\(R\frac{\partial H_{2}}{\partial s}+\frac{\partial}{\partial r}\left(\left(r+R\right)H_{1}\right)=0,\) (1)
\(\frac{1}{\rho}\frac{\partial p}{\partial s}-\frac{u^{2}}{r+R}=0,\) (2)
\(\frac{\partial u}{\partial t}+\frac{R}{r+R}u\frac{\partial u}{\partial s}+v\frac{\partial u}{\partial r}+\frac{k^{*}}{\rho}\frac{\partial N}{\partial r}-\frac{\mu_{e}}{4\pi\rho}\begin{bmatrix}\frac{H_{1}H_{2}}{r+R}+H_{2}\frac{\partial H_{1}}{\partial r}\\ \frac{RH_{1}}{r+R}\frac{\partial H_{1}}{\partial s}\\ \end{bmatrix}=\left(\nu+\frac{k^{*}}{\rho}\right)\begin{bmatrix}+\frac{\partial^{2}u}{\partial r^{2}}\\ +\frac{1}{r+R}\left(\frac{\partial u}{\partial r}-\frac{u}{r+R}\right)\\ \end{bmatrix}-\frac{1}{\rho}\left(\frac{R}{r+R}\right)\frac{\partial p}{\partial r},\) (3)
\(\frac{\partial H_{1}}{\partial t}+\frac{R}{r+R}u\frac{\partial H_{1}}{\partial s}+v\frac{\partial H_{1}}{\partial r}+\frac{H_{1}H_{2}}{r+R}-\begin{bmatrix}\frac{\text{vu}}{r+R}+H_{2}\frac{\partial v}{\partial r}\\ \frac{RH_{1}}{r+R}\frac{\partial u}{\partial s}\\ \end{bmatrix}=\mu_{e}\begin{bmatrix}+\frac{\partial^{2}H_{1}}{\partial r^{2}}-\frac{H_{1}}{\left(r+R\right)^{2}}\\ +\frac{1}{r+R}\frac{\partial H_{1}}{\partial r}\\ \end{bmatrix},\) (4)
\(\frac{\partial N}{\partial t}+\frac{\text{Ru}}{r+R}\frac{\partial N}{\partial s}+v\frac{\partial N}{\partial r}=-\frac{\gamma^{*}}{\text{jρ}}\left(\frac{1}{r+R}\frac{\partial T}{\partial r}+\frac{\partial^{2}T}{\partial r^{2}}\right)-\frac{k^{*}}{\text{jρ}}\begin{pmatrix}\frac{1}{r+R}\frac{\partial u}{\partial r}\\ +2N\\ +\frac{u}{r+R}\\ \end{pmatrix},\) (5)
\(\frac{\partial T}{\partial t}+\frac{\text{Ru}}{r+R}\frac{\partial T}{\partial s}+v\frac{\partial T}{\partial r}=\alpha\left(\frac{1}{r+R}\frac{\partial T}{\partial r}+\frac{\partial^{2}T}{\partial r^{2}}\right),\) (6)