On Generalized Fourier’s and Fick’s Laws in Bio-Convection Flow of Magnetized Burgers’ Nanofluid Utilizing Motile Microorganisms

Abstract

This article describes the features of bio-convection and motile microorganisms in magnetized Burgers’ nanoliquid flows by stretchable sheet. Theory of Cattaneo–Christov mass and heat diffusions is also discussed. The Buongiorno phenomenon for nanoliquid motion in a Burgers’ fluid is employed in view of the Cattaneo–Christov relation. The control structure of governing partial differential equations (PDEs) is changed into appropriate ordinary differential equations (ODEs) by suitable transformations. To get numerical results of nonlinear systems, the bvp4c solver provided in the commercial software MATLAB is employed. Numerical and graphical data for velocity, temperature, nanoparticles concentration and microorganism profiles are obtained by considering various estimations of prominent physical parameters. Our computations depict that the temperature field has direct relation with the thermal Biot number and Burgers’ fluid parameter. Here, temperature field is enhanced for growing estimations of thermal Biot number and Burgers’ fluid parameter.

1. Introduction

Nowadays investigators are showing huge interest in nanofluid heat transfer applications. Based on the empirical results, working fluids in various technical and medical fields have been found to have the robust features of heat transfer, mass and density during the flow. While different base liquids have similar heat transfer capability, due to poor thermal efficiency, such liquids are typically not favored for heat transport applications. In order to overcome this problem, heat efficiency of such conventional techniques can be boosted by the usage of some nanoparticles additives. Suspension of regular fluids with nano-size materials is deemed to become most efficient solution to enhance heat transport that has been used in a wide range of products in different air-conditioning systems. With this concept, convectional fluids with lower heat efficiency have been substituted by nanoliquids that are capable of transmitting further heat to devices. Compared to ordinary fluids, nanoliquids are generally extra stable in terms of heat and momentum transfer. Such nanomaterials can be enhanced by utilizing different metals such as gold, silver, copper, steel, borides, oxides and nitrides with special caution.

Choi [1] introduced the term “nanofluid”. Buongiorno [2] analyzed the concept of natural convective transport in a nanofluid. Many researchers have studied the nature of the nanofluid flow via Brownian motion and thermophoretic factors. Turkyilmazoglu [3] researched the influence of single-phase nanoliquids and its linear stabilization. Ellahi et al. [4] studied nano-sized hafnium particles. The results of second-order slip over a Poiseuille nanoliquid flow subject to the impact of Stefan blows in channels were identified by Alamri et al. [5]. Khan et al. [6] analyzed the conservation-based performance evaluation of hybrid nanoliquid-assisted composite materials. Irfan et al. [7] researched the movement of EMHD nanoliquids over fixed thickness sheets. Reddy et al. [8] analyzed the movement of chemically reacting stagnation point Powell-Eyring nanoliquids flowing through an inclined cylinder subject to energy activation and Cattaneo–Christov heat transfer. Khan et al. [9] observed the behavior of nonlinear heat radiation by using nanomaterials and activation energy. Agamid et al. [10] discussed the unsteady flow of carbon nanotubes based-fluids between two moving disks. Uddin et al. [11] addressed the actions of nanomaterials to disperse blood across a cylindrical tube utilizing a single kernel. Abbas et al. [12] investigated the non-uniform hemodynamic nanoliquid motion. Babazadeh et al. [13] discussed the process of modifying Lorentz’s force over nanoparticles flowing inside two disks. Alamri et al. [14] examined the impact of mass transport on the second-grade liquid movement. Niazmand et al. [15] discussed the impact of nanoparticles in a lamp-driven cylindrical cavity. Steady 2D convective viscous nanoliquid flow by a stretchable cylinder with a chemical process was explored by Mondal et al. [16]. Saif et al. [17] addressed hydromagnetic Jeffrey nanoliquid flow through a curved stretched surface. Nanoliquid turbulence and flow in an annular space was studied mathematically and experimentally by Abdulrazzaq et al. [18]. Saeed et al. [19] analyzed the transport of heat through a porous stretched cylinder and Darcy-Forchheimer hybrid-nanoliquid movement. Souayeh et al. [20] examined radiative heat transition and slippery flow through nanofluid. Tlili et al. [21] studied boundary-layer MHD flow subject to convection features across a wedge. Farhangmehr et al. [22] numerically investigated the MHD flow of a nanofluid over a moving surface. Many researchers have worked on nanofluids as may be seen in various publications [23,24,25,26,27,28,29,30].

