Entropy generation in radiative motion of tangent hyperbolic nanofluid in the presence of gyrotactic microorganisms and activation energy

In this work, entropy generation is optimized through the application of the second law of thermodynamics. The slip mechanisms, Brownian diffusions, and thermophoresis are elaborated using the tangent hyperbolic nanomaterial model. Magnetohydrodynamic (MHD) fluid is taken into consideration. To characterize the impact of activation energy, a unique model involving the binary chemical reaction is deployed. The effects of mixed convection that is nonlinear in nature, bioconvection, and Joule effect are all taken into consideration. The key partial differential equations (PDEs) are reduced into ordinary differential equations (ODEs) by utilizing appropriate similarity transformations and then solved numerically with the help of a built-in ‘bvp4c’ technique of MATLAB software. Varied flow parameters’ impacts on the nanoparticle volume concentration, entropy number, microorganism concentration, temperature, and velocity fields are analyzed using graphs. Various flow variables are taken into consideration to calculate the total rate of entropy generation. The obtained results show that concentration irreversibility, Joule effect irreversibility, viscous dissipation, and heat irreversibility all influence the entropy. The numerical outcomes were observed by fixing the physical parameters as $0.1<\alpha <4.0$, $0.1<M<1.2$, $0.1<Nr<2.2$, $0.1<Le<2.2$, $0.1<Nb<0.4$, $0.1<Nt<1.0$, $2.0<\mathrm{Pr}<5.0$, and $0.1<Lb<2.0$, as well as their impact on the momentum, thermal, concentration, and microorganism density profiles. From results, an increasing estimate of the variable representing chemical reaction indicates a decline in the concentration. The higher the chemical reaction variable, Hartmann number, and Weissenberg number, the higher the entropy number, while the Bejan number has a contrary behavior. Subsequently, all the outcomes are plotted in graphs and discussed in detail, when subjected to the involving physical quantities.

1 Introduction

Engineers and scientists have held a longstanding fascination with researching non-Newtonian fluids, primarily because of their behavior on stretched surfaces with heat and mass transfer. This is because of its numerous practicalities, such as paper manufacturing, wire drawing, paper manufacturing, petroleum production, and polymer sheet manufacturing. Because shear rate and shear stress are not linearly related in non-Newtonian fluids, one constitutive equation is insufficient for representing the relationship between shear rates and shear stresses. Owing to this, various frameworks have been presented in the literature, including the Maxwell model, Reiner–Rivlin model, generalized Burgers, Oldroyd-A model, Oldroyd-B model, Oldroyd-8 constants, Carreau model, and the Carreau–Yasuda model. In the current investigation, we took non-Newtonian tangent hyperbolic fluid under consideration. Effects of shear thinning are described by a four-constant pseudoplastic model of fluid, i.e., tangent hyperbolic fluid. When flow decreases, owing to increased shear stress, it is useful to evaluate the inter-particle friction in the fluid. Examples of such liquids include ketchup, paint, blood, nail polish, and whipped cream. [1] presented a numerical approach for the flow of rate-type fluid across the stretchable cylinder using MHD flow. [2] focused on thermal radiation, which is non-linear in nature, to probe the flow by a stretchable surface with Brownian diffusion and thermophoresis. The flow of the two-dimensional rate-type fluid toward a stretchable sheet with the applied magnetic effect was investigated [3]. [4] found solutions in the series form for a tangent hyperbolic fluid that is inclined uniformly in a tube. [5] analyzed the tangent hyperbolic fluid using a stretchable sheet while considering both the impacts of Soret and Dufour in MHD stagnation point flow. [6–13] provided more research articles on non-Newtonian fluids.

Recently, some energy-efficient mechanisms have been designed to minimize the quantity of wasted energy and boost energy utilization capabilities of liquid systems. The transport of heat and viscous dissipation results in energy waste, which aids the irreversibility of the liquid system. Electrochemical and chemical, evaporative cooling, natural convection, solar thermal, microchannels, air separators, chillers, gas turbines, fuel cells, and functionally graded materials are all examples of systems where entropy generation minimization (EGM) is beneficial. [14] recently analyzed the generation of entropy in nanofluids using copper and silver nanoparticles. [15] studied the increase in entropy in a burner model with various energy sources. The optimization of entropy in a nonlinear radiative flow while accounting for Joule effects and dissipation is investigated by [16].

The change in density over a region induces additive rapid movement in self-propelled microorganisms; as a result of this, there is macroscopic movement of the fluid, which is termed as bioconvection. Self-propelled microorganisms have a tendency to enhance the base fluid to create a bio-convective stream in a specific direction. The self-propelled microorganisms are categorized into different types, i.e., gyrotactic, negative gravitaxis, and chemotaxis or oxytactic microorganisms. Compared to self-propelled microorganisms, nanoparticles do not move on their own, and their movement is driven by the effect of Brownian motion and thermophoresis. If there is minimal nanoparticle concentration, then in the nanofluids, bioconvection is supposed to be feasible and it will not result in an essential increase in the thickness of the base fluid. Bioconvection with nanoparticles was originally discussed in [17, 18]. Afterward, combining nanoparticles with gyrotactic microorganisms was proposed by [19] using Buongiornos theory.[20] studied the flow of nanofluids with bioconvection in porous medium numerically. Bioconvective flow with nanoparticles through a symmetric channel was studied by [21], and a bio nanoengineering model was presented. In rapidly changing magnetic lines of force, the clot blood model was also analyzed by [22] using the Jeffrey fluid model with microorganisms and nanoparticles. A magnetized laminar flow of nanofluid along with phototactic microorganisms in a medium that is non-Darcy with pores was observed by [23]. The external magnetic effect and bioconvective flow were researched by [24] with convective boundary conditions and nanoparticles.

In the recent past, as a result of advancements in nanotechnology, the researchers’ focus has shifted to improving heat transport through the nanoparticle interaction. Because of their fascinating thermo-physical aspects, the nanoparticles, which include metallic particles of micro-size, are useful in biological, mechanical, chemical, and other engineering fields. The conductivity of heat energy and the viscosity of the fluid improve not just with the addition of nanoparticles but also with the structure of particle size, and slip features in nanoparticles play an important role as well. [25] developed nanoparticles to improve the product efficiency and fluid thermal conductivity. Heat transfer near a stagnation point in the Maxwell nanofluid flow over a porous rotating disk, highlighting the impact of shear stress and flow field alterations on heat transfer, has been investigated by [26]. Some interesting studies on nanofluids have been discussed in [27–30]. [31] analyzed the magnetohydrodynamic stagnation point flow of Williamson hybrid nanofluid over a porous sheet, considering chemical reactions, energy generation, and boundary conditions. [32] examined the unsteady 2D laminar flow of Williamson hybrid nanoliquid over a convectively heated stretching sheet, focusing on maximizing energy and mass transfer rates. [33] investigated entropy generation in the mixed convection time-dependent flow of cross-hybrid nanoliquid at a stagnation point using engine oil with $CuOandTi{O}_2$ nanoparticles.

