# Bodewadt Flow Over a Permeable Disk with Homogeneous-Heterogeneous Reactions: A Numerical Study
## 3 Numerical Results and Discussion
Therefore, the damping effect, arising as a consequence of wall suction, can overcome the instabilities in Bodewadt flow known from the published literature. Figure 1: Velocity components \((H,F,\mathrm{and}\ G\) with and without suction effects. We determined the evolution of concentration profile \(\phi\) at different values of involved parameters, as shown in Figure 3a-d. Graphs of \(\phi\) against \(\zeta\) for various values of suction-strength parameters are depicted in Figure 3a. The concentration distribution has no meaningful solution in the absence of suction, an outcome that has been briefly explained in previous reports. Hence, we pursued the numerical solution of \(\phi\) for \(A\ \geq\ 1\). The general trend in the concentration profile a constant value at the wall and its value grows monotonically with increasing axial distance. The concentration boundary layer does not exist when \(K_{s}\ =\ 0\), which is also apparent from the condition in Equation (21). The concentration boundary layer appears to decrease when \(A\) increases. Also, the profile of \(\phi\) is broader at \(A\ =\ 1\) than at \(A\ =\ 3\). Figure 3b portrays the plots of \(\phi\) as the Schmidt number \(Sc\) changes. \(\phi\) decreases sharply for increasing values of \(Sc\). Physically, for a higher Schmidt number, \(Sc\ =\ \
u/D_{a}\), mass diffusion coefficient is very small compared with the momentum diffusion. To understand the onset of homogeneous reaction parameter \(K\), Figure 3c was plotted. The curves of \(\phi\) flatten as parameter \(K\) increases. Physically, by increasing either \(K\) or \(K_{s}\), more reactants are consumed in the boundary layer, thereby decreasing \(\phi(\zeta)\). Figure 2: Velocity components (\(H,F,\) and \(G\)) and \(\phi\) for different values of \(A\). The profiles of surface concentration \(\phi\)\((0)\) are presented in Figure 4a,b. A linear decay occurs in the surface concentration when either \(K\) or \(K_{\mathrm{s}}\) becomes large. Hence, for a fixed \(K_{\mathrm{s}}\), the wall concentration gradient \(\phi^{\prime}\)\((0)\) vanishes for sufficiently high values of \(K\). Physically, the chemical reaction occurs at some distance high above the disk when \(K\) is sufficiently large. However, Figure 4b shows that \(\phi\)\((0)\) grows non-linearly as \(Sc\) increases. Figure 4: Graph of surface concentration \((\phi\)\((0))\) versus (**a**) \(K\) and (**b**) \(Sc\) at different values of heterogeneous reaction parameter \(K_{\mathrm{s}}\). Figure 3: Concentration profile \((\phi)\) for different values of embedded parameters. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{\(A\)} & \multicolumn{2}{c}{\(-F\) (0)} & \multicolumn{2}{c}{\(G^{\prime}\) (0)} \\ \cline{2-5}  & **Present** & **[12]** & **Present** & **[12]** \\ \hline
0 & 0.9419 & 0.9420 & 0.7728 & 0.7729 \\ \hline
0.5 & 0.9033 & 0.9033 & 1.0650 & 1.0652 \\ \hline
1 & 0.8351 & 0.8351 & 1.3883 & 1.3885 \\ \hline
2 & 0.6468 & 0.6468 & 2.1560 & 2.1560 \\ \hline
3 & 0.4800 & 0.4800 & 3.0611 & 3.0611 \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of values of \(F^{\prime}(0)\) and \(G^{\prime}(0)\) obtained by [PERSON] and [PERSON] [12] for various suction-strength parameters.