The occurrence of bioconvection is correlated with microorganisms and biological solutions. Such a practice is generally noted in dilute liquids in which microorganisms travel upward in a suspended liquid resulting in instability due to concentration stratification. The bioconvection mechanism is understood by Rayleigh-Bernard convection which is also visible in the reversal of the nanofluid instability caused by the haphazard motion of nanomaterials. The key difference between molecules and a biological suspension is that nanomaterials are not self-propelled and they follow Brownian motion owing to hydrodynamic instability. Introduction of swimming gyrotactic motile microorganisms into a nanoliquid is known to cause a significant change in the fluid convection behavior. Kuznetsov and Avramenko [31] have introduced the bioconvection of nanomaterials through a horizontal convection sheet in the occurrence of swimming microorganisms. Ghorai and Hill [32] additionally clarified that bioconvection can be utilized to illustrate the process of impulsive model improvement in fluids containing microorganisms like bacteria and algae. Atif et al. [33] scrutinized the homogenous micropolar bioconvective fluid flow with nanoparticles and gyrotactic microorganisms. Khan et al. [34] inspected 2D bioconvection coupled stress fluid flow with nanoparticles, magnetic field and gyrotactic motile microorganisms. Amirsom et al. [35] presented theoretical research on the 3D movement of bioconvection nanoliquids comprising gyrotactic motile microorganisms throughout a bi-axial stretched sheet. Zhang et al. [36] determined the features of activation energy in radiative rate-type nanofluid with bioconvection aspect configured by stretching/shrinking disks. Usman et al. [37] studied the model of three-dimensional nanoliquid bioconvection with a stagnation connection. Basir et al. [38] proposed nanoliquid laminar convective MHD flow through heat, mass and gyrotactic microorganism transfers. The bioconvection motion of a Carreau nanofluid over a wedge was addressed by Muhammad et al. [39]. Mansour et al. [40] presented a mathematical analysis of magneto-hydrodynamic convective flow with gyrotactic motile microorganisms in a squared lid-driven space. Li et al. [41] investigated the bioconvection flow of generalized second grade nanofluids under the effect of Wu’s slip. Waqas et al. [42] scrutinized the non-thermal radiation behavior in Oldroyd-B nanofluid flow with swimming motile microorganisms past a rotating disk. More works on bioconvection can be found in [43,44,45,46,47,48,49,50].

The objective of the present research article is to explore the bio-convection flow of a magnetized Burgers’ nanofluid subject to the presence of swimming motile microorganisms. The Buongiorno expressions for nanoliquid motion in the Burgers’ fluid are employed in view of generalized Fourier’s and Fick’s laws. Partial differential equations (PDEs) are changed into appropriate ordinary differential equations (ODEs) by suitable transformations [51,52,53,54]. To perform numerical computations of nonlinear expressions, the bvp4c solver in the commercial software MATLAB is employed. Numerical and graphical results for velocity, nanofluid temperature, nanofluid concentration, microorganisms, skin friction, Nusselt, Sherwood and motile density numbers are obtained by considering various estimations of physical numbers.

2. Mathematical Formulation

A mathematical relation is designed for 2D Burgers nanofluid flow with Cattaneo–Christov models, motile microorganisms and bioconvection past a stretching sheet. Impacts of Brownian movement and thermophoresis diffusion are also taken into account. Furthermore, stagnation point flow and thermal and solutal stratifications are scrutinized. Wall temperature of nanoparticles (T_w), wall concentration of nanoparticles (C_w) and wall motile microorganisms (N_w) are considered. Moreover, (T_∞) is the ambient temperature, (C_∞) is the ambient concentration and (N_∞) is the ambient microorganism. The velocity components are associated in the way of x-axis and y-axis which are depicted in Figure 1.

The rate-type liquid expressed via a Burgers’ liquid is:

$$\left(1+{\lambda }_1\frac{D}{Dt}+{\lambda }_2\frac{D^2}{D{t}^2}\right)S=\mu \left(1+{\lambda }_3\frac{D}{Dt}\right)A_1$$

(1)

Here (S) stands for extra stress tensor, $\left(\mu \right)$ for the dynamic viscosity of the fluid, $\left(A_1=\left(\nabla .V\right)+{\left(\nabla .V\right)}^T\right)$ for the first Rivilin-Ericksen tensor and $\left(\frac{D}{Dt}\right)$ for the upper convective derivative. The continuity and momentum expressions for present flow are:

$$\nabla .V=0,$$

(2)

$${\rho }_f\frac{dV}{d}t=-\nabla p+divS+J_1\times B,$$

(3)

where the energy law conservation for the present model is:

$$\left(\rho C_p\right)\frac{dT}{dt}-\left(\rho C_p\right)\left[D_B\nabla C.\nabla T+\frac{D_T}{D_{\infty }}{\left(\nabla T\right)}^2\right]=-\nabla .q+Q_0\left(T-T_{\infty }\right),$$

(4)

in which $\left(q\right)$ stands for heat flux satisfy the following property:

$$q+{\lambda }_1\left[\frac{\partial q}{\partial t}+V.\nabla q-q.\nabla V+\left(\nabla .V\right)q\right]=-k\nabla T,$$

(5)

where the nanoparticles concentration for the present flow model is:

$$\frac{dC}{dt}-\frac{D_T}{D_{\infty }}{\nabla }^{2}T=-\nabla .J-k_c\left(C-C_{\infty }\right),$$

(6)

in which $\left(J\right)$ stands for the mass flux satisfying the following property:

$$J+{\lambda }_c\left[\frac{\partial J}{\partial t}+V.\nabla J-J.\nabla V+\left(\nabla .V\right)J\right]=D_B\nabla C.$$

(7)

Here $\left(\nu \right)$ stands for kinematic viscosity, $\left({\lambda }_1\right)$ for the fluid relaxation time, $\left({\lambda }_2\right)$ for material parameter of the Burgers’ fluid, $\left({\lambda }_3\le {\lambda }_1\right)$ for the fluid retardation time, $\left({\lambda }_c\right)$ for the mass relaxation time, $\left({\lambda }_t\right)$ for the thermal relaxation time, $\left(T\right)$ for the liquid temperature, $\left(C\right)$ for the liquid concentration, $\left(T_{\infty }\right)$ for the ambient temperature, $\left(C_{\infty }\right)$ for the ambient concentration, $\left(D_B\right)$ for the diffusion coefficient, $\left(J_1\right)$ for the current density, $\left(q\right)$ for the heat flux and $\left(J\right)$ for the mass flux. The governing expressions for the current flow model are:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,$$