Activation energy and chemical reactions have been used in mass transportation phenomena, which have been carried out in the history of food sciences, chemistry, chemical, and mechanical industries. In 1889, Svante Arrhenius engaged in the key observations on activation energy and defined it in terms of the least quantity of energy that is required to begin a chemical process. Geothermal sciences, petroleum engineering applications, oil reserves, heat reactions, and a lot of other fields are part of the activation energy phenomenon. Some major contributions to this subject are discussed in [34–36]. [37] numerically investigated the natural convection boundary-layer flow over a truncated cone with gyrotactic microorganisms in water-based nanofluid, revealing enhanced transport properties and boundary conditions’ effects. [38] explored the bioconvection phenomenon of a nanofluid with gyrotactic microorganisms around a horizontal circular cylinder, utilizing Buongiorno’s model and finite-difference algorithm for the comprehensive analysis of transport phenomena. [39] investigated mixed convection of nanofluid with gyrotactic microorganisms over a permeable vertical surface in a porous medium, which reveals suction/injection effects, with significant impacts on heat and mass transfer. [40] also examined the mixed bio-convective stagnation point flow of a power-law nanofluid over a stretchable surface with passively controlled boundary conditions. [41] explored non-Newtonian nanofluid flow induced by a rotating stretchable disk, analyzing heat and mass transfer with Stefan blowing and Cattaneo–Christov fluxes. He also examined the flow of a non-Newtonian nanofluid in a conical gap between rotating/stationary surfaces, considering heat and mass transfer and a uniform magnetic field [42]. Some of the similar results are discussed in [43–45]. The study of the fluid flow in porous medium is of great importance in fluid mechanics. [46] analyzed the motion of Jeffrey nanofluid in porous medium with motile microorganisms between two revolving stretching disks. The study of different types of fluids in between porous medium is given in [47–50]. Furthermore, heat and mass transfer analysis of nanofluid is numerically scrutinized under the influence of magnetic field subject to marangoni boundary layer [51, 52].

The major goal of this work is to look at the optimization of entropy in the tangent hyperbolic nanofluid flow on a stretchable sheet. The uniqueness of the work under discussion is that it investigates the optimization of entropy generation in a non-Newtonian fluid model with non-linear thermal radiation, non-linear mixed convection, viscous dissipation, bioconvection, and activation energy as additional effects. To characterize activation energy’s effects, we used a unique two-step chemical reaction model. Various flow variables are utilized to compute the rate of total entropy generation. We have used a method called the shooting technique to deal with the nonlinear ordinary system. To examine the behavior of the included variables on fluid properties, graphs and tables are created. Analysis on various methodologies is shown in [53–56].

2 Mathematical model

In tangent hyperbolic nanomaterial flow, we analyzed Bejan numbers and entropy. In the current model, the impacts of non-linear heat radiation, non-linear mixed convection, activation energy, and Joule effect are studied. The stretchable sheet with the stretching rate being a generates the flow. ${\stackrel{\sim }{T}}_w,{\stackrel{\sim }{C}}_w$, and ${\stackrel{\sim }{N}}_w$ signify the temperature, concentration, and density of self-propelling microorganisms at the sheet, respectively, while ${\stackrel{\sim }{T}}_{\infty },{\stackrel{\sim }{C}}_{\infty }$, and ${\stackrel{\sim }{N}}_{\infty }$ represent the ambient temperature, concentration, and density of self-propelling microorganisms, respectively. $B_0$ magnetic strength is in the y-direction (see Figure 1). In light of these considerations, the governing equations for the given problem are Equations 1–6 [57]:

$\frac{\partial {\stackrel{\sim }{u}}_1}{\partial x}+\frac{\partial {\stackrel{\sim }{u}}_2}{\partial y}=0,(1)$

$\begin{array}{l}{\stackrel{\sim }{u}}_1\frac{\partial {\stackrel{\sim }{u}}_1}{\partial x}+{\stackrel{\sim }{u}}_2\frac{\partial {\stackrel{\sim }{u}}_1}{\partial y}=\frac{{\partial }^2{\stackrel{\sim }{u}}_1}{\partial y^2}\nu \left(1-s\right)+\frac{\partial {\stackrel{\sim }{u}}_1}{\partial y}\frac{{\partial }^2{\stackrel{\sim }{u}}_1}{\partial y^2}\sqrt{2}\nu s\mathrm{\Gamma }-\frac{\sigma }{{\rho }_f}{B_0}^2{\stackrel{\sim }{u}}_1+g\beta \left(1-C_{\infty }\right)\left(\stackrel{\sim }{T}-{\stackrel{\sim }{T}}_{\infty }\right)\\ -g\left(\frac{{\rho }_m-{\rho }_f}{{\rho }_f}\right)\left(\stackrel{\sim }{C}-{\stackrel{\sim }{C}}_{\infty }\right)-{\gamma }_{1}g\left(\frac{{\rho }_m-{\rho }_f}{{\rho }_f}\right)\left(\stackrel{\sim }{N}-{\stackrel{\sim }{N}}_{\infty }\right),\end{array}(2)$

