**2. Governing Equations**

The present study considers the mixed bioconvection flow of a non-Newtonian power-law nanofluid having motile microorganisms and obeying the Ostwald–de Waele model [18] near the stagnation point at a heated stretchable vertical surface coinciding with the plane y = 0. The flow is confined to the region y > 0, where x and y are the cartesian coordinates. The model suggested by Buongiorno is utilized for the nanofluid attitude in which the influence of thermophoresis and Brownian movement is taken into consideration. The stretchable surface is maintained at a constant temperature Tw along with the constant density of motile microorganisms Nw. The ambient temperature, concentration, and motile microorganisms are symbolized as T∞, C∞, and N∞, respectively. It is considered that the stretching velocity is given by Uw(x) = cx and the velocity of external flow in the neighborhood of the stagnation point at x = y = 0 is given by U∞(x) = ax, where a and c are positive constants. Although, in the past, this reality is eliminated, in a practical situation, there is no nanoparticle flux on the boundaries, and the values of the nanofluid fraction C adapt to the concentration distribution. This means that we consider passively controlled boundary conditions as proposed by Kuznetsov and Nield [26].

Under the above-mentioned assumptions, the physical description of the problem under consideration in Figure 1 with Boussinesq approximations can be expressed as follows [8,24]:

$$\frac{\partial \mu}{\partial x} + \frac{\partial v}{\partial y} = 0 \tag{1}$$

$$\ln \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = \mathcal{U}\_{\infty} \frac{d\mathcal{U}\_{\infty}}{dx} + \frac{1}{\rho\_f} \frac{\partial \mathbf{r}\_{xy}}{\partial y} + \frac{1}{\rho\_f} \begin{bmatrix} (1 - \mathcal{C}\_{\infty}) \text{g} \mathcal{S} (T - T\_{\infty})\\ - (\rho\_p - \rho\_f) \text{g} (\mathcal{C} - \mathcal{C}\_{\infty})\\ - (\rho\_m - \rho\_f) \text{g} \, \gamma (N - N\_{\infty}) \end{bmatrix} \tag{2}$$

$$
\mu \frac{\partial T}{\partial \mathbf{x}} + \upsilon \frac{\partial T}{\partial y} = \mathfrak{a} \frac{\partial^2 T}{\partial y^2} + \mathfrak{r} \left[ D\_B \frac{\partial C}{\partial y} \frac{\partial T}{\partial y} + \frac{D\_T}{T\_{\infty}} \left( \frac{\partial T}{\partial y} \right)^2 \right] \tag{3}
$$

$$
\mu \frac{\partial \mathcal{C}}{\partial x} + v \frac{\partial \mathcal{C}}{\partial y} = D\_B \frac{\partial^2 \mathcal{C}}{\partial y^2} + \frac{D\_T}{T\_{\infty}} \frac{\partial^2 T}{\partial y^2} \tag{4}
$$

$$u\frac{\partial N}{\partial x} + v\frac{\partial N}{\partial y} + \frac{b\mathcal{W}c}{\mathcal{C}\_{\infty}} \left[ \frac{\partial}{\partial y} \left( N \frac{\partial \mathcal{C}}{\partial y} \right) \right] = D\_m \frac{\partial^2 N}{\partial y^2} \tag{5}$$

The boundary conditions can be set in the following form [14]:

$$u = \mathcal{U}\_w(\mathbf{x}), v = 0, T = T\_{w\prime} D\_B \frac{\partial \mathcal{C}}{\partial y} + \frac{D\_T}{T\_{\infty}} \frac{\partial T}{\partial y} = 0,\\ N = N\_w \text{ at } y = 0 \tag{6a}$$

$$\mu = \mathcal{U}\_{\infty}(\mathbf{x}),\\T = T\_{\infty \prime} \ \mathbb{C} = \mathbb{C}\_{\infty \prime} \ N = N\_{\infty} \text{ as } y \to \infty \tag{6b}$$

**Figure 1.** Schematic functionality of the flow.

In the present problem, we have *∂*u/*∂*y > 0 when a/c > 1 (the ratio of free stream velocity and stretching velocity), which gives the shear stress where *τxy* = *K*(*∂u*/*∂y*) *n*. Here, K is the consistency coefficient and *n* is the power-law fluid. It is noted that, when *n* = 1, the fluid is Newtonian; when *n* < 1, the fluid is called pseudoplastic power-law fluid; and when *n* > 1, it is called dilatants power-law fluid.

Here (*u*, *v*) are the velocity components, along with the (*x*, *y*) directions. Here, *T*, *C,* and *N* are the fluid temperature, concentration, and density of motile microorganisms, respectively, and g is the gravitational acceleration. Wc denotes the maximum cell swimming speed, α stands for thermal diffusivity, β is the coefficient of thermal expansion, γ represents the average volume of a microorganism, ρ<sup>f</sup> represents the density of the fluid, ρ<sup>p</sup> denotes the density of the particles, ρ<sup>m</sup> is the density of the microorganism, *τ* = (*ρc*)*p*/(*ρc*)*<sup>f</sup>* is the nanofluid heat capacity ratio, (*ρc*)*<sup>f</sup>* is the heat capacity of the base fluid and (*ρc*)*<sup>p</sup>* is the effective heat capacity of the nanoparticle material, Dn stands for diffusivity of the microorganisms, *DB* represents the Brownian diffusion coefficient, and DT stands for the thermophoretic diffusion coefficient of the microorganisms.