(8)

$$\begin{array}{l}u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial y}+{\lambda }_1\left[u^2\frac{{\partial }^{2}u}{\partial x^2}+v^2\frac{{\partial }^{2}v}{\partial y^2}+2uvv\frac{{\partial }^{2}u}{\partial x\partial y}\right]\\ +{\lambda }_2\left[\begin{array}{l}{u}^3\frac{{\partial }^{3}u}{\partial x^3}+v^3\frac{{\partial }^{3}v}{\partial y^3}+u^2\left(\frac{\partial v}{\partial x}\frac{{\partial }^{2}u}{\partial x\partial y}-\frac{\partial u}{\partial x}\frac{{\partial }^{2}v}{\partial x^2}+\frac{\partial u}{\partial x}\frac{{\partial }^{2}u}{\partial x^2}\right)+3v^2\left(\frac{\partial v}{\partial y}\frac{{\partial }^{2}u}{\partial y^2}+\frac{\partial u}{\partial y}\frac{{\partial }^{2}u}{\partial x\partial y}\right)\\ +3uv\left(u\frac{{\partial }^{3}u}{\partial x^2\partial y}+v\frac{{\partial }^{3}u}{\partial x\partial y^2}\right)+2uv\left(\frac{\partial v}{\partial y}\frac{{\partial }^{2}u}{\partial x\partial y}+\frac{\partial v}{\partial x}\frac{{\partial }^{2}u}{\partial y^2}+\frac{\partial u}{\partial y}\frac{{\partial }^{2}u}{\partial x^2}-\frac{\partial u}{\partial y}\frac{{\partial }^{2}v}{\partial x\partial y}\right)\end{array}\right]\\ =v{\lambda }_3\left[v\frac{{\partial }^{3}v}{\partial y^3}+u\frac{{\partial }^{3}u}{\partial x\partial y^2}-\frac{\partial u}{\partial x}\frac{{\partial }^{2}u}{\partial y^2}-\frac{\partial u}{\partial y}\frac{{\partial }^{2}v}{\partial y^2}\right]+v\left[\frac{{\partial }^{2}u}{\partial y^2}\right]+\frac{d{U}_e}{dx}\\ -\frac{\sigma B_0^2}{\rho }\left[u-U_e+{\lambda }_{1}v\frac{\partial u}{\partial y}+{\lambda }_2\left(u\frac{\partial v}{\partial x}\frac{{\partial }^{2}u}{\partial y^2}-v\frac{\partial u}{\partial x}\frac{{\partial }^{2}u}{\partial y^2}+uv\frac{{\partial }^{2}u}{\partial x\partial y}+v^2\frac{{\partial }^{2}u}{\partial y^2}\right)\right]\\ +\frac{1}{{\rho }_f}\left[\begin{array}{l}\left(1-C_f\right){\rho }_f{\beta }^{\ast \ast }{g}^{\ast }\left(T-T_{\infty }\right)-\left({\rho }_p-{\rho }_f\right)g^{\ast }\left(C-C_{\infty }\right)\\ -\left(N-N_{\infty }\right)g^{\ast }\gamma \left({\rho }_m-{\rho }_f\right)\end{array}\right],\end{array}$$

(9)

$$\begin{array}{l}u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \left(\frac{{\partial }^{2}T}{\partial y^2}\right)-{\lambda }_1\left[\begin{array}{l}{u}^2\frac{{\partial }^{2}T}{\partial x^2}+v^2\frac{{\partial }^{2}T}{\partial y^2}+2uvv\frac{{\partial }^{2}T}{\partial x\partial y}+\frac{\partial T}{\partial x}\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)\\ +\frac{\partial T}{\partial y}\left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right)\end{array}\right]\\ +{\lambda }^{\ast }{}_t\tau D_B\left[u\frac{{\partial }^{2}C}{\partial x\partial y}\frac{\partial T}{\partial y}+u\frac{\partial C}{\partial y}\frac{{\partial }^{2}T}{\partial x\partial y}+v\frac{{\partial }^{2}C}{\partial y^2}\frac{\partial T}{\partial y}+v\frac{\partial C}{\partial y}\frac{{\partial }^{2}T}{\partial y^2}\right]+2{\lambda }^{\ast }{}_t\tau \frac{D_T}{D_{\infty }}\left[\begin{array}{l}u\frac{\partial T}{\partial y}\frac{{\partial }^{2}T}{\partial x\partial y}\\ +u\frac{{\partial }^{2}T}{\partial y^2}\frac{\partial T}{\partial y}\end{array}\right]\\ +\frac{Q_0}{\rho C_p}\left(T-T_{\infty }\right)+{\lambda }^{\ast }{}_t\frac{Q_0}{\rho C_p}\left(v\frac{\partial T}{\partial y}+u\frac{\partial T}{\partial x}\right)+\frac{1}{{\left(\rho c\right)}_f}\frac{\partial }{\partial y}\left(\frac{16{\sigma }^{\ast }{T}_{\infty }^3}{3k^{\ast }}\left(\frac{\partial T}{\partial y}\right)\right)\end{array}$$

(10)

$$\begin{array}{l}u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=D_B\left(\frac{{\partial }^{2}C}{\partial y^2}\right)-{\lambda }^{\ast }{}_c\left[\begin{array}{l}{u}^2\frac{{\partial }^{2}C}{\partial x^2}+v^2\frac{{\partial }^{2}C}{\partial y^2}+2uv\frac{{\partial }^{2}C}{\partial x\partial y}+\frac{\partial C}{\partial x}\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)\\ +\frac{\partial C}{\partial y}\left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right)\end{array}\right]\\ -\frac{D_T}{T_{\infty }}\left[u\frac{{\partial }^{3}T}{\partial x\partial y^2}+u\frac{{\partial }^{3}C}{\partial y^3}+\frac{{\partial }^{2}T}{\partial y^2}\right]-K{r}^2\left(C-C_{\infty }\right){\left(\frac{T}{T_{\infty }}\right)}^n\exp \left(\frac{-E_a}{kT}\right)-{\lambda }^{\ast }{}_c{k}_c\left(v\frac{\partial C}{\partial y}+u\frac{\partial C}{\partial x}\right),\end{array}$$