$\begin{array}{l}{\stackrel{\sim }{u}}_1\frac{\partial \stackrel{\sim }{T}}{\partial x}+{\stackrel{\sim }{u}}_2\frac{\partial \stackrel{\sim }{T}}{\partial y}=\frac{{\partial }^2\stackrel{\sim }{T}}{\partial y^2}\frac{k}{{\left(\rho c_p\right)}_f}+\left\{D_B\frac{\partial \stackrel{\sim }{C}}{\partial y}\frac{\partial \stackrel{\sim }{C}}{\partial y}+\frac{D_T}{{\stackrel{\sim }{T}}_{\infty }}{\left(\frac{\partial \stackrel{\sim }{T}}{\partial y}\right)}^2\right\}\frac{{\left(\rho c_p\right)}_s}{{\left(\rho c_p\right)}_f}+\left[3{\stackrel{\sim }{T}}^2{\left(\frac{\partial \stackrel{\sim }{T}}{\partial y}\right)}^2+{\stackrel{\sim }{T}}^3\frac{{\partial }^2\stackrel{\sim }{T}}{\partial y^2}\right]\\ \times \frac{1}{{\left(\rho c_p\right)}_f}\frac{16{\sigma }^{*}}{3k^{*}}+\frac{\sigma }{{\left(\rho c_p\right)}_f}{B_0}^2{{\stackrel{\sim }{u}}_1}^2+{\left(\frac{\partial {\stackrel{\sim }{u}}_1}{\partial y}\right)}^2\frac{\mu }{{\left(\rho c_p\right)}_f}\left(1-s\right)+\frac{\partial {\stackrel{\sim }{u}}_1}{\partial y}{\left(\frac{\partial {\stackrel{\sim }{u}}_1}{\partial y}\right)}^2\times \frac{s\mu \mathrm{\Gamma }}{{\left(\rho c_p\right)}_f\sqrt{2}},\end{array}(3)$

${\stackrel{\sim }{u}}_1\frac{\partial \stackrel{\sim }{C}}{\partial x}+{\stackrel{\sim }{u}}_2\frac{\partial \stackrel{\sim }{C}}{\partial y}=\frac{{\partial }^2\stackrel{\sim }{C}}{\partial y^2}{D}_B+\frac{{\partial }^2\stackrel{\sim }{T}}{\partial y^2}\frac{D_T}{{\stackrel{\sim }{T}}_{\infty }}-{k_r}^2{\left(\frac{\stackrel{\sim }{T}}{{\stackrel{\sim }{T}}_{\infty }}\right)}^n\mathit{exp}\left[\frac{-E_a}{\kappa \stackrel{\sim }{T}}\right]\left({\stackrel{\sim }{C}}_w-{\stackrel{\sim }{C}}_{\infty }\right),(4)$

${\stackrel{\sim }{u}}_1\frac{\partial \stackrel{\sim }{N}}{\partial x}+{\stackrel{\sim }{u}}_2\frac{\partial \stackrel{\sim }{N}}{\partial y}+\frac{b{W}_c}{\left(\stackrel{\sim }{C}-{\stackrel{\sim }{C}}_{\infty }\right)}\left[\frac{\partial }{\partial y}\left(\stackrel{\sim }{N}\frac{\partial \stackrel{\sim }{C}}{\partial y}\right)\right]=D_m\left(\frac{{\partial }^2\stackrel{\sim }{N}}{\partial y^2}\right).(5)$

Figure 1

Figure 1. Geometrical representation of the problem.

Following are the flow constraints used with the current flow model:

$\begin{array}{l}{\stackrel{\sim }{u}}_1=U_w=ax,{\stackrel{\sim }{u}}_2=0,\stackrel{\sim }{T}={\stackrel{\sim }{T}}_w,\stackrel{\sim }{C}={\stackrel{\sim }{C}}_w,\stackrel{\sim }{N}={\stackrel{\sim }{N}}_{w}aty=0,\\ \stackrel{\sim }{u}\to 0,T\to T_{\infty },C\to C_{\infty },N\to N_{\infty }asy\to \infty ,\end{array}(6)$

where $\left({\stackrel{\sim }{u}}_1,{\stackrel{\sim }{u}}_2\right)$ denote the velocity in the $\left(x,y\right)$ direction, respectively; $\left(s\right)$ being power law index, $\left(\nu \right)$ is the kinematic viscosity, $\left(\mathrm{\Gamma }\right)$ is notified as the Williamson parameter, $\left(\sigma \right)$ denotes the electrical conductivity, $\left({\rho }_f\right)$ represents the density of the fluid, $\left(C_p\right)$ is the specific heat, $\left(\stackrel{\sim }{T}\right)$ is the temperature of the fluid, $\left({\sigma }^{*}\right)$ represents the Stefan–Boltzmann constant, $\left(\mu \right)$ is the dynamic viscosity, $\left(k^{*}\right)$ is the coefficient of mean absorption, $\left(D_B\right)$ is the coefficient of Brownian diffusion, $\left(k\right)$ is the thermal conductivity, $\left(D_T\right)$ represents the coefficient of thermophoretic diffusion, $\left({k_r}^2\right)$ is the chemical reaction rate constant $\left(n\right)$ denotes the fitted rate constants, $\left(\stackrel{\sim }{C}\right)$ is the fluid concentration, $\left(E_a\right)$ is activation energy, $\kappa =8.61105eV/K$ is the Boltzmann constant, $\left({\left(\frac{\stackrel{\sim }{T}}{{\stackrel{\sim }{T}}_{\infty }}\right)}^n\mathit{exp}\left[\frac{E_a}{\kappa \stackrel{\sim }{T}}\right]\right)$ represents the modified Arrhenius parameter, $\left(\stackrel{\sim }{N}\right)$ is the microorganism concentration, $\left(D_m\right)$ shows the diffusion coefficient of microorganisms, $\left(b\right)$ denotes the chemotaxis constant, and $\left(W_c\right)$ is the maximum swimming speed of cell.

For present flow, suitable transformations Equation 7 are derived from [53]

$\xi =\sqrt{\frac{a}{\nu }},{\stackrel{\sim }{u}}_1=ax{f}^{\prime }\left(\xi \right),{\stackrel{\sim }{u}}_2=-\sqrt{a\nu }f\left(\xi \right),\theta =\frac{\stackrel{\sim }{T}-{\stackrel{\sim }{T}}_{\infty }}{{\stackrel{\sim }{T}}_w-{\stackrel{\sim }{T}}_{\infty }},\varphi =\frac{\stackrel{\sim }{C}-{\stackrel{\sim }{C}}_{\infty }}{{\stackrel{\sim }{C}}_w-{\stackrel{\sim }{C}}_{\infty }},\chi =\frac{\stackrel{\sim }{N}-{\stackrel{\sim }{N}}_{\infty }}{{\stackrel{\sim }{N}}_w-{\stackrel{\sim }{N}}_{\infty }}.(7)$

The governing equations with the aforementioned variables yield these Equations 8–12 in dimensionless forms:

$\left(1-s\right)f^{‴}+f^{″}{f}^{‴}s{W}_e-f^{\prime 2}+{ff}^{‴}-f^{\prime }M+\lambda \left(\theta -Nr\varphi -Rb\chi \right)=0,(8)$