It is observed that the continuity equation is automatically determined by specifying the upgraded stream function, such that *u* = (*∂ψ*/*∂y*), *v* = −(*∂ψ*/*∂x*), and exhibits the following non-dimensional variables:

 $\theta = \frac{T - T\_{\infty}}{T\_{\text{w}} - T\_{\text{m}}}$ ,  $\Phi = \frac{C - C\_{\infty}}{C\_{\infty}}$ ,  $\chi = \frac{N - N\_{\infty}}{N\_{\text{w}} - N\_{\text{m}}}$ , 
$$\psi = \left(\frac{K/\rho\_f}{\epsilon^{1 - 2\pi}}\right)^{1/(n+1)} x^{2n/(n+1)} F(\eta), \ \eta = \mathcal{y} \left(\frac{\epsilon^{2 - n}}{K/\rho\_f}\right)^{1/(n+1)} x^{(1-n)/(1+n)} \tag{7}$$

Equations (1)–(5) take the following non-dimensional form [8,9]:

$$nF''^{(n-1)}F'' + \frac{2n}{1+n}FF'' - F'^2 + \varepsilon^2 + \lambda\theta - Nr\phi - Rb\ \chi = 0\tag{8}$$

$$\theta'' + \Pr\left(\frac{2n}{1+n}F\theta' + Nb\theta'\phi' + Nt\theta'^2\right) = 0\tag{9}$$

$$
\phi'' + \mathrm{Sc}\frac{2n}{n+1}F\phi' + \frac{\mathrm{Nt}}{\mathrm{Nb}}\theta'' = 0\tag{10}
$$

$$
\chi'' + Sb \frac{2n}{1+n} F \chi' - Pe \left(\phi \chi' + (\chi + \sigma) \phi''\right) = 0 \tag{11}
$$

$$F'(0) = 1,\\ F(0) = 0, \theta(0) = 1,\\ Nb\phi'(0) + Nt\theta'(0) = 0,\\ \chi(0) = 1 \tag{12a}$$

$$F'(\infty) = \varepsilon, \theta(\infty) = 0, \phi(\infty) = 0, \chi(\infty) = 0 \tag{12b}$$

where the prime denotes differentiation with respect to *η* and *λ* = *Grx* Re<sup>2</sup> *x* is the dimensionless mixed convection parameter, *Nr* <sup>=</sup> (*ρp*−*ρf*)*C*<sup>∞</sup> *<sup>ρ</sup><sup>f</sup> <sup>β</sup>*(*Tw*−*T*∞)(1−*C*∞) stands for the buoyancy ratio parameter, *Grx* <sup>=</sup> (1−*C*∞)*ρ<sup>f</sup> <sup>g</sup>β*(*Tw*−*T*∞) (*K*/*ρf*) <sup>2</sup> is the local Grashof number, Re*<sup>x</sup>* <sup>=</sup> (*cx*) <sup>2</sup>−*nxn K*/*ρ<sup>f</sup>* is the local Reynolds number, *Nb* = *<sup>τ</sup>DBC*<sup>∞</sup> *cx*2Re−2/(*n*+1) *<sup>x</sup>* is the Brownian motion parameter, *Nt* = *<sup>τ</sup>DT*(*Tw*−*T*∞) *<sup>T</sup>*∞*cx*2Re−2/(*n*+1) *<sup>x</sup>* is the thermophoresis parameter, Pr = *cx*<sup>2</sup> *<sup>α</sup>* Re−2/(*n*+1) *<sup>x</sup>* is the Prandtl number and *Sc* = *cx*<sup>2</sup> *DB* Re−2/(*n*+1) *<sup>x</sup>* is the Schmidt number, *Rb* <sup>=</sup> (*Nw*−*N*∞)*γ*(*ρm*−*<sup>ρ</sup> <sup>f</sup>*) *<sup>ρ</sup> <sup>f</sup> <sup>β</sup>*(*Tw*−*T*∞)(1−*C*∞) is the bioconvection Rayleigh number, and *ε* = *a*/*c* stands for the ratio of the velocity parameter. *Sb* = *cx*2Re −2 1+*n x Dm* denotes the bioconvection Schmidt number, *<sup>σ</sup>* <sup>=</sup> *<sup>N</sup>*<sup>∞</sup> *Nw*−*N*<sup>∞</sup> is the bioconvection constant, and *Pe* = *bWc Dm* represents the bioconvection Peclet number.

Various engineering quantities of interest like local skin friction *Cf*, the local Nusselt number *Nux*, and the local density of the motile microorganisms' number *Nnx* are explored for the present nanofluid flow model. These quantities are defined as follows:

$$\mathcal{C}\_{f} = \frac{2\pi\_{w}}{(c\chi)^{2}\rho\_{f}} ; \; N u\_{\rm x} = \frac{q\_{w}\chi}{k\_{f}(T\_{\rm w} - T\_{\infty})} ; \; N n\_{\rm x} = \frac{q\_{n}\chi}{D\_{\rm n}(T\_{\rm w} - T\_{\infty})} \tag{13}$$

For more clarification, the expressions of the shear stress *τw*, wall flux *qw* (i.e., heat flux), and *qn* (i.e., motile microorganisms density flux) are given as follows:

$$\pi\_w = K \left( \frac{\partial u}{\partial y} \right)\_{y=0}; \; q\_w = -k\_f \left( \frac{\partial T}{\partial y} \right)\_{y=0}; \; q\_n = -D\_n \left( \frac{\partial N}{\partial y} \right)\_{y=0} \tag{14}$$

By invoking the transformations of Equation (8), Equations (15) and (16) are reduced as follows:

$$\frac{1}{2}\mathbb{C}\_{f}\mathrm{Re}\_{x}^{1/(1+n)} = (F''(0))^{n};\ \mathrm{Nu}\_{x}\mathrm{Re}\_{x}^{-1/(1+n)} = -\theta'(0);\ \mathrm{Nu}\_{x}\mathrm{Re}\_{x}^{-1/(1+n)} = -\chi'(0)\tag{15}$$
