**2. Problem Formulation**

Consider a two-dimensional MHD nanofluid flow conveying nano-sized particles and gyrotactic micro-organisms across a curved surface of radius *R*, as seen in Figure 1. Indeed, the curved sheet is stretched linearly along the *s*−direction with a variable velocity *Uw* = *as*, in which a velocity slip condition *Uslip* is imposed at the fluid-solid interface, where *a* is a positive constant. In addition, a uniform magnetic flux density *B*<sup>0</sup> is applied radially on the developed electrically conducting nanofluid flow.

**Figure 1.** Flow configuration.

The following physical suppositions are adopted:


Based on the aforementioned assumptions, the governing conservation equations are written in the steady state as follows [37]:

$$
\frac{\partial v}{\partial r} + \left(\frac{1}{r+R}\right)v + \left(\frac{R}{r+R}\right)\frac{\partial u}{\partial s} = 0,\tag{1}
$$

$$\left(\frac{1}{r+R}\right)u^2 = -\frac{1}{\rho}\frac{\partial p}{\partial r},\tag{2}$$

$$\begin{split} \upsilon \frac{\partial u}{\partial r} + \left(\frac{\mathbb{R}}{r+\mathbb{R}}\right) u &\quad \frac{\partial u}{\partial s} + \left(\frac{1}{r+\mathbb{R}}\right) u v \\ &= -\frac{1}{\rho} \left(\frac{\mathbb{R}}{r+\mathbb{R}}\right) \frac{\partial p}{\partial s} + \upsilon \left[\frac{\partial^2 u}{\partial r^2} + \left(\frac{1}{r+\mathbb{R}}\right) \frac{\partial u}{\partial r} - \left(\frac{1}{r+\mathbb{R}}\right)^2 u\right] \\ &\quad - \frac{\sigma}{\rho} \, B\_0^2 \, u \, \end{split} \tag{3}$$

$$
\mu \frac{\partial T}{\partial r} + \left(\frac{R}{r+R}\right) u \frac{\partial T}{\partial s} = \frac{k}{(\rho \mathbb{C}\_P)} \left[ \left(\frac{1}{r+R}\right) \frac{\partial T}{\partial r} + \frac{\partial^2 T}{\partial r^2} \right] + \tau \left[ D\_B \frac{\partial \mathbb{C}}{\partial r} \frac{\partial T}{\partial r} + \frac{D\_T}{T\_\infty} \left(\frac{\partial T}{\partial r}\right)^2 \right], \tag{4}
$$

$$\upsilon \frac{\partial \mathcal{C}}{\partial r} + \left(\frac{R}{r+R}\right) \mu \frac{\partial \mathcal{C}}{\partial s} = D\_B \left[ \left(\frac{1}{r+R}\right) \frac{\partial \mathcal{C}}{\partial r} + \frac{\partial^2 \mathcal{C}}{\partial r^2} \right] + \frac{D\_T}{T\_\infty} \left[ \left(\frac{1}{r+R}\right) \frac{\partial T}{\partial r} + \frac{\partial^2 T}{\partial r^2} \right], \tag{5}$$

$$
\upsilon \frac{\partial N}{\partial r} + \left(\frac{R}{r+R}\right) \mu \frac{\partial N}{\partial s} = D\_m \left[ \left(\frac{1}{r+R}\right) \frac{\partial N}{\partial r} + \frac{\partial^2 N}{\partial r^2} \right] - \frac{b\mathcal{W}\_c}{\mathbb{C}\_w - \mathbb{C}\_\infty} \frac{\partial}{\partial r} \left( N \frac{\partial \mathcal{C}}{\partial r} \right), \tag{6}
$$

where the symbols *u* and *v* denote the *s*− and *r*−velocity components. *p* means the pressure, *σ* is the nanofluid electrical conductivity. (*ρCP*) reflects the nanofluid heat capacitance. *DT* represents the thermophoresis coefficient. *T*, *Tw* and *T*<sup>∞</sup> designate the nanofluid temperature, the wall temperature, and the ambient temperature, respectively. *DB* refers to the coefficient of Brownian diffusion. *C*, *Cw* and *C*<sup>∞</sup> indicate the nanofluid concentration (i.e., nanoparticles' volume fraction), the wall concentration, and the ambient concentration, respectively. *Dm* signifies the coefficient of motile micro-organisms' diffusion. *N*, *Nw* and *N*∞ stand for the motile micro-organisms' concentration, the motile micro-organisms' concentration at the wall, and the motile micro-organisms' concentration at the ambient region, respectively. *b* symbolizes the chemotaxis constant. *Wc* marks the maximum cell speed.