$\begin{array}{l}{\theta }^{″}\frac{1}{P_r}+f{\theta }^{\prime }+{\left(1+\theta \left({\theta }_w-1\right)\right)}^2\left(\left(\theta \left({\theta }_w-1\right)+3{\theta }^{\prime 2}\left({\theta }_w-1\right)+1\right){\theta }^{″}\right)\frac{R}{P_r}\\ +\left(1-s\right)Ec{f}^{″2}\frac{s}{2}WeEc{\left(f^{″}\right)}^3+{\theta }^{\prime 2}Nt+{\theta }^{\prime }{\varphi }^{\prime }Nb+f^{\prime 2}MEc=0,\end{array}(9)$

${\varphi }^{″}+\frac{Nt}{Nb}{\theta }^{″}+{\varphi }^{\prime }Scf-\varphi \mathit{exp}\left[\frac{-E}{1+\delta \theta }\right]Sc{{\sigma }_1}^2{\left(1+\delta \theta \right)}^n=0,(10)$

${\chi }^{″}+{\chi }^{\prime }Lbf-Pe\left({\chi }^{\prime }{\varphi }^{\prime }+{\varphi }^{″}\left(\chi +{\delta }_1\right)\right)=0,(11)$

with

$\begin{array}{l}{f}^{\prime }\left(\infty \right)\to 0,f\left(0\right)=0,f^{\prime }\left(0\right)=1,\theta \left(\infty \right)\to 0,\theta \left(0\right)=1,\\ \varphi \left(\infty \right)\to 0,\varphi \left(0\right)=1,\chi \left(0\right)=1,\chi \left(\infty \right)\to 0,\end{array}(12)$

where the Hartmann number is denoted by $\left(M=\frac{\sigma {B_0}^2}{\rho a}\right)$, Prandtl number $\left(\mathit{Pr}=\frac{{\left(\rho c_p\right)}_f\nu }{k}\right)$, $\left(R=\frac{16{\sigma }^{*}{\stackrel{\sim }{T}}_{\infty }^3}{3k{k}^{*}}\right)$ shows the radiation parameter, $\left(W_e=\frac{a^{\frac{2}{3}}\sqrt{2}\mathrm{\Gamma }U}{\sqrt{\nu }}\right)$ represents the Weissenberg number, $\left(Ec=\frac{a^2{U}^2}{c_p\left(T_f-T_{\infty }\right)}\right)$ is the Eckert number, $\left(\lambda =\frac{Gr}{R{e}_x^2}\right)$ denotes the variable for mixed convection, temperature Grashof number, $Nr=\left(\frac{g{\beta }_c\left(C_w-C_{\infty }\right)}{g{\beta }_T\left(T_w-T_{\infty }\right)}\right)$ is the buoyancy force, $Rb=\left(\frac{{\gamma }_1\left({\rho }_m-{\rho }_f\right)\left(N_w-N_{\infty }\right)}{{\rho }_f\beta \left(1-C_{\infty }\right)\left(T_w-T_{\infty }\right)}\right)$ is the bioconvection Rayleigh number, the Schmidt number is represented by $\left(Sc=\frac{\nu }{D_B}\right)$, the chemical reaction rate constant is $\left({\sigma }_1=\frac{k_r^2}{a}\right)$, $\left(\delta =\frac{{\stackrel{\sim }{T}}_w-{\stackrel{\sim }{T}}_{\infty }}{{\stackrel{\sim }{T}}_{\infty }}\right)$ is the temperature relative parameter, $\left(\upsilon =\frac{E_a}{\kappa {\stackrel{\sim }{T}}_{\infty }}\right)$ represents the activation energy parameter, $\left(Nt=\frac{{\left(\rho c_p\right)}_s{D}_T\left({\stackrel{\sim }{T}}_w-{\stackrel{\sim }{T}}_{\infty }\right)}{{\stackrel{\sim }{T}}_{\infty }{\left(\rho c_p\right)}_f\nu }\right)$ is the parameter for the thermophoresis effect, $\left(Nb=\frac{{\left(\rho c_p\right)}_s{D}_T\left({\stackrel{\sim }{C}}_w-{\stackrel{\sim }{C}}_{\infty }\right)}{{\stackrel{\sim }{T}}_{\infty }\nu {\left(\rho c_p\right)}_f}\right)$ is the Brownian motion parameter, $\left(Lb\right)$ is the bioconvection Lewis parameter, $\left(Pe=\frac{b{W}_c}{D_m}\right)$ shows the Peclet number, $\left({\delta }_1=\frac{{\stackrel{\sim }{N}}_{\infty }}{{\stackrel{\sim }{N}}_w-{\stackrel{\sim }{N}}_{\infty }}\right)$ is the motile microorganism difference parameter, and $\left({\theta }_w=\frac{{\stackrel{\sim }{T}}_w}{{\stackrel{\sim }{T}}_{\infty }}\right)$ is the temperature ratio parameter.

The following are the mathematical expressions (Equations 13–16) for the motile density number, local Sherwood number, skin friction coefficient, and the local Nusselt number:

$C_f=\frac{2{\tau }_w}{\rho {U_w}^2},N{u}_x=\frac{x{q}_w}{k\left({\stackrel{\sim }{T}}_w-{\stackrel{\sim }{T}}_{\infty }\right)},S{h}_x=\frac{x{q}_m}{k\left({\stackrel{\sim }{C}}_w-{\stackrel{\sim }{C}}_{\infty }\right)},N{n}_x=\frac{x{q}_n}{k\left({\stackrel{\sim }{N}}_w-{\stackrel{\sim }{N}}_{\infty }\right)},(13)$

with

$\begin{array}{l}{\tau }_w\mu \left(1-s\right){\left(\frac{\mathit{\partial }{\stackrel{\sim }{u}}_1}{\partial y}\right)}_{y=0}+\mu \frac{s\mathrm{\Gamma }}{\sqrt{2}}{\left(\frac{\mathit{\partial }{\stackrel{\sim }{u}}_1}{\partial y}\right)}_{y=0}^3,q_w=-{\left(\frac{\mathit{\partial }\stackrel{\sim }{T}}{\mathit{\partial }y}\right)}_{y=0}k+{\left(q_r\right)}_w,\\ q_m=-{\left(\frac{\mathit{\partial }\stackrel{\sim }{C}}{\mathit{\partial }y}\right)}_{y=0}{D}_B,q_n=-{\left(\frac{\mathit{\partial }\stackrel{\sim }{N}}{\mathit{\partial }y}\right)}_{y=0}{D}_m,\end{array}(14)$

where

${\left(q_r\right)}_w=\frac{-16{\sigma }^{*}{\stackrel{\sim }{T}}^3}{3k^{*}}{\left(\frac{\partial \stackrel{\sim }{T}}{\partial y}\right)}_{y=0}.(15)$

