## 2 Mathematical Formulation
The impact of the Newtonian heating parameter \(\gamma_{1}\) on temperature profile is illustrated in Figure 8 for both flat and cylindrical surfaces. We observed that the Newtonian heating parameter has a direct impact on the temperature profile. Furthermore, it is seen that the magnitude of the temperature profile is higher for cylindrical surfaces as compared with a flat plate for increasing values of the parameter \(\gamma_{1}\Figure 9 and 10 demonstrate the impact of \(Hp^{+}\) = 0.1, 0.2, 0.3 (heat generation) parameter and \(Hp^{-}=-0.1\), \(-0.2\), \(-0.3\) (heat absorption) parameter on temperature profile for both cylindrical and flat plate cases. From Figure 9, we observed that higher values of heat generation parameters produce heat energy, and as a result, the temperature of the fluid enhances. Such a trend is similar in the case of the flat and cylindrical surfaces. Figure 10 depicts the impact of heat absorption parameters on fluid temperature. We noticed that the temperature of Casson fluid over a flat and cylindrical surface decreases the function of the heat generation parameter. \[\text{div}\ \overline{\
abla} =0,\]6 Effect of \(K\) on \(\theta(\eta)\). \[\text{div}\ \overline{\
abla} =0,\]7 Effect of \(Pr\) on \(\theta(\eta)\) for plate/cylinder. \[\text{div}\ \overline{\
abla} =0,\]8 Impact of \(Hp^{-}\) on \(\theta(\eta)\) for plate/cylinder. Tables 1 and 2 analyze the impacts of emerging parameters on skin friction coefficient for plate and cylindrical surfaces, respectively. Particularly, Table 1 reports the skin friction coefficient variations at the flat surface for different values of mixed convection parameter \(\lambda\) and Casson fluid parameter \(\beta\), and the magnetic field parameter \(\gamma\). We noticed that the skin friction coefficient tends to decrease (in an absolute sense) for positive values of the Casson fluid parameter and the mixed convection parameter. Furthermore, the outcomes are in contrast for a magnetic field parameter \(\gamma\) that is for higher values of magnetic field  
\[\text{div}\ \overline{\
abla} =0,\]9 Effect of \(Hp^{+}\) on \(\theta(\eta)\) for plate/cylinder. parameter the skin friction coefficient enhances. Table 2 shows the impact of the Casson fluid parameter, mixed convection, and magnetic field parameter on the skin friction coefficient at the cylindrical surface. We observed that for both the positive Casson fluid parameter and mixed convection parameter, the skin friction coefficient declines, whereas the opposite is the case for magnetic field parameter. From both Tables 1 and 2, we have seen that the magnitude for cylindrical surface is enlarged as compared with flat plate. The influences of the Prandtl number, heat generation and heat absorption parameters, and the Newtonian heating parameter on heat transfer rate on both flat plate and cylindrical surfaces are presented in Tables 3 and 4, respectively, with a fixed Casson fluid parameter \(\beta=0.3\), mixed convection parameter \(\lambda=0.1\), and magnetic field parameter \(\gamma=0.3\). From Tables 3 and 4, we seen that the heat transfer rate for both the plate and cylinder increases significantly (in an absolute sense) for higher values of \(Hp^{+}\) and Newtonian heating parameter \(\gamma_{1}\), whereas \(Pr\) and \(Hp^{-}\) have a reducing impact on the heat transfer rate. Table 5 presents comparative values of Nusselt number with the existing literature. We found an excellent match that validates the present results.