$\frac{1}{r}\cdot \frac{\partial }{\partial r}\left[r\left(\mu \frac{\partial v_z}{\partial r}\right)\right]-\frac{\partial p}{\partial z}=\frac{\partial \left(\rho v_z\right)}{\partial t}\frac{\partial \rho }{\partial t}+v_z\frac{\partial \rho }{\partial z}=-\rho \frac{\partial v_z}{\partial z}=0$

$\frac{1}{r}\frac{\text{d}{E}_z}{\text{d}r}+\frac{{\text{d}}^2{E}_z}{\text{d}{r}^2}+k^2{E}_z=0$ alternatively $\frac{1}{r}\frac{\text{d}{E}_{\theta }}{\text{d}r}+\frac{{\text{d}}^2{E}_{\theta }}{\text{d}{r}^2}-\frac{E_{\theta }}{r^2}+k^2{E}_{\theta }=0$

$E_z=A_z\ast J_0\left(k\cdot r\right)+B_z\ast Y_0\left(k\cdot r\right)$ alternatively $E_{\theta }=A_{\theta }\ast J_1\left(k\cdot r\right)+B_{\theta }\ast Y_1\left(k\cdot r\right)$

$-v_z\left(r\right)\frac{\partial C_A}{\partial z}+\frac{\partial }{\partial z}\left(\mathfrak{D}\frac{\partial C_A}{\partial z}\right)+\frac{1}{r}\frac{\partial }{\partial r}\left[r\left(\mathfrak{D}\frac{\partial C_A}{\partial r}\right)\right]-\left(-r_A\right)=\frac{\partial C_A}{\partial t}$

$-v_z\left(r\right)\frac{\partial \left(\rho c_{p}T\right)}{\partial z}+\frac{\partial }{\partial z}\left(\mathcal{K}\frac{\partial T}{\partial z}\right)+\frac{1}{r}\frac{\partial }{\partial r}\left[r\left(\mathcal{K}\frac{\partial T}{\partial r}\right)\right]+\left(-r_A\right)\left(-\Delta H_r\right)+\{\begin{array}{l}\pi f{\epsilon }_0{\epsilon }^{″}{\left|E_z\right|}^2\\ \pi f{\epsilon }_0{\epsilon }^{″}{\left|E_{\theta }\right|}^2\end{array}=\rho c_p\frac{\partial T}{\partial t}$

$-r_A=\frac{{\left(-r_A\right)}_{\max }{C}_A{C}_B}{C_A{C}_B+k_B{C}_A\left(1+\frac{C_A}{k_{iA}}\right)+k_A{C}_B\left(1+\frac{C_B}{k_{iB}}\right)}$

$t=0$

$v_z=0$

$C_A=0$

$\forall r,\forall z$

$r=0$

$\frac{\partial v_z}{\partial r}=0$

$\frac{\partial C_A}{\partial r}=0$

$\frac{\partial T}{\partial r}=0$

$\frac{\text{d}{E}_z}{\text{d}r}=0$

alternatively

$\frac{\text{d}{E}_{\theta }}{\text{d}r}=0$

$\forall z,\forall t>0$

$r=R$

$v_z=0$

$\frac{\partial C_A}{\partial r}=0$

$\frac{\partial T}{\partial r}=0$

$E_z=E_{z0}$

alternatively

$E_{\theta }=E_{\theta 0}$

$\forall z,\forall t>0$

$z=0$

$p=p_0$

$C_A=C_{A0}$

$T=T_0$

$\forall r,\forall t>0$

$z=L$

$p=p_L$

$\frac{\partial C_A}{\partial z}=0$

$\frac{\partial T}{\partial z}=0$

$\forall r,\forall t>0$