# Entropy Optimization in MHD Nanofluid Flow over an Exponential Stretching Sheet
## 6 Results and Discussion  
In this section, we present tabulated results and graphical representations of different features of the fluid flow and heat and mass transfer under the effects of several parameters. The quantities of engineering interest, such as skin fraction and Nusselt and Sherwood numbers are computed and presented in Table 1. The fluid flow equations were solved numerically using the multi-domain bivariate spectral quasi-linearization method for the chosen parameter values. A comparison with previously computed results is depicted in Table 2. These results are comparable to those of [PERSON] et al. [19] and [PERSON] [31], which show that this method is robust and gives accurate results. Figure 4a,c show increases in velocity and concentration with higher values of the variable viscosity parameter (\(\beta_{1}\)). This is true from the Poiseuille law, as the pressure and viscosity within the flow region increase, leading to the observed increase in both flow rate and concentration. The fluid temperature reduces, as most of the thermal energy is lost through convection and conduction to the walls of the sheet, as depicted in Figure 4b. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\mathbf{\xi}\) & \(\mathbf{Nt}\) & \(\mathbf{Nb}\) & \(\mathbf{Nr}\) & \(\mathbf{Pr}\) & \(\mathbf{Cf_{x}}\) & \(\mathbf{Nu_{x}}\) & \(\mathbf{Sh_{x}}\) \\ \hline
5 & & & & & 3.0874257 & \(-1.5088474\) & 0.4653439 \\
10 & 0.001 & 0.001 & 0.5 & 6.8 & 4.3131008 & \(-1.9976737\) & 0.9329592 \\
15 & & & & & 5.2604334 & \(-2.3949035\) & 1.3198495 \\
20 & & & & & 6.0620646 & \(-2.7411796\) & 1.6592534 \\ \hline  & 0.001 & & & & 5.187825 & 2.154772 & \(-0.008580\) \\
10 & 0.3 & 0.001 & 0.5 & 6.8 & 5.488176 & 1.989699 & \(-3.770309\) \\  & 0.5 & & & & 5.656798 & 1.881303 & \(-5.678613\) \\  & 0.7 & & & & 5.803213 & 1.776072 & \(-7.140323\) \\ \hline  & & 0.001 & & & 6.003369 & 2.503334 & \(-0.068861\) \\
15 & 0.001 & 0.1 & 0.5 & 6.8 & 5.992718 & 2.466485 & \(-0.033776\) \\  & & 0.15 & & & 5.988566 & 2.520528 & \(-0.027644\) \\  & & 0.2 & & & 5.970386 & 3.233199 & \(-0.013723\) \\ \hline
10 & 0.001 & 0.001 & 0.001 & 0.1 & 6.8 & 5.198892 & 2.432598 & \(-0.011509\) \\  & & & 0.2 & & 5.196294 & 2.362772 & \(-0.010779\) \\  & & & 0.2 & & 5.193907 & 2.301212 & \(-0.010132\) \\  & & & 0.3 & & 5.191718 & 2.246864 & \(-0.009559\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Skin friction, and heat and mass transfer coefficients for \(\beta_{1}=0.5,Gr_{T}=Gr_{C}=0.5\), \(\lambda=-0.09\), \(Ec=0.05\), \(Le=0.4\), and \(R_{1}=0.3\). \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{Pr}\) & **[PERSON] et al. [19]** & **[PERSON] [31]** & **Present Results** \\  & **Finite-Difference Method** & **Keller Box Method** & **MD-BSQLM** \\ \hline
0.1 & 0.142003 & 0.140034 & 0.1400343 \\
1.0 & 0.332173 & 0.332057 & 0.3320571 \\
10.0 & 0.728310 & 0.728141 & 0.7281412 \\
100.0 & 1.572180 & 1.571831 & 1.5718323 \\
1000.0 & 3.388090 & 3.387083 & 3.3870854 \\
10,000.0 & 7.300800 & 7.297402 & 7.2974001 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the values of \(-\theta^{\prime}(0,0)\) for different values of \(\mathbf{Pr}\). Table 1 gives the computed skin friction and heat and mass transfer coefficients for different values of the selected parameters. It is noted that the coefficient of skin friction increases for higher values of the non-dimensional number (\(\xi\)), thermal radiation (\(Nr\)), and thermophoresis parameters (\(Nt\)). The dynamic pressure on the surface of the sheet is increased due to the force generated by an increased temperature gradient. The heat transfer coefficient was observed to decrease with higher values of \(Nr\) and \(Nt\) and increase with \(\xi\). The exponentially stretching sheet of the vibrating sheet emits thermal energy through conduction of the fluid particles, which causes the Nusselt number to decrease.