(11)

$$u\frac{\partial N}{\partial x}+v\frac{\partial N}{\partial y}+\frac{b{W}_c}{\left(C_w-C_{\infty }\right)}\left[\frac{\partial }{\partial y}\left(N\frac{\partial C}{\partial y}\right)\right]=D_m\frac{{\partial }^{2}N}{\partial y^2},$$

(12)

$$\begin{array}{l}u=U_w=cx, v=0, -k\frac{\partial T}{\partial y}=h_f\left(T_w-T\right),-D_B\frac{\partial C}{\partial y}=h_g\left(C_w-C\right),\\ -D_m\frac{\partial N}{\partial y}=h_n\left(N_w-N\right) at y=0,\end{array}$$

(13)

$$v\to 0, \frac{\partial u}{\partial y}\to 0,T\to T_w,C\to C_w,N\to N_w as y\to \infty$$

(14)

Here $\left(\alpha =\frac{k}{\rho Cp}\right)$ stands for the thermal diffusivity, $\left(\rho \right)$ for the density, $\left(p\right)$ for the pressure, $\left(C_p\right)$ for the specific heat capacity and $\left(k\right)$ for the liquid thermal conductivity. The appropriate transformations are expressed by:

$$\begin{array}{l}\psi =x\sqrt{c\nu }f\left(\zeta \right),u=cx{f}^{\prime }\left(\zeta \right), v=-\sqrt{c\nu }f\left(\zeta \right), \theta \left(\zeta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},\\ \varphi \left(\zeta \right)=\frac{C-C_{\infty }}{C_w-C_{\infty }}, \chi \left(\zeta \right)=\frac{N-N_{\infty }}{N_w-N_{\infty }}, \zeta =y\sqrt{\frac{c}{\nu }}\end{array}\}.$$

(15)

PDEs are altered to the following ODEs by utilizing the above transformations:

$$\begin{array}{l}{f}^{‴}+f{f}^{″}-{\left(f^{\prime }\right)}^2+{\alpha }_1\left[2f{f}^{\prime }{f}^{″}-f^2{f}^{‴}\right]-{\alpha }_2\left[3f^2{\left(f^{″}\right)}^2-2f{\left(f^{\prime }\right)}^2{f}^{″}-f^3{f}^{iv}\right]\\ +{\alpha }_3\left[{\left(f^{″}\right)}^2-f{f}^{iv}\right]-M^2\left[{\alpha }_{2}f{f}^{‴}-{\alpha }_{1}f{f}^{″}+f^{\prime }\right]+K^2+M^{2}K+\lambda \left(\theta -Nr\varphi -Nc\chi \right) =0,\end{array}$$

(16)

$$\begin{array}{l}\left(1+\frac{4}{3}Rd\right){\theta }^{″}+\mathrm{Pr}\left(f{\theta }^{\prime }+Nb{\theta }^{\prime }{\varphi }^{\prime }+Nt{\left({\theta }^{\prime }\right)}^2\right)+\delta \theta +\delta {\lambda }_{T}f{\theta }^{\prime }\\ +\mathrm{Pr}{\lambda }_T\left(\begin{array}{l}-f{f}^{\prime }{\theta }^{\prime }-f^2{\theta }^{″}-2Ntf{\theta }^{\prime }{\theta }^{″}\\ -Nbf{\theta }^{\prime }{\varphi }^{″}-Nbf{\theta }^{″}{\varphi }^{\prime }\end{array}\right)=0,\end{array}$$

(17)

$$\begin{array}{l}{\varphi }^{″}+Le\mathrm{Pr}f{\varphi }^{\prime }-Le\mathrm{Pr}{\sigma }^{\ast }{\left(1+{\delta }_0\theta \right)}^n\exp \left(\frac{-E}{1+{\delta }_0\theta }\right)\varphi -k_1{\lambda }_{C}f{\varphi }^{\prime }\\ +Le\mathrm{Pr}{\lambda }_C\left[-f{f}^{\prime }{\varphi }^{\prime }-f^2{\varphi }^{″}-\frac{Nt}{Nb}{\theta }^{‴}f+\frac{Nt}{Nb}{\theta }^{″}\right]=0,\end{array}$$

(18)

$${\chi }^{″}+Lb{\chi }^{\prime }f-Pe\left[{\varphi }^{″}\left(\chi +\delta \right)+{\chi }^{\prime }{\varphi }^{\prime }\right]=0,$$

(19)

$$\begin{array}{l}f=0, f^{\prime }=1, {\theta }^{\prime }=-{\gamma }_1\left(1-\theta \left(\zeta \right)\right),{\varphi }^{\prime }=-{\gamma }_2\left(1-\varphi \left(\zeta \right)\right),\\ {\chi }^{\prime }=-{\gamma }_3\left(1-\chi \left(\zeta \right)\right) at \zeta =0,\end{array}$$

(20)

$$f^{\prime }\to K,f^{″}\to 0,\theta \to 0, \varphi \to 0, \chi \to 0 as \zeta \to \infty ,$$