The above quantities are expressed as follows in non-dimensional forms:

$\begin{array}{l}\frac{1}{2}R{e}^{\frac{1}{2}}{C}_f=f^{″}\left(0\right)\left(1-s\right)+\frac{sWe}{2}{\left(f^{″}\left(0\right)\right)}^2,N{u}_{x}R{e}^{-0.5}=-{\theta }^{\prime }\left(0\right)\left[1+R{\theta }_w^3\right],\\ S{h}_{x}R{e}^{-0.5}=-{\varphi }^{\prime }\left(0\right),N{n}_{x}R{e}^{-0.5}=-{\chi }^{\prime }\left(0\right),\end{array}(16)$

where $C_f,Nu,Sh,Nn$, and ${R_e}_x$ are the coefficients of skin friction, the local Sherwood, the local Nusselt, the Reynolds number $\left(R{e}_x=\frac{a^{2}x}{\nu }\right)$, and motile density, respectively.

3 Entropy generation

For tangent hyperbolic fluid, entropy generation is [58]

$\begin{array}{l}{S}_G=\underset{\text{Thermal\hspace{0.17em}irreversibility}}{\underbrace{\frac{k}{{\stackrel{\sim }{T}}_{\infty }^2}\left[1+\frac{16{\sigma }^{*}{\stackrel{\sim }{T}}^3}{3k{k}^{*}}\right]}}+\underset{\text{Fluid\hspace{0.17em}Friction\hspace{0.17em}irreversibility}}{\underbrace{\frac{\mu }{{\stackrel{\sim }{T}}_{\infty }}\mathrm{\Phi }}}+\underset{\text{Joule\hspace{0.17em}dissipation\hspace{0.17em}irreversibility}}{\underbrace{\frac{\sigma }{{\widehat{\stackrel{\sim }{T}}}_{\infty }}{B}_0^2\left(u_1^2\right)}}+\underset{\text{Concentration\hspace{0.17em}irreversibility}}{\underbrace{\frac{RD}{{\stackrel{\sim }{T}}_{\infty }}\left(\frac{\partial \stackrel{\sim }{T}}{\partial y}\frac{\partial \stackrel{\sim }{C}}{\partial y}\right)+\frac{RD}{{\stackrel{\sim }{C}}_{\infty }}{\left(\frac{\partial \stackrel{\sim }{C}}{\partial y}\right)}^2}}\\ \left.+\underset{\text{Microorganisms\hspace{0.17em}concentration\hspace{0.17em}irreversibility}}{\underbrace{\frac{RD}{{\stackrel{\sim }{N}}_{\infty }}{\left(\frac{\partial \stackrel{\sim }{N}}{\partial y}\right)}^2+\frac{RD}{{\stackrel{\sim }{T}}_{\infty }}\left(\frac{\partial \stackrel{\sim }{N}}{\partial y}\frac{\partial \stackrel{\sim }{T}}{\partial y}\right),}}\right\}\end{array}(17)$

where

$\mathrm{\Phi }={\left(\frac{\partial u_1}{\partial y}\right)}^2\left(1-s\right)+\frac{s\mathrm{\Gamma }}{\sqrt{2}}\left(\frac{\partial u_1}{\partial y}\right){\left(\frac{\partial u_1}{\partial y}\right)}^2.(18)$

Now Equation 17 becomes using Equation 18

$\begin{array}{l}{S}_G=\underset{\text{Thermal\hspace{0.17em}irreversibility}}{\underbrace{\frac{k}{{\stackrel{\sim }{T}}_{\infty }^2}\left[1+\frac{16{\sigma }^{*}{\stackrel{\sim }{T}}^3}{3k{k}^{*}}\right]}}+\underset{\text{Fluid\hspace{0.17em}Friction\hspace{0.17em}irreversibility}}{\underbrace{\frac{\mu }{{\stackrel{\sim }{T}}_{\infty }}\left[\left(1-s\right){\left(\frac{\partial u_1}{\partial y}\right)}^2+\frac{s\mathrm{\Gamma }}{\sqrt{2}}\left(\frac{\partial u_1}{\partial y}\right)\left(\frac{\partial u_1}{\partial y}\right)\right]}}+\underset{\text{Joule\hspace{0.17em}dissipation\hspace{0.17em}irreversibility}}{\underbrace{\frac{\sigma }{\widehat{{\stackrel{\sim }{T}}_{\infty }}}{B}_0^2\left(u_1^2\right)}}\\ +\underset{\text{Nanoparticle\hspace{0.17em}concentration\hspace{0.17em}irreversibility}}{\underbrace{\frac{RD}{{\stackrel{\sim }{C}}_{\infty }}{\left(\frac{\partial \stackrel{\sim }{C}}{\partial y}\right)}^2+\left(\frac{\partial \stackrel{\sim }{C}}{\partial y}\frac{\partial \stackrel{\sim }{T}}{\partial y}\right)\frac{RD}{{\stackrel{\sim }{T}}_{\infty }}}}\left.+\underset{\text{Microorganisms\hspace{0.17em}concentration\hspace{0.17em}irreversibility}}{\underbrace{{\left(\frac{\partial \stackrel{\sim }{N}}{\partial y}\right)}^2\frac{RD}{{\stackrel{\sim }{N}}_{\infty }}+\frac{RD}{{\stackrel{\sim }{T}}_{\infty }}\left(\frac{\partial \stackrel{\sim }{T}}{\partial y}\frac{\partial \stackrel{\sim }{N}}{\partial y}\right)}.}\right\}\end{array}(19)$

Equation 19 contains five factors: i) viscosity dissipation in the entropy generation for tangent hyperbolic fluid, ii) thermal irreversibility, iii) nanoparticle concentration irreversibility, iv) Joule effect irreversibility, and v) microorganism concentration irreversibility. The dimensionless form are in Equation 20

