# Bodewadt Flow Over a Permeable Disk with Homogeneous-Heterogeneous Reactions: A Numerical Study
## 3 Numerical Results and Discussion  
The influence of homogeneous-heterogeneous reactions on the Bodewadt flow with suction was the main focus of this research. Following [PERSON] and [PERSON] [12], we relied on MATLAB's (MathWorks, Massachusetts, USA) routine bvp4c to estimate velocity, temperature, and concentration distributions above the disk. During the computational procedure, a small value of \(\zeta_{\infty}\) was and then it was gradually increased while observing the residuals of \(F,G\), and \(\phi\) at infinity so that they remained less than \(10^{-6}\). The value of \(\zeta_{\infty}=20\) appears sufficient to meet the specified tolerance criterion. Like the thermal energy equation, the mass transfer problem in Bodewadt flow can exhibit a physically compatible solution only when a wall permeability condition is imposed. A comparison of the results for \(F^{\prime}(0)\) and \(G^{\prime}(0)\) with those estimated by [12], and such comparison seems convincing (Table 1). The asymptotic approximation of \(\phi\) (in Equation (30)) agrees well with the computational results for larger values of \(A\) (Table 2). Table 3 includes the numerical normalized skin friction data, far field axial velocity, and wall concentration gradient. Wall transpiration provides increases the skin friction encountered at the disk, flow rate, and species concentration at the disk. However, the suction effects contribute largely toward improvement in the wall cooling rate, which is vital for many technical processes. Table 4 shows that wall concentration increases with increasing values of \(Sc\). This is in compliance with higher values of \(Sc\) indicating relatively lower mass diffusion, thereby resulting in a thinner concentration boundary layer and higher wall concentration gradient. Increasing either \(K\) or \(K_{\mathrm{s}}\) results in more consumption of the reactants, thereby lowering surface concentration. Figure 1a,b show the evolution of velocity curves \((H,F,\mathrm{and}\ G)\) with and without the suction effect. The axial velocity remains constant at a certain distance above the disk, while its magnitude is reduced near the disk due to viscous drag. For the conservation principle to hold, the axial flow remains upward in pure Bodewadt flow. Velocity components have oscillatory profiles with amplitude inversely proportional to the axial distance. Note that the axial velocity at infinity becomes negative when sufficiently high suction velocity is imposed. Figure 2a-d shows the distribution of velocity components \((H,F,\mathrm{and}\ G)\) and concentration profile \((\phi)\) for a variety of suction strength parameters. The amplitude of oscillations considerably decreases when boundary suction is introduced in the model. The strength of the axial flow grows with increasing values of \(A\). However, the inward radial flow induced by the pressure gradient slows with increasing \(A\). The circumferential flow also decelerates with the inclusion of wall permeability. Figure 2a shows that the axial velocity profile \(H\) reaches some constant value \(H_{0}\) when a sufficiently high value of \(A\) is considered. The results in Figure 2a reveal that a higher volume of fluid is drawn toward the disk when a larger \(A\) is used. The asymptotic solution in Equation (25) shows that a solution for concentration would be possible only when \(H(\infty)\) is negative. For this reason, \(\phi\) has a physically inconsistent solution when \(A\ =\ 0\), in which \(H(\infty)\) has a positive value (Figure 1a). However, for \(A\ =\ 1\) where \(H\ (\infty)\) is negative (Figure 1b), a physically consistent solution for \(\phi\) is accomplished. In conclusion, the axially upward flow is a result of an unphysical solution to the advection-diffusion problem. This physical inconsistency can be overcome by reversing the direction of the axial flow via the application of sufficiently high suction velocity.