(21)

where $\left(\mathrm{Pr}\right)$ stands for the Prandtl number, $\left(Nt\right)$ for the thermophoresis parameter, $\left({\lambda }_T\right)$ for the thermal relaxation parameter, $\left(Nb\right)$ for the Brownian motion parameter, $\left(Nc\right)$ for the bioconvection Rayleigh parameter, $\left(Rd\right)$ for the thermal radiation parameter, $\left({\alpha }_1,{\alpha }_3\right)$ for the Deborah numbers, $\left({\alpha }_2\right)$ for the Burgers’ fluid parameter, $\left(M\right)$ for the magnetic parameter, $\left(Lb\right)$ for the bioconvection Lewis parameter, $\left(Pe\right)$ for the Peclet parameter, $\left(\delta \right)$ for the microorganism difference number, $\left(Le\right)$ for the Lewis number, $\left({\lambda }_C\right)$ for the mass relaxation parameter, $\left(E\right)$ for the activation energy, $\left(Nr\right)$ for the buoyancy ratio number, $\left({\delta }_0\right)$ for the temperature difference parameter and $\left({\sigma }^{\ast }\right)$ for the chemical reaction parameter, which are defined as follows:

$$Pe=\frac{b{W}_c}{D_m},Lb=\frac{\nu }{D_m},\delta =\frac{N_{\infty }}{N_w-N_0},\},$$

$$\begin{array}{l}\mathrm{Pr}=\frac{\nu }{\alpha },Nt=\frac{\tau D_T\left(T_w-T_{\infty }\right)}{\nu T_{\infty }},{\lambda }_T=c{\lambda }^{\ast }{}_t,Nb=\frac{\tau D_B\left(C_w-C_{\infty }\right)}{\nu },\delta =\frac{\nu Q_0}{c\left(\rho c\right)f}\},\\ Rd=\frac{4{\sigma }^{\ast }{T}_{\infty }^3}{k^{\ast }}\end{array}$$

$$Le=\frac{\alpha }{D_B},{\lambda }_C=a{\lambda }^{\ast }{}_c, E=\frac{E_a}{k{T}_{\infty }}{\delta }_0=\frac{T_w-T_0}{T_{\infty }},{\sigma }^{\ast }=\frac{\nu K{r}^2}{a}\},$$

$$\begin{array}{l}{\alpha }_1=c{\lambda }_1,{\alpha }_2=c{\lambda }_2,{\beta }_3=c{\lambda }_3\\ M={\left(\frac{\sigma l{B}_0^2}{\rho f{U}_0}\right)}^{\frac{1}{2}},\lambda \left(=\frac{\left(1-C_{\infty }\right)\gamma \left(T_w-T_0\right){\beta }^{\ast \ast }}{a{U}_w}\right),\\ Nr=\frac{\left({\rho }_p-{\rho }_f\right)\left(C_w-C_o\right)}{\left(1-C_{\infty }\right)\left(T_w-T_0\right)},Nc=\frac{\gamma g\left({\rho }_m-{\rho }_f\right)\left(N_w-N_0\right)}{\left(1-C_{\infty }\right)\left(T_w-T_0\right){\beta }^{\ast \ast }}\end{array}\},$$

(22)

3. Numerical Approach

Due to the accuracy and efficiency, the nonlinear ODEs (16)–(19) with boundary restrictions (20) and (21) are solved numerically by using the bvp4c solver in the MATLAB computational software. Fourth-order collocation method is described via the bvp4c technique. Furthermore, the Lobatto-IIIa relation is considered to have a tolerance factor of ${10}^{-7}$. Initially the set of nonlinear ODEs are converted into a first order initial value problem. Let:

$$\begin{array}{l}f=p_1,f^{\prime }=p_2,f^{″}=p_3,f^{‴}=p_4,f^{iv}={p^{\prime }}_4,\theta =p_5,{\theta }^{\prime }=p_6,\\ {\theta }^{″}={p^{\prime }}_6,\varphi =p_7,{\varphi }^{\prime }=p_8,,{\varphi }^{″}={p^{\prime }}_8,\chi =p_9,{\chi }^{\prime }=p_{10},{\chi }^{″}={p^{\prime }}_{10}\end{array}\},$$

(23)

$${p^{\prime }}_4,=\frac{\begin{array}{l}{p}_4+p_1{p}_3-{\left(p_2\right)}^2+{\alpha }_1\left[2p_1{p}_2{p}_3-p_1^2{p}_4\right]-{\alpha }_2\left[3p_1^2{\left(p_3\right)}^2-2p_1{\left(p_2\right)}^2{p}_3\right]\\ +{\alpha }_3\left[{\left(p_3\right)}^2\right]-M^2\left[{\alpha }_2{p}_1{p}_4-{\alpha }_1{p}_1{p}_3+p_2\right]+K^2+M^{2}K+\lambda \left(p_5-Nr{p}_8-Nc{p}_9\right) \end{array}}{\left({\alpha }_3{p}_1-{\alpha }_2{p}_1^3\right)}$$

(24)

$${p^{\prime }}_6,=\frac{\mathrm{Pr}\left(p_1{p}_6+Nb{p}_6{p}_8+Nt{\left(p_6\right)}^2\right)+\delta p_5+\delta {\lambda }_T{p}_1{p}_5+\mathrm{Pr}{\lambda }_T\left(\begin{array}{l}-p_1{p}_2{p}_6\\ -Nb{p}_1{p}_6{p^{\prime }}_8\end{array}\right)}{\left(\mathrm{Pr}{\lambda }_T{f}^2+\mathrm{Pr}{\lambda }_{T}2Nt{p}_1{p}_6+\mathrm{Pr}{\lambda }_{T}Nb{p}_1{p}_8-1-\frac{4}{3}Rd\right)}$$