$\begin{array}{l}{N}_g=\left[R{\left(\theta \left({\theta }_w-1\right)+1\right)}^3+1\right]{\alpha }_1{{\theta }^{\prime }}^2+\left(1-s\right){f^{″}}^{2}Br+{\left(f^{″}\right)}^{3}Br\frac{We}{2}+BMr{f^{\prime }}^2+L\frac{{\alpha }_2}{{\alpha }_1}{{\varphi }^{\prime }}^2+\\ \left.L{\theta }^{\prime }{\varphi }^{\prime }+L^{*}\frac{{\alpha }_3}{{\alpha }_1}{{\chi }^{\prime }}^2+L^{*}{\theta }^{\prime }{\chi }^{\prime },\right\}\end{array}(20)$

$\begin{array}{l}{\alpha }_1=\frac{\Delta \stackrel{\sim }{T}}{{\stackrel{\sim }{T}}_{\infty }}=\frac{{\stackrel{\sim }{T}}_w-{\stackrel{\sim }{T}}_{\infty }}{{\stackrel{\sim }{T}}_{\infty }},{\alpha }_2=\frac{\Delta \stackrel{\sim }{C}}{{\stackrel{\sim }{C}}_{\infty }}=\frac{{\stackrel{\sim }{C}}_w-{\stackrel{\sim }{C}}_{\infty }}{{\stackrel{\sim }{C}}_{\infty }},{\alpha }_3=\frac{{\stackrel{\sim }{N}}_w-{\stackrel{\sim }{N}}_{\infty }}{{\stackrel{\sim }{N}}_{\infty }}=\frac{\Delta \stackrel{\sim }{N}}{{\stackrel{\sim }{N}}_{\infty }},Br=\frac{\mu a^2}{k\Delta \stackrel{\sim }{T}},\\ N_g=\frac{{\stackrel{\sim }{T}}_{\infty }{S}_G\nu }{ak\Delta \stackrel{\sim }{T}},L=\frac{RD\left({\stackrel{\sim }{C}}_w-{\stackrel{\sim }{C}}_{\infty }\right)}{k},\left.L^{*}=\frac{RD\left({\stackrel{\sim }{N}}_w-{\stackrel{\sim }{N}}_{\infty }\right)}{k},\right\}\end{array}(21)$

where Equation 21 represents ${\alpha }_1,{\alpha }_2,{\alpha }_3,Br,L,L^{*},\text{and\hspace{0.17em}}{N}_g$ differences in the temperature variable, Brinkman number, concentration difference parameter, diffusion parameter due to nanoparticles and microorganisms’ concentration, and local entropy generation. The dimensionless Bejan number is expressed in Equation 22 and mathematical form is in Equation 23

$Be=\frac{\text{Entropy}\text{generation}\text{due}\text{to}\text{mass}\text{and}\text{heat}\text{transfer}}{\text{Total}\text{entropy}\text{generation}},(22)$

$\left.Be=\frac{\left[1+R{\left(\theta \left({\theta }_w-1\right)+1\right)}^3\right]{{\theta }^{\prime }}^2{\alpha }_{1}L{\theta }^{\prime }{\varphi }^{\prime }+L\frac{{\alpha }_2}{{\alpha }_1}{{\varphi }^{\prime }}^2+L{\theta }^{\prime }{\varphi }^{\prime }+L^{*}\frac{{\alpha }_3}{{\alpha }_1}{{\chi }^{\prime }}^2+L^{*}{\theta }^{\prime }{\chi }^{\prime }}{\begin{array}{l}\left(1-s\right)Br{f^{″}}^2+\left[1+R{\left(\theta \left({\theta }_w-1\right)+1\right)}^3\right]{{\theta }^{\prime }}^2{\alpha }_1+\frac{We}{2}{\left(f^{″}\right)}^{3}Br+BMr{f^{\prime }}^2+\frac{{\alpha }_2}{{\alpha }_1}L{{\varphi }^{\prime }}^2+{\theta }^{\prime }{\varphi }^{\prime }L+L^{*}\frac{{\alpha }_3}{{\alpha }_1}{{\chi }^{\prime }}^2+L^{*}{\theta }^{\prime }{\chi }^{\prime }\end{array}}\right\}.(23)$

4 Numerical algorithm

It is a well-known observation that a number of physical models contain differential equations which are nonlinear depending on the included constraints that can be solved using numerical approaches. In the suggested numerical approaches, the shooting method is perhaps a more reliable strategy for achieving adequate accuracy. After generating dimensionless forms, the bvp4c built-in package is used to tackle numerically. This approach is reliable and efficient, having an error tolerance of 10⁻⁶. As a result, we used this strategy to target Equations 8–12. The collection of these mathematical equations is turned into first-order differential equations using dummy variables as in Equations 24–28:

$\begin{array}{l}f=z_1,f^{\prime }=z_2,f^{″}=z_3,f^{″}=z_3^{\prime },\theta =z_4,{\theta }^{\prime }=z_5,{\theta }^{″}=z_5^{\prime },\\ \varphi =z_6,{\varphi }^{\prime }=z_7,{\varphi }^{″}=z_7^{\prime },\chi =z_8,{\chi }^{\prime }=z_9,{\chi }^{″}=z_9^{\prime }.\end{array}$

Then,

$z_3^{\prime }=\frac{\left[z_2^2-z_1{z}_3+M{z}_2-\lambda z_4\left(1+{\beta }_t{z}_4\right)-\lambda z_6{N}^{*}\left(1+{\beta }_c{z}_6\right)-\lambda z_8{N}^{**}\left(1+{\beta }_n{z}_8\right)\right]}{\left(1-s+sWe{z}_3\right)},(24)$

$z_5^{\prime }=\frac{\left[\begin{array}{l}-3R{\left(z_4\left({\theta }_w-1\right)+1\right)}^2\left(z_5^2\left({\theta }_w-1\right)\right)-\mathit{Pr}{z}_1{z}_5-\left(1-s\right)E{c}^2{z}_3^5\frac{s}{2}WePr\\ -NtPr{z}_5^2-NbPr{z}_5{z}_7-M{z}_2^{2}EcPr\end{array}\right]}{\left[1+R{\left(z_4\left({\theta }_w-1\right)+1\right)}^2\left(z_4\left({\theta }_w-1\right)+1\right)\right]},(25)$

$z_7^{\prime }=-\frac{Nt}{Nb}{z}_5^{\prime }-Sc{z}_1{z}_7+Sc{\sigma }_1^2{\left(1+\delta z_4\right)}^n\mathit{exp}\left[\frac{-\upsilon }{1+\delta z_4}\right]z_6,(26)$

$z_9^{\prime }=-Lb{z}_1{z}_9+Pe\left[z_7^{\prime }\left(z_8+{\delta }_1\right)+z_7{z}_9\right].(27)$

