1. Introduction
Nanofluids have many applications, including the transport of generated heat in micro-electronics, power generation, manufacturing, and the transportation of ethylene glycol, water, engine oil, etc. The conventional heat transfer of the fluids is poor due to low heat conductivity and it does not transfer the heat produced from a specific surface to the fluid effectively. To enhance heat transfer rates, nanofluids are preferred in industries, and are applied in heat exchangers and electronic cooling systems. Choi [
1] introduced the notion of nanofluids in 1995. Masuda et al. [
2] analyzed the phenomenon in which nanofluids have a characteristic feature thermal conductivity enhancements, which indicate the possibility of using nanofluids in advanced nuclear systems [
3]. Buongiorno [
4] studied an analytical model for convective transport in nanofluids by considering Brownian diffusion and thermophoresis. He developed an explanation for abnormal convective heat transfer enhancements observed in nanofluids. Moreover, he observed that Brownian diffusion and thermophoresis were the most important nanoparticle/base-fluid slip mechanisms. Kuznetsov et al. [
5] deliberated the natural convective boundary-layer flow of a nanofluid past a vertical plate. In recent research studies, Izadi et al. [
6] studied the porous metal for cooling CPUs on MHD to enhance nanofluid thermal transport. In this study, viscous dissipation and Darcy–Brinkman–Forchheimer model effects were considered to find the fluid stream of nanofluids. Raza et al. [
7] observed the thermal radiation magnetohydrodynamic stream of non-Newtonian nanofluids across a stretching surface. Jamshed [
8] deliberated a computational study of magnetohydrodynamic effects on Maxwell nanofluids. He studied the entropy generation on the MHD stream of Maxwell nanofluids across an infinite horizontal sheet. Koriko et al. [
9] deliberated a stream of bioconvection on magnetohydrodynamic nanofluids across a vertical channel with nanoentities and gyrotactic microorganisms. Li et al. [
10] described the thermal and mass transportation past an exponentially porous extending sheet on magnetohydrodynamic Williamson nanofluid flows. They viewed this analysis for two various conditions of thermal transport described as prescribed exponential sheet temperatures and prescribed exponential order thermal fluxes. Abbas et al. [
11] viewed the entropy-optimized MHD nanofluid stream as fully realized. Dawar et al. [
12] retrieved the velocity slips and non-uniform heat sources on MHD micropolar nanofluid flows. Shi et al. [
13] examined bioconvective nanofluid streams across stretching sheets with swimming microbes. Ashraf et al. [
14] deliberated the nanofluid flows occurring across a slender elastic surface. Recently, published research articles on nanofluids are mentioned in references [
15,
16,
17,
18,
19].
Flows through porous media are of principal interest because these are quite prevalent. Such a flow has applications in a broad range of scientific fields, from geophysics to chemical engineering. Numerous technological applications need to flow through porous media that are saturated with fluid, and this need is growing as applications of geothermal energy and astronomical issues increase. An improved comprehension of the fundamentals of mass, energy, and momentum transport in porous media may also be beneficial for several other applications, such as the cooling of nuclear reactors, the underground disposal of nuclear waste, operating petroleum reservoirs, building insulation, food processing, casting, and welding in manufacturing processes. Working fluid heat production (source) or absorption (sink) effects are significant in several porous medium applications. Representative studies dealing with these effects have been addressed by authors such as Gupta and Sridhar [
20], Subhas and Veena [
21], and Prasad et al. [
22]. The impacts of variable viscosity and thermal conductivity on an unsteady two-dimensional laminar flow of viscous incompressible conducting fluids across a semi-infinite vertical porous moving plate are studied by Seddeek and Salama [
23]. Zheng et al. [
24] analyzed the heat transfer of nanofluid past an extending surface with a porous medium. Dessie and Kishan [
25] studied the impacts on variable viscosity and viscous dissipation across stretching sheets embedded in a porous medium. The study of visco-elastic fluid flow through porous media has gained importance in recent years; see Refs, e.g., [
26,
27].
