**2. Materials and Methods**

An incompressible electrically conducting viscous nanofluid flow via a vertical cone with bioconvection is explored in two dimensions as an axisymmetric, steady, natural convective ferrofluid flow. Furthermore, it is postulated that temperature and concentration are non-uniform at the surface due to the influence of heat generation/absorption, chemical reaction and viscous dissipation. The flow is electrically magnetized by a magnetic dipole, and a Darcy–Forchheimer porous medium model is also used. Thermal radiation exists as a unidirectional flux in the transverse to the cone surface (*s*-direction). In comparison to the *s*direction, the radiation heat flux in the *x*-direction is considered neglected. The *x*-axis of the chosen coordinate system corresponds to the direction of flow over the cone surface. *Tw* is taken to be the temperature at the cone's surface (*s* = 0), and the concentration is governed by the condition *DB <sup>∂</sup><sup>C</sup> <sup>∂</sup><sup>s</sup>* <sup>+</sup> *DT T*∞ *∂T <sup>∂</sup><sup>s</sup>* = 0 at the cone's surface, where *T*<sup>∞</sup> is the temperature and *C*∞ is the concentration and *N*∞ density of microorganisms in the ambient nanofluid.

The boundary layer equation [12,14,21] based on the assumptions stated above are the equations of continuity and momentum as well as energy, concentration, and microorganisms:

$$\frac{\partial (ru)}{\partial x} + \frac{\partial \{(rw)\}}{\partial s} = 0,\tag{1}$$

$$w\frac{\partial u}{\partial s} + u\frac{\partial u}{\partial x} = \frac{\mu\_f}{\rho\_f}\frac{\partial^2 u}{\partial s^2} - \frac{\mu\_f}{k\_o^\*}u - \frac{\rho\_f \mathbf{C}\_b}{\sqrt{k\_o^\*}}u^2 + \lambda\_o M \frac{\partial H}{\partial x}$$

$$+ g\left[\beta\_T(T - T\_\infty) + \beta\_C(\mathbf{C} - \mathbf{C}\_\infty) + \beta\_N(N - N\_\infty)\right] \cos(\alpha),\tag{2}$$

$$w\frac{\partial T}{\partial \mathbf{s}} + u\frac{\partial T}{\partial \mathbf{x}} = a\_f \frac{\partial^2 T}{\partial \mathbf{s}^2} + \frac{\mu\_f}{(\rho c\_p)\_f} \left(\frac{\partial u}{\partial \mathbf{s}}\right)^2 + \left(u\frac{\partial H}{\partial \mathbf{x}} + w\frac{\partial H}{\partial \mathbf{s}}\right) \frac{\lambda\_0}{(\rho c\_p)\_f} T \frac{\partial M}{\partial T}$$

$$\tau \left[D\_B \frac{\partial C}{\partial \mathbf{s}} \frac{\partial T}{\partial \mathbf{s}} + \frac{D\_T}{T\_\infty} \left(\frac{\partial T}{\partial \mathbf{s}}\right)^2\right] - \frac{1}{(\rho c\_p)\_f} \frac{\partial q\_r}{\partial \mathbf{s}},\tag{3}$$

$$w\frac{\partial \mathcal{C}}{\partial \mathbf{s}} + u\frac{\partial \mathcal{C}}{\partial \mathbf{x}} = D\_B \frac{\partial^2 \mathcal{C}}{\partial \mathbf{s}^2} + \frac{D\_T}{T\_{\infty}} \frac{\partial^2 T}{\partial \mathbf{s}^2} - K\_I (\mathcal{C} - \mathcal{C}\_{\infty}),\tag{4}$$

$$
\hbar \frac{\partial N}{\partial \mathbf{s}} + \mu \frac{\partial N}{\partial \mathbf{x}} + \frac{bN\mathsf{V}\_{\mathbf{c}}}{\mathsf{C}\_{w} - \mathsf{C}\_{\infty}} \left( \frac{\partial \mathsf{C}}{\partial \mathbf{s}} \frac{\partial N}{\partial \mathbf{s}} + N \frac{\partial^{2} \mathsf{C}}{\partial \mathbf{s}^{2}} \right) = D\_{n} \frac{\partial^{2} N}{\partial \mathbf{s}^{2}},\tag{5}
$$

with initial boundary conditions

$$\mu = 0, \text{ w} = \mathcal{W}\_{\text{W}}, \ T = T\_{\text{W}}, \ -D\_{B} \frac{\partial \mathcal{C}}{\partial \mathbf{s}} = \frac{D\_{T}}{T\_{\infty}} \frac{\partial T}{\partial \mathbf{s}}, \ N = N\_{\text{w}} \text{ at} \quad \mathbf{s} = \mathbf{0}, \tag{6}$$

$$u \to 0, \quad w \to 0, \quad T \to T\_{\infty \prime} \quad \mathbb{C} \to \mathbb{C}\_{\infty \prime} \quad N \to N\_{\infty \prime} \quad \text{as} \quad s \to \infty,\tag{7}$$

where (*u*,*w*) are the velocity components in the *x*-direction (radial) and *s*-direction (transverse), respectively; *T*, *C*, *N* are the temperature, concentration, and gyrotactic microorganism, respectively; the diffusion coefficients named Brownian, thermophoresis, and microorganism correspond to *DB*, *DT*, and *Dn*, respectively; while *τ* is the ratio of heat capacitance, fluid density is *ρ<sup>f</sup>* , thermal conductivity of fluid is *k <sup>f</sup>* , electrical conductivity of fluid is *σ* , the dynamic viscosity is *μ<sup>f</sup>* , thermal diffusivity of base fluid is *α<sup>f</sup>* , magnetic permeability is *λo*, heat capacitance of fluid is (*ρcp*)*<sup>f</sup>* , first order chemical reaction parameter is *Kr*, speed of gyrotactic cell is *Wc*, and *b* is chemotaxis.
