$\mu \left(\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial v_z}{\partial r_s}\right)\right)+\frac{P_{in}-P_{out}}{L}=\rho \frac{\partial v_z}{\partial t}$

$-2v_m\left[1-{\left(\frac{r}{R}\right)}^2\right]\frac{\partial C_s}{\partial z}+\mathfrak{D}\frac{{\partial }^2{C}_s}{\partial z^2}+\mathfrak{D}\left[\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial C_s}{\partial r}\right)\right]-\left(-r_s\right)=\frac{\partial C_s}{\partial t}$

$-2\rho cp{v}_m\left[1-{\left(\frac{r}{R}\right)}^2\right]\frac{\partial T}{\partial z}+k\frac{{\partial }^{2}T}{\partial z^2}+k\left[\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial T}{\partial r}\right)\right]-\left(-r_s\right)\left(-\Delta H_r\right)+16\mu v_m^2\frac{r^2}{R^4}=\rho cp\frac{\partial T}{\partial t}$

where $\left(-r_s\right)=\frac{K_3{C}_E{C}_S}{k_m\left(1+\frac{C_{S0}-C_S}{k_I}\right)+C_S+\frac{C_S^2}{k_{III}}}$

$t=0$

$v_z=0,C_S=0,T=T_0$

$\forall z,\forall r$

$r=0$

$\frac{\partial v_z}{\partial r}=0,\frac{\partial C_s}{\partial r}=0,\frac{\partial T}{\partial r}=0$

$\forall z,\forall t>0$

$r=R$

$v_z=0,\frac{\partial C_s}{\partial r}=0,\frac{\partial T}{\partial r}=0$

$\forall z,\forall t>0$

$z=0$

$C_s=C_{s0},T=T_0$

$\forall r,\forall t>0$

$z=L$

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

$\forall r,\forall t>0$