## 2 Problem Formulation  
Steady two-dimensional, nonlinear, laminar magnetohydrodynamic convective flow of incompressible viscous nanofluid over a stretching sheet with variable viscosity and variable thermal conductivity are considered. Gyrotactic micro-organisms are suspended in the nanofluid. The \(\overline{x}\)-axis is along the stretching surface while \(\overline{y}\)-axis is normal to it. It is assumed that the stretching velocity \(u_{w}(\overline{x})=a\overline{x}\) where \(a\) is a constant. The fluid is electrically-conducting owing to the presence of magnetic nanoparticles. A constant transverse magnetic field \(B_{0}\) is imposed normal to the stretching sheet. Since magnetic Reynolds number is small, the induced magnetic field is thereby deemed to have been negated. The assumption is made that the temperature of the melting surface is \(T_{m}\) while the temperature in the free stream is \(T_{\infty}\) such that \(T_{m}<T_{\infty}\). The nanoparticle volume fraction, \(C_{w}\), is assumed constant at the stretching surface and at the free stream is denoted as \(C_{\infty}\). Motile microorganisms \(n_{w}\) are assumed to be at the wall. \(n_{\infty}\) is assumed as microorganism at ambient values. The physical model and associated Cartesian coordinate system are illustrated in Figure 1.  
The governing equations for continuity, momentum, energy, nanoparticle volume fraction (concentration) and microorganism may be stated following Mabood, Mastroberardino [48] and Animasaun [51] with modifications for bioconvection species conservation and variable properties as:  
Continuity:  
\[\frac{\partial\overline{u}}{\partial\overline{x}}+\frac{\partial\overline{v}}{ \partial\overline{y}}=0 \tag{1}\]  
Momentum:  
\[\overline{u}\frac{\partial\overline{u}}{\partial\overline{x}}+\overline{v} \frac{\partial\overline{u}}{\partial\overline{y}}=\frac{1}{\rho_{\infty}} \frac{\partial}{\partial\overline{y}}\bigg{[}\mu(T)\frac{\partial\overline{u} }{\partial\overline{y}}\bigg{]}-\frac{\sigma B_{0}^{2}}{\rho_{f}}\overline{u} \tag{2}\]  
Energy:  
\[\overline{u}\frac{\partial T}{\partial\overline{x}}+\overline{v}\frac{ \partial T}{\partial\overline{y}}=\frac{1}{\rho_{\infty}c_{p}}\frac{\partial }{\partial\overline{y}}\bigg{[}k(T)\frac{\partial T}{\partial\overline{y}} \bigg{]}-\tau D_{B}(C)\frac{\partial T}{\partial\overline{y}}\frac{\partial C }{\partial\overline{y}}+\tau\frac{D_{T}}{T_{\infty}}\bigg{(}\frac{\partial T }{\partial\overline{y}}\bigg{)}^{2}+\frac{1}{\rho_{\infty}c_{p}}\mu(T)\bigg{(} \frac{\partial\overline{u}}{\partial\overline{y}}\bigg{)}^{2} \tag{3}\]  
Concentration:  
\[\overline{u}\frac{\partial C}{\partial\overline{x}}+\overline{v}\frac{ \partial C}{\partial\overline{y}}=\frac{\partial}{\partial\overline{y}} \bigg{[}D_{B}(C)\frac{\partial C}{\partial\overline{y}}\bigg{]}+\frac{D_{T}}{T _{\infty}}\frac{\partial^{2}T}{\partial\overline{y}^{2}} \tag{4}\]  
Microorganism:  
\[\overline{u}\frac{\partial\overline{n}}{\partial\overline{x}}+\overline{v} \frac{\partial\overline{n}}{\partial\overline{y}}+\frac{\overline{b}W_{c}}{C _{w}-C_{\infty}}\bigg{[}\frac{\partial}{\partial\overline{y}}\bigg{(} \overline{n}\frac{\partial C}{\partial\overline{y}}\bigg{)}\bigg{]}=\frac{ \partial}{\partial\overline{y}}\bigg{[}D_{m}(C)\frac{\partial\overline{n}}{ \partial\overline{y}}\bigg{]} \tag{5}\]  
The relevant boundary condition is  
\[\overline{u}=\overline{u}_{w}+\overline{u}_{slip},\ T=T_{m},\ k(T)\bigg{(} \frac{\partial T}{\partial\overline{y}}\bigg{)}_{y=0}=\rho_{\infty}[\lambda+c _{s}(T_{m}-T_{0})]v(\overline{x},0),\ C=C_{w},\ n=n_{w}\ \text{at}\ \overline{y}=0, \tag{6}\]  
\(\overline{u}\to 0,\ T\to T_{\infty},\ C\to C_{\infty},\ n_{\infty}\to 0\) as \(\overline{y}\to\infty\). Following studies by [PERSON] et al.