## 2 Mathematical Formulation
The BCs for the considered model are:  
\[\overline{u}=\overline{u}_{w}=\frac{U_{0}\overline{x}}{L},\;\overline{v}=0, \;\frac{\partial\overline{T}}{\partial\overline{r}}=-h_{s}\overline{T},\; \text{at}\;\overline{r}=\overline{R}, \tag{7}\]  
\[\overline{u}\to 0,\overline{T}\rightarrow\infty,\;\text{as}\;\overline{r} \rightarrow\infty,\]  
where \(\overline{u}(\overline{x})=\frac{U_{0}\overline{x}}{L}\) is the linear velocity of the stretching cylinder, \(U_{0}\) denotes the reference velocity, \(L\) is referred to as the characteristic length, \(h_{s}\) indicates the heat transfer coefficient, and \(\overline{R}\) represents the radius of the cylinder. The relationship of the stream function \(\overline{\psi}\) with the velocity components is defined as:  
\[\overline{u}=\frac{1}{\overline{r}}\frac{\partial\overline{\psi}}{\partial \overline{r}},\;\overline{v}=-\frac{1}{\overline{r}}\frac{\partial\overline{ \psi}}{\partial\overline{x}}, \tag{8}\]  
Now, to transform governing Equations (4)-(6) into nonlinear ODEs, we introduced suitable set of transformations [31], defined as:  
\[\text{div}\ \overline{\
abla} =0,\]0 (9) \[\text{div}\ \overline{\
abla} =0,\]1  
\[\text{div}\ \overline{\
abla} =0,\]2  
Boundary conditions are transformed as:  
\[\text{div}\ \overline{\
abla} =0,\]3  
\[\text{div}\ \overline{\
abla} =0,\]4  
Here, \(K,\lambda,\Pr,\gamma,Gr,\gamma_{1}\), and \(Hp\) denote the curvature parameter, mixed convection parameter, Prandtl number, magnetic field parameter, Grashof number, Newtonian heating parameter, and heat generation/absorption parameters, respectively. The mathematical relations for such involved parameters are:  
\[\begin{array}{l}K=\frac{1}{K}\sqrt{\frac{U_{0}}{U_{0}}},\;\lambda=\frac{Gr} {\text{Re}^{2}x},\;\Pr=\frac{v_{Gr}}{K_{1}},\gamma=\sqrt{\frac{\sigma B^{2}U_ {0}}{\rho U_{  
Figure 4 shows that the rising value of the magnetic field parameter \(\gamma=0.1,0.5\) and \(0.9\) causes a reduction in the magnitude of fluid velocity \(f^{\prime}(\eta)\), for both the stretched plate and cylindrical cases. This is due to the resistive force, named Lorentz force, which is enhanced when the magnetic field parameter is increased. This force offers resistance to the fluid flow, and consequently, velocity reduces. It is also observed that for the flow field over a cylindrical surface, the velocity magnitude is higher in strength as compared with the velocity of fluid over a flat surface. Figures 5-11 illustrate the effects of physical parameters on temperature \(\theta(\eta)\). Figure 5 reveals that fluid temperature \(\theta(\eta)\) rises for increments in curvature parameter \(K=0.1,0.3\) and \(0.7\). This is because we noticed that for higher values of curvature parameter, the contact surface area reduces, and the resistance offered by the cylindrical surface reduces. This leads to an increase in the velocity of particles. The kinetic energy has a direct impact on the higher velocity. The higher velocity corresponds to a larger kinetic energy. The large value of average velocity results in higher temperature. The impacts of Prandtl numbers \(Pr=1.2,1.4\) and \(1.6\) on the temperature profile for the plate and cylinder is presented in \[\text{div}\ \overline{\
abla} =0,\]5 It is evident from the figure that variation in the Prandtl number causes a reduction in fluid temperature \(\theta(\eta)\), and a decline in the thermal boundary layer thickness is also observed for both the plate and cylinder. Here, the higher value of the Prandtl number causes a reduction in the thermal conductivity of fluid, and hence, the temperature profile for both the cylindrical and flat surface reduces significantly. It is observed from Figure 7 that increments in values of the parameter \(\gamma=0.1,0.7\) and \(1.0\) render an increase in temperature \(\theta(\eta)\). When the magnetic field increases, the Lorentz force increases. The resistance offered by the magnetic field causes particles to produce heat energy as a result, and the temperature for fluid flow over a flat and cylindrical surface increases.