# A Compact Numerical Scheme for the Heat Transfer of Mixed Convection Flow in Quantum Calculus
## 5 Applications to Engineering Problems
Under the assumptions of boundary layer theory, the governing equations of the flow can be expressed as  
\[\frac{\partial u^{*}}{\partial x^{*}}+\frac{\partial v^{*}}{\partial y^{*}}=0 \tag{62}\]  
\[\frac{\partial u^{*}}{\partial t^{*}}=v\frac{\partial^{2}u^{*}}{\partial y^{* }}+g\Big{(}\Lambda_{1}(T-T_{\infty})+\Lambda_{2}(T-T_{\infty})^{2}\Big{)} \tag{63}\]  
\[\frac{\partial T}{\partial t^{*}}=\alpha\frac{\partial^{2}T}{\partial y^{*2}} \tag{64}\]  
subject to the boundary conditions  
\[u^{*}(t^{*},y^{*})=U_{w}=ax,\ T(t^{*},y^{*})=T_{w}\text{ when }y^{*}=0 \tag{65}\]  
\[u^{*}(t^{*},y^{*})\to 0,\ T(t^{*},y^{*})\to T_{\infty}\text{ when }y^{*}\to\infty \tag{66}\]  
and for oscillatory plate, the boundary conditions are expressed as  
\[u(t,-0.2)=\frac{1}{\Big{(}1+e^{\sqrt{\frac{2}{5}}(-0.2)}-\frac{5\rho}{6}t \Big{)}^{2}}\ \&\ u(t,0.8)=\frac{1}{\Big{(}1+e^{\sqrt{\frac{2}{5}}(0.8)}-\frac{5\rho}{6}t \Big{)}^{2}}, \tag{59}\]0  
\[u(t,-0.2)=\frac{1}{\Big{(}1+e^{\sqrt{\frac{2}{5}}(-0.2)}-\frac{5\rho}{6}t \Big{)}^{2}}\ \&\ u(t,0.8)=\frac{1}{\Big{(}1+e^{\sqrt{\frac{2}{5}}(0.8)}-\frac{5\rho}{6}t \Big{)}^{2}}, \tag{59}\]1  
\[u(t,-0.2)=\frac{1}{\Big{(}1+e^{\sqrt{\frac{2}{5}}(-0.2)}-\frac{5\rho}{6}t \Big{)}^{2}}\ \&\ u(t,0.8)=\frac{1}{\Big{(}1+e^{\sqrt{\frac{2}{5}}(0.8)}-\frac{5\rho}{6}t \Big{)}^{2}}, \tag{59}\]2 Impact of \(q\) on the solution of the problem in Example 1. The transformations for linearly stretching sheet are  
\[u(t,-0.2)=\frac{1}{\Big{(}1+e^{\sqrt{\frac{2}{5}}(-0.2)}-\frac{5\rho}{6}t \Big{)}^{2}}\ \&\ u(t,0.8)=\frac{1}{\Big{(}1+e^{\sqrt{\frac{2}{5}}(0.8)}-\frac{5\rho}{6}t \Big{)}^{2}}, \tag{59}\]3  
and the transformations for oscillatory sheet are  
\[u(t,-0.2)=\frac{1}{\Big{(}1+e^{\sqrt{\frac{2}{5}}(-0.2)}-\frac{5\rho}{6}t \Big{)}^{2}}\ \&\ u(t,0.8)=\frac{1}{\Big{(}1+e^{\sqrt{\frac{2}{5}}(0.8)}-\frac{5\rho}{6}t \Big{)}^{2}}, \tag{59}\]4  
The governing Equations (62)-(64) are transformed to the following non-dimensional set of equations and consider the spatial \(q\)-derivative instead of the classical spatial derivative:  
\[u(t,-0.2)=\frac{1}{\Big{(}1+e^{\sqrt{\frac{2}{5}}(-0.2)}-\frac{5\rho}{6}t \Big{)}^{2}}\ \&\ u(t,0.8)=\frac{1}{\Big{(}1+e^{\sqrt{\frac{2}{5}}(0.8)}-\frac{5\rho}{6}t \Big{)}^{2}}, \tag{59}\]5  
\[u(t,-0.2)=\frac{1}{\Big{(}1+e^{\sqrt{\frac{2}{5}}(-0.2)}-\frac{5\rho}{6}t \Big{)}^{2}}\ \&\ u(t,0.8)=\frac{1}{\Big{(}1+e^{\sqrt{\frac{2}{5}}(0.8)}-\frac{5\rho}{6}t \Big{)}^{2}}, \tag{59}\]6  
subject to the boundary conditions  
\[ \begin{array}{c}u(t,\;y)=1,\;T(t,y)=1,\;when\;y=0\\ u(t,\;y)\to 0,\;T(t,y)\to 0,\;when\;y\rightarrow\infty\end{array} \right\}, \tag{72}\]  
\[ \begin{array}{c}u(t,\;y)=cos(t),\;T(t,y)=1,\;when\;y=0\\ u(t,\;y)\to 0,\;T(t,y)\to 0,\;when\;y\rightarrow\infty\end{array} \right\} \tag{73}\]  
and for oscillatory plate, dimensionless boundary conditions are expressed as  
\[ \begin{array}{c}u(t,\;y)=cos(t),\;T(t,y)=sin(t),\;when\;y=0\\ u(t,\;y)\to 0,\;T(t,y)\to 0,\;when\;y\rightarrow\infty\end{array} \right\} \tag{74}\]  
The linearized form of Equation (70) with Equation (71) using boundary conditions (72)-(74) are solved numerically using the proposed scheme. A comparison of the two schemes is shown in Figure 4 over independent variables \(y\) in finding the relative error. Figure 4 indicates that the proposed scheme for spatial direction with the Runge-Kutta scheme for time direction produces less relative error than the one obtained by the Crank-Nicolson method in time direction for this particular problem. Figures 5 and 6 show the solution to [PERSON] first and second problems by applying Runge-Kutta and Crank-Nicolson schemes in time direction. For this case, Crank-Nicolson time discretization performs better than the Runge-Kutta time discretization scheme. Figure 7 shows the variation of \(q\) on the velocity and temperature profiles. Both profiles decay by decreasing the value of \(q\). The impact of mixed convection variable \(\lambda\) on the velocity profile is shown in Figure 8.  
The velocity escalates by incrementing the mixed convection variable \(\lambda\).