For the flow problem the appropriate boundary conditions are:

$$\begin{cases} \mu = \mathcal{U}\_{\text{slip}} + \mathcal{U}\_{\text{w}\prime} \; v = 0, & T = T\_{\text{w}\prime} \; \mathcal{C} = \mathcal{C}\_{\text{w}\prime} \; N = N\_{\text{w}} \; \text{at } r = 0, \\\ \quad \mu \to 0, \; \frac{\partial u}{\partial r} \to 0, \; T \to T\_{\text{cos}\prime} \; \mathcal{C} = \mathcal{C}\_{\text{cos}\prime} \; N = N\_{\text{w}} \; \text{as } r \to \infty \end{cases} \tag{7}$$

Here the velocity slip is given by:

$$
\hbar L\_{slip} = \beta \left( \frac{\partial u}{\partial r} - \frac{u}{r + R} \right) \tag{8}
$$

where *β* denotes the slip length. *β* = 0 denotes the no-slip boundary condition. The following new variables are defined to simplify the flow equations:

$$\begin{cases} \frac{\eta}{r} = \sqrt{\frac{a}{\upsilon}}, F' = \frac{u}{L\_w}, F = -\frac{(r+R)}{R\sqrt{av}}\upsilon \; , \; \theta = \frac{T-T\_{\infty}}{T\_w - T\_{\infty}\prime} \; \bigg| \\\ \qquad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \frac{C-C\_{\infty}}{\Gamma\_w - \mathrm{C}\_{\infty}\prime} \; P = \frac{p}{\rho \mathrm{II}\_w^2} \end{cases} \tag{9}$$

Using Equation (9), Equations (1)–(6) takes the following non-dimensional form as follows:

$$P' - \frac{{\bf F'}^2}{\left(\eta + A\right)} = 0,\tag{10}$$

$$F''' + \frac{F''}{\left(\eta + A\right)} - \frac{F'}{\left(\eta + A\right)^2} + \frac{AFF''}{\left(\eta + A\right)} + \frac{AFF'}{\left(\eta + A\right)^2} - \frac{AF'^2}{\left(\eta + A\right)} - \frac{2AP}{\left(\eta + A\right)} - MF' = 0,\tag{11}$$

$$a' \qquad \text{bz-} A \cap \partial'$$

$$
\theta'' + \frac{\theta'}{(\eta + A)} + \frac{\Pr AF\theta'}{(\eta + A)} + \Pr Nt \,\theta'^2 + \Pr Nb \,\theta'\phi' = 0,\tag{12}
$$

$$
\phi'' + \frac{\phi'}{\left(\eta + A\right)} + \frac{ScAF\phi'}{\left(\eta + A\right)} + \frac{Nt}{Nb} \left[\theta'' + \frac{\theta'}{\left(\eta + A\right)}\right] = 0,\tag{13}
$$

$$\chi'' + \frac{\chi'}{(\eta + A)} + \frac{SbAF\chi'}{(\eta + A)} + Pb\left[\chi'\phi' + (\tau\_0 + \chi)\phi''\right] = 0\tag{14}$$

.

The related boundary conditions in the non-dimensional form are given by:

$$\begin{cases} F = 0, \; F' = 1 + \beta\_1 \left( F'' - \frac{1}{A} F' \right), \; \theta = \phi = \chi = 1 \text{ at } \eta = 0, \\\ \quad F' = F'' = \theta = \phi = \chi = 0 \quad \text{as } \; \eta \to \infty \end{cases} \tag{15}$$

The parameters in non-dimensional forms are represented in Table 1 below:

**Table 1.** Non-dimensional parameters and their related expressions.


The physical quantities of interest are represented as follows:

$$\mathcal{C}\_{f} = \frac{\mathfrak{r}\_{rs}}{\rho l I\_{w}^{2}}, \text{ where } \mathfrak{r}\_{w} = \mu \left( \frac{\partial u}{\partial r} - \frac{u}{r + R} \right)\_{r = 0}^{} \tag{16}$$

$$N\mu\_s = \frac{sq\_{\overline{w}}}{k(T\_{\overline{w}} - T\_{\infty})}, \text{ where } q\_{\overline{w}} = -k \left(\frac{\partial T}{\partial r}\right)\_{r=0} \tag{17}$$

$$Sh\_s = \frac{sq\_\circ}{D\_B(\mathbb{C}\_w - \mathbb{C}\_\infty)}, \text{ where } q\_\circ = -D\_B \left(\frac{\partial \mathbb{C}}{\partial r}\right)\_{r=0} \tag{18}$$

$$M n\_s = \frac{sq\_m}{D\_m(N\_w - N\_\infty)}, \text{ where } q\_m = -D\_m \left(\frac{\partial N}{\partial r}\right)\_{r=0} \tag{19}$$

where *τ<sup>w</sup>* represents the surface shear stress. *qw* denotes the wall heat flux. *qj* designates the wall mass flux. *qm* stands for the wall motile micro-organisms' flux. The non-dimensional forms of Equations (16)–(19) are given as:

$$\mathcal{L}\_f Re\_s^{\frac{1}{2}} = F''(0) - \frac{F'(0)}{A} \, \tag{20}$$

$$N\mu\_s \text{Re}\_s^{-\frac{1}{2}} = -\theta'(0),\tag{21}$$

$$\operatorname{Sh}\_{\text{s}} \text{Re}\_{\text{s}}^{-\frac{1}{2}} = -\phi'(0),\tag{22}$$

$$M n\_{\sf s} \text{Re}\_{\sf s}^{-\frac{1}{2}} = -\chi'(0). \tag{23}$$