The transformed boundary conditions are

$\begin{array}{l}{z}_1\left(0\right)=0,z_2\left(\infty \right)\to 0,-1+z_2\left(0\right)=0,-1+z_4\left(0\right)=0,z_4\left(\infty \right)\to 0,\\ z_6\left(0\right)-1=0,z_6\left(\infty \right)\to 0,z_8\left(0\right)-1=0,z_8\left(\infty \right)\to 0.\end{array}(28)$

5 Discussion

This part describes the visual representation of physical characteristics such as velocity distribution $f^{\prime }$, temperature $\theta$, nanoparticle concentration $\varphi$, and motile gyrotactic $\chi$. In the process of modifying the related physical parameter, the quantities that are left have been allotted constant values such as $M=0.5,s=\lambda ={\beta }_c=We={\sigma }_1=\delta =Ec=0.1,Sc=Nb=Nt=N^{*}=0.01,\mathrm{Pr}=2$, and ${\theta }_w=1.1$. Figures 2–4 examine the behavior of a variety of physical factors, including the Hartmann number $M$, Weissenberg number $We$, first-order velocity slip $\alpha$, buoyancy ratio parameter $Nr$, second-order slip parameter $\beta$, and mixed convection variable $\lambda$. The effects of the Hartmann number $M$ and $We$ Weissenberg number on the velocity graph $f^{\prime }$ are depicted in Figure 2. A declining behavior is prominent for velocity distribution by increasing both $M$ and $We$. For higher values of (M = 0.1, 0.4, 8, 1.2), Figure 2 shows a decreasing trend of $f^{\prime }\left(\xi \right)$. Because $M$ depends on the Lorentz force, the inter-particle resistance increases as $M$ increases, which reduces $f^{\prime }\left(\xi \right)$. We observe that as $We$ increases, fluid motion gradually decreases. Given that the fraction of relaxation time by a particular processing time is denoted by $We$, larger $We$ will give rise to a longer relaxation period, which will lessen the velocity profile. Figure 3 discusses the first-order slip parameter $\alpha$. In addition, the mixed convection parameter $\lambda$ for the profile of velocity $f^{\prime }\left(\xi \right)$. First, it should be noted that the first order is the feature of the medium that is linked to the pathways between fluid flows. The values for the velocity declines as we change first-order slip parameters. Moreover, it is noticed that the thickness of the boundary, which is entitled to first-order slip factor, is decreasing. The velocity profile increases with the increase in the mixed convection parameter $\lambda .$ Physically, the buoyant force overcomes the force, owing to the increase in inertia as the parameter for mixed convection, and as a result, a leading increase is noticed for $f^{\prime }\left(\xi \right)$.

Figure 2

Figure 2. Illustration of $M$ and $We$ on $f^{\prime }$.

Figure 3

Figure 3. Illustration of $\alpha$ and $\lambda$ on $f^{\prime }$.

The parameter for the buoyancy ratio $Nr$ and the parameter for the second-order slip $\beta$ for the dimensionless distribution of velocity $f^{\prime }\left(\xi \right)$ are graphed in Figure 4. The profile for velocity decreases as both of these factors increase.

Figure 4

Figure 4. Illustration of $Nr$ and $\beta$ on $f^{\prime }$.

The impacts of the Prandtl number $\mathit{Pr}$, Biot number $Bi$, heat sink/source $Q$, parameter for thermophoresis $Nt$, temperature ratio parameter ${\theta }_w$, and Eckert number $Ec$ on $\theta$ temperature field are depicted in Figures 5–7. Figure 5 shows the physical impact caused by $\mathit{Pr}$ and $Bi$ on temperature field $\theta$. It is easy to observe that $\theta$ is the decreasing function of ($\mathit{Pr}$ = 2.0, 3.0, 4.0, 5.0). Furthermore, the thermal Biot number $Bi$ has a connection to the coefficient of heat transfer that causes the increase in the temperature distribution of nanoparticles. The Eckert number $Ec$ and the characteristics of parameters for the temperature ratio ${\theta }_w$ and on the profile for temperature $\theta$ are demonstrated in Figure 6. In fluid temperature $\theta$, an increasing trend is prominent for the increase in the values of parameters for the temperature ratio (${\theta }_w$ = 1.5, 1.6, 1.7, 1.8). For increasing values of $Ec$, it displays that the fluid temperature shows an increase. The reason for it being higher $Ec$ fluid friction results in an interconversion of mechanical energy and thermal energy, which peaks the temperature $\theta$ of the fluid.

Figure 5

Figure 5. Illustration of $\mathit{Pr}$ and $Bi$ on $\theta$.

Figure 6

Figure 6. Illustration of ${\theta }_w$ and $Ec$ on $\theta$.

Figure 7

Figure 7. Illustration of $Q$ and $Nt$ on $\theta$.

The effects of the $\left(Q<0\right)$ heat sink or $\left(Q>0\right)$ heat source and thermophoresis parameter $Nt$ are shown in Figure 7. In the boundary layer, when a heat source is present, it increases energy, which results in the increase in fluid’s temperature. The existence of a heat sink in the boundary layer absorbs energy, which, in turn, lowers the temperature of the fluid. These behaviors are depicted in Figure 7. For larger $Nt$, particles of fluid increase from hot to cold areas of the system. The temperature profile increases as a result of an increase in thermophoresis force.

To examine the changes in concentration distribution $\varphi$ versus different values of the $E$ activation energy, $\mathit{Pr}$ Prandtl number, $Le$ Lewis number, and $Nb$ Brownian movement, Figures 8, 9 are formulated. Figure 8 presents the graph of the Prandtl number $\mathit{Pr}$ and Lewis number $Le$ on $\varphi$. Increases in $\mathit{Pr}$ and $Le$ result in decreases in the profile of concentration of nanoparticles. The ramifications of yet two other crucial parameters, the parameters for Brownian motion $Nb$ and activation energy $E$, are shown in Figure 9. Although the concentration of nanoparticles decreases with $Nb$, but for $E$, the profile for the concentration improves.

Figure 8

Figure 8. Illustration of $\mathit{Pr}$ and $Le$ on $\varphi$.

Figure 9

Figure 9. Illustration of $PeE$ and $Nb$ on $\varphi$.