Many materials, such as colors, ketchup, lubricants, dirt, some paints, blood at low shear rates, and certain care items involving the flow of non-Newtonian fluids, have been extensively investigated. Non-Newtonian fluids are utilized in a various industrial applications, including composite processing, polymer depolarization production, boiling, fermentation, bubble columns, bubble absorption, and plastic foam processes. In addition, the effects of heat and mass transfer in non-Newtonian fluid [
28,
29] have great importance in engineering applications such as the thermal design of industrial equipment, food stuffs, slurries, etc. Moreover, Navier–Stoke’s theory is insufficient when there are some complex rheological fluids. Thus, there is a need to study non-Newtonian fluids [
30,
31]. Since not a single model exhibits all properties of fluids, therefore, many non-Newtonian models have been presented by various authors (Refs. [
32,
33]). The first time Powell and Eyring’s theories were introduced was in 1944. Bilal et al. [
34] described the thermal transport of Eyring–Powell fluids through a stretching surface with the influence of mixed convection. here, the fluid is viscous, laminar, and incompressible. Akbar et al. [
35] deliberated that MHD flow impacts Eyring-Powell fluid streams across an extending sheet. Ibrahim et al. [
36] presented the stream of Eyring–Powell nanofluids through a porous medium. Javed et al. [
37] described the Eyring–Powell fluid across a boundary layer flow across an extending surface for non-Newtonian fluids by using the Keller box method. Via a porous stretching sheet, Vishalakshi et al. [
38] quizzed the non-Newtonian MHD fluid stream in 3D, taking into account heat radiation and mass transpiration. By considering the consequences of heat radiation and absorption, the MHD Casson and Williamson liquids over stretching surfaces were discussed by Saravana et al. [
39]. Sarada et al. [
40] analyzed the influence of MHD on thermal transference characteristics of non-Newtonian liquids over a stretched surface. Ajeeb et al. [
41] elaborated on the convective heat transference of non-Newtonian multiwall carbon nanotubes and aided in tuning nanofluid properties.
The heat transfer analysis of boundary layer flows with radiation is also important in electrical power generation, astrophysical flows, solar power technology, space vehicle re-entry, and other industrial areas. The extensive literature dealing with flows in the presence of radiation effects is now available. Because of this, Raptis et al. [
42] studied the effect of thermal radiation on the MHD flow of a viscous fluid past a semi-infinite stationary plate. Hayat et al. [
43] analyzed the influence of thermal radiation on the MHD flow of a second-grade fluid. Cortell [
44] analyzed the numerical analysis for flow and heat transfers in a viscous fluid and thermal radiation by extension over a nonlinearly stretched sheet. Rashidi et al. [
45] deliberated the buoyancy impacts on the MHD flow of nanofluid across a stretching surface with thermal radiation. In recent studies, numerous investigators analyzed thermal radiation in recent years (Refs [
46,
47]).
Bioconvection is a natural mechanism in which microorganisms move in a single-celled or colony-like arrangement at random. In still water, gyrotactic microorganisms move upstream against gravity, which starts to cause the top area of the suspension to be denser than the bottom. Bioconvection is employed in biofuels, fertilisers, bioreactors, oil recovery, biosensors, biomicrosystem plant products, and enzymes, among many other uses. Bioconvection technology has been used in a wide range of products in both organic and powered industries. A model for collective movement and pattern formation in layered suspensions of negatively geotactic microorganisms was presented by Childress [
48]. The effect of gyrotaxis on the linear stability of a suspension of swimming, negatively buoyant microorganisms was examined by [
49]. MHD boundary layer flows with heat and mass transfers of an electrically conducting water-based nanofluid containing gyrotactic microorganisms were analyzed by [
50]. Alqarni et al. [
51] deliberated the thermal transfer of a bioconvection stream of micropolar nanofluids possessing velocity slips. Zadeh et al. [
52] examined the heat and mass transportation flow and nanofluid flow over an extending sheet in the presence of bioconvection and investigated these phenomena numerically. Chu et al. [
53] described the effects of bioconvection magnetohydrodynamic third-grade fluids across an extended sheet. Alshomrani [
54] numerically studied a nanofluid with magnetic dipoles and bioconvection flows. Asjad et al. [
55] studied bioconvection flows over a vertical surface. Jawad et al. [
56] deliberated entropy generations on magnetohydrodynamic bioconvections of Casson nanofluids across a rotating disk. Across the upper-surface revolution, Mair and Khan et al. [
57] analyzed magnetohydrodynamic bioconvections. Ali et al. [
15] analyzed bioconvection flows through an inclined surface.
