# Bodewadt Flow Over a Permeable Disk with Homogeneous-Heterogeneous Reactions: A Numerical Study
## 2 Problem Statement  
Suppose that a viscous fluid revolves about the vertical axis with uniform angular velocity \(\omega\), sufficiently high above a stationary surface aligned with the plane \(z\ =\ 0\). The considered surface is permeable with \(V\) representing wall transpiration velocity. In line with [2], flow field is assumed to be steady, laminar, and incompressible, and it comprises chemically reacting species. Because the problem exhibits cylindrical symmetry, we work in cylindrical coordinates \((r,\varphi,z)\), with the corresponding velocity components being denoted by \(\left(V_{r},V_{\varphi},V_{z}\right)\). We consider a cubic autocatalytic reaction [15]:  
\[\alpha+2\beta\to 3\beta;\ rate=k_{c}ab^{2}, \tag{1}\]whereas a single, isothermal, and first-order reaction occurs at the catalyst surface:  
\[\alpha\rightarrow\beta;\ rate=k_{s}a \tag{2}\]  
where \(a\) and \(b\) represent the concentrations of chemical species \(\alpha\) and \(\beta\), respectively; and \(k_{c}\) and \(k_{s}\) are constants. Given the above assumptions, the constitutive relationships result in the following equations:  
\[\frac{\partial V_{r}}{\partial r}+\frac{V_{r}}{r}+\frac{\partial V_{z}}{ \partial z}=0, \tag{3}\]  
\[\rho\bigg{(}V_{r}\frac{\partial V_{r}}{\partial r}-\frac{V_{\varphi}^{2}}{r} +V_{z}\frac{\partial V_{r}}{\partial z}\bigg{)}=-\frac{\partial p}{\partial r }+\mu\bigg{(}\frac{\partial^{2}V_{r}}{\partial r^{2}}+\frac{\partial}{\partial r }\bigg{(}\frac{V_{r}}{r}\bigg{)}+\frac{\partial^{2}V_{r}}{\partial z^{2}} \bigg{)}, \tag{4}\]  
\[\rho\bigg{(}V_{r}\frac{\partial V_{\varphi}}{\partial r}+\frac{V_{r}V_{ \varphi}}{r}+V_{z}\frac{\partial V_{\varphi}}{\partial z}\bigg{)}=\mu\bigg{(} \frac{\partial^{2}V_{\varphi}}{\partial r^{2}}+\frac{\partial}{\partial r} \bigg{(}\frac{V_{\varphi}}{r}\bigg{)}+\frac{\partial^{2}V_{\varphi}}{\partial z ^{2}}\bigg{)}, \tag{5}\]  
\[\rho\bigg{(}V_{r}\frac{\partial V_{z}}{\partial r}+V_{z}\frac{\partial V_{z}}{ \partial z}\bigg{)}=-\frac{\partial p}{\partial z}+\mu\bigg{(}\frac{\partial^ {2}V_{z}}{\partial r^{2}}+\frac{1}{r}\frac{\partial V_{z}}{\partial r}+\frac{ \partial^{2}V_{z}}{\partial z^{2}}\bigg{)}, \tag{6}\]  
\[V_{r}\frac{\partial a}{\partial r}+V_{z}\frac{\partial a}{\partial z}=D_{ \alpha}\bigg{(}\frac{\partial^{2}a^{2}}{\partial r^{2}}+\frac{1}{r}\frac{ \partial a}{\partial r}+\frac{\partial^{2}a}{\partial z^{2}}\bigg{)}-k_{c}ab^{ 2}, \tag{7}\]  
\[V_{r}\frac{\partial b}{\partial r}+V_{z}\frac{\partial b}{\partial z}=D_{ \beta}\bigg{(}\frac{\partial^{2}b}{\partial r^{2}}+\frac{1}{r}\frac{\partial b }{\partial r}+\frac{\partial^{2}b}{\partial z^{2}}\bigg{)}+k_{c}ab^{2}. \tag{8}\]  
where \(\rho\) denotes the fluid density, \(\mu\) is the dynamic viscosity, \(D_{\alpha}\) and \(D_{\beta}\) are the respective diffusion coefficients, and \(p\) represents pressure. Outside the boundary layer, the centrifugal force is balanced by radial pressure gradient. Mathematically,  
\[\frac{\partial p}{\partial r}=\rho r\omega^{2}. \tag{9}\]  
The flow problem is subjected to the following conditions:  
\[ \begin{array}{ll}V_{r}=0,&V_{\varphi}=0,&V_{z}=-V,&D_{\alpha}\frac{ \partial t}{\partial z}=k_{s}a,&D_{\beta}\frac{\partial b}{\partial z}=-k_{s}a \text{ at }z=0,\\ &V_{r}\to 0,&V_{\varphi}\to r\omega,&a\to a,&b\to 0 \text{ as }z\rightarrow\infty.\end{array}  \tag{10}\]  
Using von-Karman variables together with new functions \(\phi(\zeta)\) and \(h(\zeta)\):  
\[ \begin{array}{ll}(V_{r},V_{\varphi},V_{z})=\Big{(}r\omega F(\zeta),r\omega G (\zeta),&\sqrt{\
u\omega}H(\zeta)\Big{)},&(a,b)=(a_{0}\phi(\zeta),&a_{0}h( \zeta)),\end{array}  \tag{11}\]  
where \(\zeta=z(\omega/\
u)^{1/2}\) is the similarity variable.