(25)

$$\begin{array}{l}{p^{\prime }}_8=-Le\mathrm{Pr}{p}_1{p}_8+Le\mathrm{Pr}{\sigma }^{\ast }{\left(1+{\delta }_0{p}_5\right)}^n\exp \left(\frac{-E}{1+{\delta }_0{p}_5}\right)p_7+k_1{\lambda }_C{p}_1{p}_8\\ -Le\mathrm{Pr}{\lambda }_C\left[-p_1{p}_2{p}_8-p_1^2,{p^{\prime }}_8-\frac{Nt}{Nb}{p^{\prime }}_6{p}_1+\frac{Nt}{Nb}{p^{\prime }}_6\right],\end{array}$$

(26)

$${\chi }^{″}=-Lb{p}_{10}{p}_1+Pe\left[{p^{\prime }}_8\left(p_9+\delta \right)+p_9{p}_8\right]=0,$$

(27)

with:

$$\begin{array}{l}{p}_1=0, p_2=1,p_8=-{\gamma }_1\left(1-p_7\left(\zeta \right)\right),p_8=-{\gamma }_2\left(1-p_7\left(\zeta \right)\right),\\ p_{10}=-{\gamma }_3\left(1-p_9\left(\zeta \right)\right) at \zeta =0,\end{array}$$

(28)

$$p_2\to K,p_3\to 0,p_5\to 0, p_7\to 0, p_9\to 0 as \zeta \to \infty .$$

(29)

4. Results and Discussion

In this segment, the physical significance of prominent parameters like thermophoresis parameter $Nt\left(0.1\le Nt\le 2.2\right),$ magnetic parameter $M\left(0.1\le M\le 1.2\right),$ Burgers fluid parameter ${\alpha }_2\left(0.1\le {\alpha }_2\le 2.2\right),$ Deborah number ${\alpha }_1\left(0.1\le {\alpha }_1\le 1.5\right),$ bioconvection Rayleigh number $Nc\left(0.1\le Nc\le 2.2\right),$ Deborah number ${\alpha }_3\left(0.1\le {\alpha }_3\le 2.2\right),$ buoyancy ratio parameter $Nr\left(0.1\le Nr\le 2.0\right),$ velocity ratio parameter $K\left(0.2\le K\le 1.6\right),$ mixed convection parameter $\lambda \left(0.1\le \lambda \le 2.2\right),$ thermal relaxation parameter ${\lambda }_T\left(0.1\le {\lambda }_T\le 2.2\right),$ thermal Biot number ${\gamma }_1\left(0.1\le {\gamma }_1\le 1.0\right),$ Prandtl number $Pr\left(2.0\le Pr\le 5.0\right),$ activation energy parameter $E\left(0.1\le E\le 2.0\right),$ solutal Biot number ${\gamma }_2\left(0.1\le {\gamma }_2\le 1.6\right),$ thermal radiation number $Rd\left(0.1\le Rd\le 1.6\right),$ Brownian motion number $Nb\left(0.1\le Nb\le 0.8\right),$ Lewis parameter $Le\left(1.2\le Le\le 2.4\right),$ microorganism Biot number ${\gamma }_3\left(0.1\le {\gamma }_3\le 1.6\right),$ Peclet number $Pe\left(0.1\le Pe\le 1.2\right)$ and bioconvection Lewis number $Lb\left(1.0\le Lb\le 3.0\right)$ against velocity, temperature, volumetric concentration of nanoparticles and motile microorganisms is deliberated through Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15. Figure 2 represents the performance of velocity field $f^{\prime }$ for growing magnetic parameter $M$ and Burgers’ parameter ${\alpha }_2$. Clearly the velocity of fluid $f^{\prime }$ reduces for growing estimations of the magnetic number $M$ and the Burgers’ parameter ${\alpha }_2$. This scenario is observed for different values of both numbers. By raising the magnetic number, a resistive force is developed, which causes decay in the motion of the Burgers’ nanofluid. The aspects of the Deborah number ${\alpha }_1$ and bioconvection Rayleigh number $Nc$ versus velocity flow field $f^{\prime }$ are plotted in Figure 3. Reduction is observed in the velocity of the Burgers’ nanofluid $f^{\prime }$ due to the increased values of the Deborah number ${\alpha }_1$. Furthermore, it is analyzed that the decay of velocity $f^{\prime }$ is due to rises in the bioconvection Rayleigh parameter $Nc$. Features of the velocity $f^{\prime }$ due to the Deborah number for retardation ${\alpha }_3$ and the buoyancy ratio parameter $Nr$ are exhibited in Figure 4. The velocity of the Burgers’ nanofluid $f^{\prime }$ increases for a larger Deborah number for retardation time parameter ${\alpha }_3$. It is also depicted that the velocity field $f^{\prime }$ diminishes with a higher amount of the buoyancy ratio number $Nr$. The performance of the velocity ratio number K and mixed convection parameter $\lambda$ against the velocity $f^{\prime }$ is plotted in Figure 5. Here the velocity of Burgers’ nanofluid $f^{\prime }$ is enhanced by increasing the values of the velocity ratio parameter K and mixed convective number $\lambda$. The consequences of the thermal relaxation number ${\lambda }_T$ and Burgers’ number ${\alpha }_2$ changes over temperature field $\theta$ are examined in Figure 6. An increment in the thermal relaxation parameter ${\lambda }_T$ corresponds to a weaker temperature field $\theta$. It is also noted that the temperature field $\theta$ upsurges for a higher magnitude of the Burgers’ parameter ${\alpha }_2$. The thermal relaxation parameter is directly proportional to the thermal relaxation time. A greater thermal relaxation parameter indicates a higher relaxation time which produces a weaker temperature field. Figure 7 depicts the effects of the thermal Biot number ${\gamma }_1$ and thermophoresis parameter $Nt$ versus the temperature field $\theta$. It is noted that the temperature field $\theta$ upsurges with a larger thermal Biot number ${\gamma }_1$. Here it is also witnessed that the temperature field $\theta$ augments with rising variation of the thermophoresis number $Nt$. A growing thermophoresis number depicts a stronger thermophoretic force which produces a stronger temperature field. Figure 8 shows the physical appearance of the Prandtl number $Pr$ and the thermal radiation parameter $Rd$ against the temperature field $\theta$. It can be verified that the temperature distribution $\theta$ is retarded by varying the Prandtl number Pr. Here, it is also noted that the temperature $\theta$ is increased with higher estimations of the thermal radiation parameter $Rd$. Figure 9 examines the variation in nanoparticles concentration $\varphi$ for different estimations of the thermophoresis parameter $Nt$ and the activation energy parameter $E$. It is noted that the volumetric concentration of nanoparticles $\varphi$ is upgraded by increasing the estimations of the thermophoresis parameter $Nt$. It is also analyzed that the volumetric concentration of nanoparticles $\varphi$ is enlarged with larger estimations of the activation energy parameter $E$. Figure 10 elucidates the behavior of the solutal Biot number ${\gamma }_2$ and the buoyancy ratio parameter $Nr$ with the concentration of nanoparticles $\varphi$. Here the concentration of nanoparticles $\varphi$ is enhanced for a larger solutal Biot number ${\gamma }_2$ and the buoyancy ratio parameter $Nr$. Features of the effect of the Brownian motion number $Nb$ and Burgers’ number ${\alpha }_2$ on the volumetric concentration of nanoparticles $\varphi$ are plotted in Figure 11. It is illustrated that the nanoparticles concentration $\varphi$ is decayed with a larger Brownian motion number $Nb$. It is also analyzed that the concentration field $\varphi$ is knocked down due to the Burgers’ parameter ${\alpha }_2$. Brownian motion develops due to the occurrence of nanoparticles and results in the decay of the nanoparticle concentration thickness. The characteristics of the Prandtl number Pr and Lewis number versus volumetric concentration of nanoparticles $\varphi$ are shown in Figure 12. It is observed that the concentration field $\varphi$ diminishes for a higher Prandtl number $Pr$ and Lewis number $Le$. The Lewis number is inversely proportional to the Brownian diffusivity. A greater Lewis number indicates a lower Brownian diffusivity which produces a weaker nanoparticles concentration field. Figure 13 shows the microorganism’s field $\chi$ for various values of the Burgers’ parameter ${\alpha }_2$ and the buoyancy ratio parameter $Nr$. An increment in the Burgers’ parameter ${\alpha }_2$ and buoyancy ratio parameter $Nr$ leads to a stronger microorganism profile $\chi$. Figure 14 displays the characteristics of the microorganism Biot number ${\gamma }_3$ and bioconvection Rayleigh number $Nc$ against the swimming motile microorganism profile $\chi$. The swimming motile microorganism field $\chi$ is increased as the enlarging microorganism Biot number ${\gamma }_3$ and bioconvection Rayleigh number $Nc$ increase. The outcomes of the microorganism profile $\chi$ versus Peclet $Pe$ and bioconvection Lewis $Lb$ parameters are shown in Figure 15. The swimming motile microorganism field $\chi$ declines with the Peclet number $Pe$ and the bioconvection Lewis number $Lb$. From