By studying the above literature survey, we concluded that there are no research studies on the MHD flow of Eyring–Powell nanofluids with gyrotactic microorganisms across a slender elastic surface. Motivated by the aforementioned wide scope of non-Newtonian fluid and nanofluid applications, therefore, we decided to evaluate the current elaborated fluid problem. thus, the primary purpose of the present investigation is to analyze the outcomes of distinct parameters and nanoparticles on the dynamics of Eyring–Powell fluids across a slender elastic surface. Recently, research on non-Newtonian fluids has fascinated young researchers in extending the research scope due to large-scale applications in different industries. To improve the base fluid’s stability to avoid sedimentation, we also incorporated microorganisms. By using similarity modifications, the relevant nonlinear PDEs are converted into a system of ODEs. The graphical outcomes of velocity, temperature, the concentration of nanoparticles and microorganisms, the skin friction factor, Nusselt number, and Sherwood number are conducted for different inputs of physical parameters.
2. Physical Model and Mathematical Formulation
Consider the heat and mass transfer steady boundary layer flow of Eyring–Powell nanofluids and electrical conduction in the existence of magnetic field
in which the induced magnetic field can be ignored reasonably over an impermeable elastic surface
with variable thickness. The slit through which the sheet is pulled in the fluid is called the origin. The physical model of the problem is shown in
Figure 1. The x-axis is aligned with the motion’s direction, whereas the y-axis is perpendicular to it. We pretend that the sheet is not flat and is instead defined as
. Another assumption is that the nanoparticle does not affect microbial velocities. A concentration of nanoparticles of less than 1% is a reasonable assumption. As a result, bioconvection stability is defined as a suspension of solid nanoparticles in a liquid medium. Nanoparticles are uniformly distributed throughout the base fluid. By the utilization of these assumptions, the governing equations for the conservation of mass, linear momentums, energy, nanoparticle concentration, and the density of motile microorganisms in vector form are as follows.
Here,
) includes velocity components in
directions, respectively. Furthermore, the symbols in the above equations are defined in the nomenclature. The extra stress tensor of an Eyring–Powell model [
58] is expressed as follows.
The governing equations can be written as follows [
34,
59,
60].
Equation of continuity:
Equation of momentum:
Heat equation:
Nanoparticle concentration equation:
Concentration of microorganisms diffusion equation:
The boundary conditions of the present problem are as follows.
By assuming that thermophysical variables
have a linear temperature dependence, we obtain the following.
The radiation under Rosseland approximations can be written as follows [
61].
may be identified by extending in a Taylor series around
while ignoring higher terms.
Thus, substituting Equation (
8) in Equation (
7), we obtain the following.
A mathematical technique that is suitable for some cases to convert partial differential equations into ordinary differential equation is called similarity transformation. This technique is sophisticated and is the most powerful and systematic for transforming PDEs into ODEs, and it is utilized widely in non-linear dynamical systems, especially in the field of computational fluid dynamics boundary value problems. To transform the above-mentioned governing equations into dimensionless forms, we introduce the following similarity variables, which guided us to avail non-dimensional forms of leading equations [
59].
In the observation of above defined similarity variables, Equation (
7) is identically satisfied, and Equations (8)–(11) can be written as follows:
which are subject to the boundary conditions in Equation (
12):
where
,
,
,
,
,
and
,
,
,
,
,
,
, and
,
.
We set the following possible changes for additional simplifications.
As a result, the nonlinear differential Equations (18)–(21) are changed to the following:
along with modified boundary constraints.
5. Results and Discussion
This part aims to study the influence of parameters contained in the current problem on the momentum, temperature, and concentration of nanoparticles and microorganisms. To verify the exactness of the present computed outcomes with accessible distributed data, a comparison is completed between current computed outcomes and those in the available literature in a limiting case.
