Convergence of the OHAM
The interval on -curve becomes parallel to the -
axis if recognized as the set of admissible values of
non-zero auxiliary parameters ,, and for
which the solutions series converges. Figure 18,
Figure 19, Figure 20 and Figure 21, in Appendix,
the range of acceptable values of
are ,
, and
. Obviously, from the -curves for this
problem, we obtained the approximate optimal
values of at - order of
approximations as
and .
4 Conclusion
The impact of triple stratification on the bio-
convective flow of a fluid containing tiny particles
and a Cattaneo-Christov heat and mass flux has
been duly considered in Eq. 13 – Eq. 16 subject to
the dimensionless conditions Eq. 16 and Eq. 17. The
governing dimensionless parameters in Eq. 19
together with assigned values were analytically
resolved and from the obtained results, the
following conclusions are made:
1. Velocity of the thixotropic fluid increases as the
thixotropic parameters and are raised at the
lowest layer of stratification.
2. Velocity of fluid also declines for larger values
of magnetic parameter caused as a result of the
appearance of Lorentz force which arises from
the interaction between charged particles and
both electric and magnetic fields.
3. Convective heat transfer declines between
different fluid layers as a result of the influence
of strong solutal stratification parameter .
When the thermal relaxation parameter is
increased, the fluid's temperature decreases.
4. Large values of the gyrotactic microorganism’s
parameter are predicted to result in a noticeable
decline in density of the motile microorganism
distribution.
For consideration of future research, the current
model can be extended to hybrid and ternary hybrid
nanofluids in the presence of quartic autocatalytic
reaction over various geometries which has
enormous applications in industries and engineering
fields.
Acknowledgement:
The authors are thankful to the reviewers and editor-
in-chief for their support, observation and ideas
towards the improvement of this article.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed to the present
research, at all stages from the formulation of the
problem to the final findings and solution.
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Scientific Article or Scientific Article Itself
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Conflict of Interest
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that are relevant to the content of this article.
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APPENDIX
Fig. 1: Schematic diagram of the problem
Fig. 2: Effects ofand on velocity profiles
Fig. 3: Effects of on temperature
P
r
o
f
i
l
e
s
Fig. 4: Contribution of and on concentration
of nanoparticles profile
Fig. 5: Contribution of on velocity profile
Fig. 6: Contribution of on temperature profile
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Fig. 7: Contribution of on concentration of motile
microorganism profile
Fig. 8: Contribution of on density of
nanoparticles profile
Fig. 9: Contribution of on temperature profile
Fig. 10: Contribution of on concentration of
nanoparticles profile
Fig. 11: Contribution of on temperature of
nanoparticles profiles
Fig. 12: Contribution of on concentration Profiles
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Fig. 13: Contribution of on temperature of
nanoparticles profiles
Fig. 14: Contribution of concentration profiles
Fig. 15: Contribution ofon temperature of
nanoparticles profiles
Fig. 16: Contribution of concentration profiles
Fig. 17: Contribution of on density of motile
micro-organisms profile
Fig. 18: -curve of obtained at 10th order of
approximation
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Fig. 19: -curve of obtained at 10th order of
approximation
Fig. 20: -curve of obtained at 10th order of
approximation
Fig. 21: -curve of obtained at 10th order of
approximation
Table 1. Numerical values of skin friction
coefficients and reduced Nusselt number for various
values of
Table 2. Numerical values of skin friction
coefficients and reduced Nusselt number for various
values of
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