Table 1. Variations of $-f^{″}(0)$ for $M,$ $K$, $\lambda ,$ $Nr,$ $Nc,$ ${\alpha }_1,$ ${\alpha }_2 and {\alpha }_3$.

$\mathit{M}$	$\mathit{K}$	$\mathit{\lambda }$	$\mathit{N}\mathit{r}$	$\mathit{N}\mathit{c}$	${\mathit{\alpha }}_1$	${\mathit{\alpha }}_2$	${\mathit{\alpha }}_3$	$-{\mathit{f}}^{″}\left(0\right)$
0.1 0.5 1.0	0.1	0.1	0.1	0.1	0.2	0.2	0.2	0.2867 0.9398 1.0320
0.4	0.2 0.5 0.8	0.1	0.1	0.1	0.2	0.2	0.2	0.8569 0.6030 0.2641
0.4	0.1	0.2 0.8 1.6	0.1	0.1	0.2	0.2	0.2	0.9142 0.8803 0.8365
0.4	0.1	0.1	0.2 1.0 2.0	0.1	0.2	0.2	0.2	0.9203 0.9239 0.9284
0.4	0.1	0.1	0.1	0.2 1.0 2.0	0.2	0.2	0.2	0.9204 0.9242 0.9291
0.4	0.1	0.1	0.1	0.1	0.1 0.5 1.0	0.2	0.2	0.9158 0.9112 0.9591
0.4	0.1	0.1	0.1	0.1	0.2	0.1 0.5 1.0	0.2	0.9090 0.9506 0.9991
0.4	0.1	0.1	0.1	0.1	0.2	0.2	0.1 0.5 1.0	0.9463 0.9112 0.8681

, it can be noted that the skin friction coefficient $-f^{″}\left(0\right)$ increases for $M$ and $Nr$. From

Table 2. Variations of $-{\theta }^{\prime }\left(0\right)$ for $M,$ $K$, $Nb,$ $Nt$ ${\gamma }_1$.