Table 1 and
Table 2 are displayed to authenticate the current numerical results of
and
with the previously published literature in Wakif [
59] (skin friction) and [
59,
65] (Nusselt number), and these results achieved excellent agreements. We take the values of parameters as
, and
The results of an investigation on Eyring–Powell nanofluid transportation caused by a stretched slender sheet of variable thickness are presented here. The fluctuation of skin friction factor was mentioned in
Table 3. Because the physical nature of these parameters opposes the fluid stream, it, therefore, increases the resistance at the surface, increases parameters
M,
K,
,
,
, and
, and
increased the magnitude of skin friction due to the accelerated flow. The skin fraction factor is reduced in a reciprocal manner by the thermal buoyancy parameter
. The Nusselt number (local heat transfer rate) seems to increase immediately with the Prandtl number; however, it decreases reciprocally against excessive values of
,
,
, and
according to the contents of
Table 4. Thermophoresis causes the Sherwood number to drop, whereas increased
,
, and
caused it to increase (see
Table 5).
The impact of magnetic parameter
M on momentum
and temperature boundary layer
is depicted in
Figure 2a,b; in that order, growing values of
M lower the curve of momentum but increase the curve of the thermal boundary layer, according to inspection. The increase in
M is coupled with a stronger opposing force (Lorentz force) on the stream, resulting in a slowing of the speed and increase in temperature. Meanwhile, the fluid’s loss of kinetic energy is offset by a gain in heat energy, causing an increase in temperature. The effects of stronger porosity parameter
K on
and
are scrutinized in
Figure 3a,b, respectively. It is mentioned that
K is inversely related to the permeability
of the porous medium, which causes a decrement in momentum and increase in temperature. The sketches for momentum and thermal as influenced by thermal buoyancy parameter are shown in
Figure 4a,b. Due to increasing buoyancy effects, the velocity increases directly with
. The fluid’s heat energy is used to reduce the temperature throughout this procedure.
Figure 5a,b and
Figure 6a,b portray the impacts of the buoyancy ratio parameter and Rayleigh number on the velocity and temperature profile. Both of these parameters decelerated the velocity profile and intensified the temperature. These parameters are inversely related to temperature differences: When temperature increases quicker, thermal buoyancy becomes less of a reason for velocity retardation.
Figure 7a,b demonstrate the impact of fluid material parameters
and
on temperatures. The flow speed is quicker, and the thermal boundary is lower when parameter domains are boosted. This is because they are related inversely to viscosity. The boosted
and
mean lesser viscosity; hence, faster speeds result. When flow speed enhanced, the thermal profile is decremented. The impact of the wall thickness parameter
on momentum and thermal boundary layer is observed in
Figure 8a,b. Horizontal velocity
increases as
increases. When
,
and it declines with the increasing strength of
, as stated by boundary condition
. Then,
should increase. With
, the temperature of the fluid likewise increases, resulting in a greater blowing impact.
Figure 9a,b demonstrated the influence of material parameter
on momentum profile
. We observed from the figure that fluid velocities increase with higher
. As it is directly in relation to the velocity of the surface and because of the greater stretching speed, the fluid’s velocity is also higher, and the temperature of the fluid decreases with an increase in
. The fluctuation in temperatures and concentrations of nanoparticles is shown in
Figure 10a,b to comprehend the influence of Brownian motion parameter
. The higher the
, the quicker the random movement of the nanoparticles, which results in increased heat diffusion and an increase in temperature. In addition, the rapid motion causes the concentration function to deteriorate. When particles transition from a heated to a cold state, a thermophoretic impact occurs. In
Figure 11a,b, the impacts of the thermophoresis parameter
on thermal and concentration boundary layer are displayed. The gradual migration of nanoparticles from hotter environments causes the temperature and concentration in the boundary layer area to increase.
Figure 12a,b delineated the higher temperature
of the fluid in direct response to
(variable thermal conductivity parameter) and
(Radiation parameter). The increments in
include larger thermal conductivities; similarly, the higher inputs of
mean larger radiation heat diffusion, which cause an increase in temperature. As the Lewis number
is reciprocated with respect to mass diffusivity, its higher values cause a decline in concentration curve
, as shown in
Figure 13a. Similarly, against bioconvection Lewis number
, the concentration profile of microorganisms
is reduced (see
Figure 13b) because
is in an inverse relation to the diffusivity of microorganisms.
Figure 14a,b with a rescaled microorganism distribution incorporates the nature of Peclet number
and density ratio
. When Peclet number
and density ratio
increases, the microorganism profile decreases.