## 2 Mathematical Formulation  
We considered the two-dimensional convective flow of non-Newtonian fluid past a stretching cylinder. The Casson fluid model is taken in this case as a non-Newtonian fluid. The magnetic field \(B_{0}\) is induced along the radial direction of the cylinder. Moreover, the Newtonian heating condition is also imposed at the surface. We assumed that the temperature at the cylindrical surface was higher than the temperature outside the surface (i.e., ambient temperature), where \(\overline{T}>\overline{T}_{\infty}\). The geometry of problem is given in Figure 1. For viscous cases, the basic conservation equations, continuity, momentum, and thermal energy are taken as. \[\text{div}\ \overline{\
abla} =0,\] (1) \[\rho\frac{D\overline{\
abla}}{D\overline{t}} =\overrightarrow{\
abla}.\overline{\mathbf{T}}+\mu\overrightarrow{ \
abla}^{2}\overline{\mathbf{\
abla}}+\overline{\mathbf{J}}\times\mathbf{B},\] (2) \[\rho c_{p}\frac{d\overline{T}}{d\overline{t}} =K_{1}\overline{\
abla}^{2}\overline{T}.\] (3) With the following assumptions:  
* For axisymmetric flow, there is no tangential velocity (\(\overline{n}_{\theta}=0\));
* Taking velocity field as \(\overline{\
abla}=[\overline{v}(\overline{x},\overline{r}),\ 0,\overline{n}( \overline{x},\overline{r})]\);
* Approximation of the boundary layer is incorporated here; accordingly, the pressure gradient is zero along the flow \(p=p_{0}\), and variation in velocity along the radial direction is much smaller than in the axial direction, and the scale analysis for momentum equation with boundary layer theory is defined as: \(\overline{u}\gg\overline{v},\ \frac{\partial\overline{u}}{\partial \overline{v}}>>\frac{\partial\overline{u}}{\partial\overline{v}},\frac{ \partial\overline{v}}{\partial\overline{v}},\frac{\partial\overline{v}}{ \partial\overline{v}}\Rightarrow\frac{\partial p}{\partial\overline{r}}=0\). By using all the above assumptions and approximation, Equations (1)-(3), with the assumption of Casson fluid, can be reduced to the following component form:  
\[\frac{\partial(\overline{r}\;\overline{u})}{\partial\overline{x}}+\frac{\partial (\overline{r}\;\overline{v})}{\partial\overline{r}}=0, \tag{4}\]  
\[\overline{u}\frac{\partial\overline{u}}{\partial\overline{x}}+\overline{v} \frac{\partial\overline{u}}{\partial\overline{r}}=v\bigg{(}1+\frac{1}{ \beta}\bigg{)}\bigg{(}\frac{\partial^{2}\overline{u}}{\partial\overline{r}^{2 }}+\frac{1}{\overline{r}}\frac{\partial\overline{u}}{\partial\overline{r}} \bigg{)}-\sigma\frac{B_{0}{}^{2}}{\rho}\overline{u}+g\beta^{*}{}_{T}( \overline{T}-\overline{T}_{\infty}), \tag{5}\]  
\[\overline{u}\frac{\partial\overline{T}}{\partial\overline{x}}+\overline{v} \frac{\partial\overline{T}}{\partial\overline{r}}=\frac{K_{1}}{\rho c_{p}} \bigg{(}\frac{\partial^{2}\overline{T}}{\partial\overline{r}^{2}}+\frac{1}{ \overline{r}}\frac{\partial\overline{T}}{\partial\overline{r}}\bigg{)}+ \frac{Q_{0}}{\rho c_{p}}(\overline{T}-\overline{T}_{\infty}). \tag{6}\]  
Here, \(\overline{u}\) is the \(\overline{x}\)component of velocity and \(\overline{v}\) is the \(\overline{r}\)components of velocity taken along axial direction and radial direction, respectively, \(v\) denotes the kinematic viscosity, \(\sigma\) indicates electric conductivity, and \(\rho\) is the fluid density. \(B_{0}\) represents the magnetic field constant, \(g\) is used for the gravitational acceleration, \(Q_{0}\) represents the heat generation/absorption coefficient, and \(\beta^{*}{}_{T}\) indicates the thermal expansion coefficient. \(\beta,\overline{T},\overline{T}_{\infty}\) represent the Casson fluid parameter, surface temperature, and ambient temperature, respectively.