$\mathit{M}$	$\mathit{K}$	$\mathit{N}\mathit{b}$	$\mathit{N}\mathit{t}$	$\mathit{R}\mathit{d}$	${\mathit{\gamma }}_1$	$-{\mathit{\theta }}^{\prime }\left(0\right)$
0.1 0.5 1.0	0.1	0.3	0.2	0.4	0.3	0.2060 0.2048 0.2035
0.4	0.2 0.5 0.8	0.3	0.2	0.4	0.3	0.2068 0.2114 0.2149
0.4	0.1	0.1 0.4 0.8	0.2	0.4	0.3	0.2059 0.2026 0.1982
0.4	0.1	0.3	0.1 0.4 0.8	0.4	0.3	0.2078 0.2033 0.1968
0.4	0.1	0.3	0.2	0.1 0.8 1.6	0.3	0.3130 0.1961 0.1824
0.4	0.1	0.3	0.2	0.4	0.1 0.5 1.0	0.0871 0.2791 0.3787

, it can be concluded that the local Nusselt number $-{\theta }^{\prime }\left(0\right)$ declines with growing variations of $Nb$ and $Rd$. From

Table 3. Variations of $-{\varphi }^{\prime }\left(0\right)$ for $M,$ $K$, $Nb,$ $Nt$ ${\gamma }_2.$

$\mathit{M}$	$\mathit{K}$	$\mathit{N}\mathit{b}$	$\mathit{N}\mathit{t}$	$\mathit{R}\mathit{d}$	${\mathit{\gamma }}_2$	$-{\mathit{\varphi }}^{\prime }\left(0\right)$
0.1 0.5 1.0	0.1	0.3	0.2	0.4	0.3	0.2303 0.2295 0.2287
0.4	0.2 0.5 0.8	0.3	0.2	0.4	0.3	0.2310 0.2377 0.2346
0.4	0.1	0.1 0.4 0.8	0.2	0.4	0.3	0.2087 0.2399 0.2451
0.4	0.1	0.3	0.1 0.4 0.8	0.4	0.3	0.2424 0.2236 0.2033
0.4	0.1	0.3	0.2	0.1 0.8 1.6	0.3	0.2278 0.2316 0.2346
0.4	0.1	0.3	0.2	0.4	0.1 0.8 1.6	0.0860 0.4802 0.7137

, it is analyzed that the local Sherwood number $-{\varphi }^{\prime }\left(0\right)$ increases with $K$. The microorganism density number $-{\chi }^{\prime }\left(0\right)$ reduces for $Pe$ while it is enhanced with growing estimations of ${\gamma }_3$ (see

Table 4. Variations of $-{\chi }^{\prime }\left(0\right)$ for $M,$ $K$, $\lambda ,$ $Nr,$ ${\gamma }_3,$ ${\gamma }_3,$ $Pe and Lb$.

$\mathit{M}$	$\mathit{K}$	$\mathit{\lambda }$	$\mathit{N}\mathit{r}$	${\mathit{\gamma }}_3$	$\mathit{P}\mathit{e}$	$\mathit{L}\mathit{b}$	$-{\mathit{\chi }}^{\prime }\left(0\right)$
0.1 0.5 1.0	0.1	0.1	0.1	0.3	0.1	0.1	0.2786 0.2771 0.2755
0.4	0.2 0.5 0.8	0.1	0.1	0.3	0.1	0.1	0.2796 0.2855 0.2907
0.4	0.1	0.2 0.8 1.6	0.1	0.3	0.1	0.1	0.2776 0.2781 0.2788
0.4	0.1	0.1	0.2 1.0 2.0	0.3	0.1	0.1	0.2775 0.2774 0.2773
0.4	0.1	0.1	0.1	0.1 0.8 1.6	0.1	0.1	0.0900 0.4251 0.5793
0.4	0.1	0.1	0.1	0.3	0.2 0.5 1.0	0.1	0.2757 0.2565 0.2265
0.4	0.1	0.1	0.1	0.3	0.1	1.0 2.0 3.0	0.2373 0.2973 0.3184

).

Table 5. Comparative outcomes of $-{\theta }^{\prime }\left(0\right)$ for various estimations of $Pr$ when $\lambda =Nr=Nc=E=Le=Lb=0$.

Pr	Iqbal et al. [30]	Present Results
0.7	0.45312	0.45312
2.0	0.90894	0.90894
7.0	1.88986	1.88985

is constructed to validate the presented numerical solution with previously determined solution in a limiting situation. It is witnessed that the presented numerical solution shows good agreement with the solution previous reported by Iqbal et al. [30] in a limiting situation.

5. Conclusions

The effects of bio-convection and motile microorganisms in a magnetized Burgers’ nanofluid flow due to a stretching sheet are studied. Cattaneo–Christov double diffusion theory is also discussed. The Buongiorno phenomenon for nanoparticles motion in a Burgers’ fluid is employed in view of the Cattaneo–Christov relations. The velocity field of a Burgers’ nanofluid is a diminishing function of the Burgers’ fluid parameters. The velocity field of a Burgers’ nanofluid decays for buoyancy ratio and bioconvection Rayleigh numbers while it is enhanced for a mixed convective number. A reduction in temperature is analyzed for growing estimations of the thermal relaxation parameter. An increment in the thermal Biot, thermophoresis and radiation parameters leads to a stronger temperature field. Nanoparticles concentration is reduced for growing Brownian motion and Lewis numbers. The microorganism profile is enhanced by growing estimations of the microorganism Biot and bioconvection Rayleigh numbers. The microorganism field is reduced by increasing estimations of the Peclet and bioconvection Lewis numbers.