The Peclet number $Pe$ and Lewis number $Lb$ for bioconvection effects on the self-propelled microorganism profile $\chi$ are shown in Figure 10. By altering Pe, a retarding behavior is discovered; this tendency establishes because $Pe$ develops a reverse relationship with microorganism diffusivity. $\chi$ reduced as a result of this inverse relationship. A distinguishing motile microorganism profile is found when the value of $Lb$ increases. The physical reasoning is that a bigger change in $Lb$ is connected with the reduced diffusion of self-propelled microorganisms, and $\chi$ decays as a result of this. Figures 11, 12 examine the behavior of a variety of physical factors, including the Hartmann number $M$, power law index $s$, and $Nb$ parameters for Brownian motion on entropy generation $N_g$. Salient features of the Hartmann number $M$ and Weissenberg number $We$ for entropy generation $N_g$ are displayed in Figure 11. The increase in resistance for increasing $M$ entropy generation shows an increasing trend. Relaxation time increases for higher $We$, which indicates increased heat loss due to increased resistance between liquid particles. Consequently, entropy generation increases. Figure 12 is plotted to examine the entropy generation for greater values of $s$, which is the power law index and Brownian motion parameter $Nb$. For the increase in $s$ and $Nb$, entropy generation $N_g$ shows decreasing behavior.

Figure 10

Figure 10. Illustration of $Pe$ and $Lb$ on $\mathit{\chi }$.

Figure 11

Figure 11. Illustration of $M$ and $We$ on $N_g$.

Figure 12

Figure 12. Illustration of $S$ and $Nb$ on $N_g$.

6 Streamlines

The streamlines of the flow problem visualize the flow pattern and movement of the molecules, as displayed in Figures 13A–D. The laminar pattern is visible among all profiles, and no intersection is found. These streamlines describe the movement, velocity distribution, heat transport efficiency, and interaction with the surface body.

Figure 13

Figure 13. (A) Streamlines of the momentum profile. (B) Streamlines of the thermal profile. (C) Streamlines of the concentration profile. (D) Streamlines of the microorganism profile.

7 Tables

In the tabulated values in Table 1, we see the comparison of our computed results with the already in the tabulated values we see the comparison of our computed results with the already results shows the great accord with available literature.

Table 1

Table 1. Comparison of $f^{″}\left(0\right)$ with available results.

8 Conclusion

The aim of this contribution is to study the flow of tangent hyperbolic nanomaterials for entropy optimization, owing to a stretchable surface. Some key findings are as follows:

• For the first-order velocity slip factor, the velocity distribution increases, but the Hartmann number, Weissenberg number, buoyancy ratio constant, and second-order slip parameter show a declining trend.

• The thermal Biot number, parameter for the temperature ratio, Eckert number, and parameter for thermophoresis enhanced the temperature of nanoparticles, while a decreasing trend is noticed for the Prandtl number.

• Activation energy improved the concentration of nanoparticles, while it reduces the concentration for the Prandtl number, Lewis number, and Brownian parameter.

• The distribution for self-propelling gyrotactic microorganisms declines for larger Lewis and Prandtl numbers.

• Entropy generation reduces for a larger power law index and parameters for Brownian motion; however, a negative relationship is observed for larger Weissenberg and Hartmann numbers.

In the future, this approach may be extended to include the various heat and mass flux models, as well as the observation of changes in transport rates due to geometrical changes.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.

Author contributions

YW: writing–review and editing and methodology. MC: conceptualization and writing–original draft. NM: software and writing–original draft. MT: data curation and writing–original draft. MB: validation and writing–review and editing. MI: supervision and writing–original draft.

Funding

The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. This research project is supported by the 2022 university-level scientific research team project, 2018 university-level quality engineering project, and 2023 Project of Natural Science Foundation of Colleges and Universities in Anhui Province (kytd202207; 2018jpkcx17; and 2023AH030085).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Nomenclature

Symbols of parameters

Description with SI units

$x,y$ Cartesian coordinates $\left(-\right)$

$D_m$ microorganism’s diffusion coefficient $\left(-\right)$

${\stackrel{\sim }{u}}_1,{\stackrel{\sim }{u}}_2$ velocity components $\left(m/s\right)$

$\mathrm{Pr}$ Prandtl number $\left(-\right)$

$k$ nanofluid thermal conductivity $\left(W/mK\right)$

$E_a$ activation energy variable $\left(-\right)$

$C_f$ skin friction parameter $\left(-\right)$

$C_{\infty }$ free stream concentration $\left(mol/m^3\right)$

$k^{*}$ coefficient of mean absorption (--)

$T_{\infty }$ ambient temperature $\left(T\right)$

$D_B$ Brownian motion coefficient $\left(m^2/s\right)$

$\stackrel{\sim }{N}$ microorganism’s density $\left(-\right)$

$B_0$ magnetic parameter $\left(Tesla\right)$

$S{h}_x$ local Sherwood number $\left(-\right)$

$\stackrel{\sim }{C}$ concentration of particles $\left(mol/m^3\right)$

$Nb$ Brownian motion parameter $\left(-\right)$

$Sc$ Schmidt number $\left(-\right)$

$b$ chemotaxis constant $\left(-\right)$

$\stackrel{\sim }{T}$ temperature of particles $\left(K\right)$

$M$ Hartmann number $\left(-\right)$

$N{u}_x$ local Nusselt number $\left(-\right)$

$Pe$ Peclet number $\left(-\right)$

$D_T$ thermophoresis coefficient $\left(m^2/s\right)$

${\stackrel{\sim }{C}}_w$ surface concentration $\left(mol/m^3\right)$

${\stackrel{\sim }{T}}_w$ surface temperature $\left(T\right)$

$N_{\infty }$ ambient microorganisms $\left(-\right)$

$Nt$ thermophoresis parameter $\left(-\right)$

$W_e$ Weissenberg parameter $\left(-\right)$

$R$ radiation parameter $\left(-\right)$

$Lb$ bioconvection Lewis number $\left(-\right)$

Greek symbols

${\rho }_f$ density of the fluid $\left(kg/m^3\right)$

$\mu$ dynamic viscosity $\left(kg{m}^{-1}{s}^{-1}\right)$

${\sigma }^{*}$ Stefan–Boltzmann constant $\left(-\right)$

$\alpha$ first-order slip parameter $\left(-\right)$

${\delta }_1$ motile microorganism difference parameter $\left(-\right)$

$\beta$ second-order slip parameter $\left(-\right)$

$\lambda$ mixed convection parameter $\left(-\right)$

$\nu$ kinematic viscosity $\left(m^2/s\right)$

${\theta }_w$ temperature ratio parameter $\left(-\right)$

$\sigma$ electrical conductivity $\left(S/m\right)$