## 3 Solving Procedures  
For convenience, the dimensionless variables for the governing equation and boundary conditions are introduced as \(m=a/M\), \(n_{1}=d/M\), \(k=k_{\rm h}/k_{\rm v}\), \(x_{\rm D}=x/a\), \(z_{\rm D}=z/M\), \(R_{\rm D}=R/a\), \(t_{\rm D}=k_{\rm h}t/\mu_{e}a^{2}\), \(s_{\rm D}(t_{\rm D},x_{\rm D},z_{\rm D})=\frac{k_{\rm h}}{\partial}s(t,x,z)\). Incorporating the dimensionless variables into Equations (1)-(5), the governing equation and solving conditions can be transformed into the following:  
\[\frac{\partial^{2}s_{\rm D}}{\partial x_{\rm D}^{2}}+\frac{m^{2}}{k}\frac{ \partial^{2}s_{\rm D}}{\partial z_{\rm D}^{2}}=\frac{\partial s_{\rm D}}{ \partial t_{\rm D}}\ \ \ \ (0\leq x_{\rm D}\leq R_{\rm D},\ 0\leq z_{\rm D}\leq 1) \tag{6}\]  
\[\left.\frac{\partial s_{\rm D}}{\partial x_{\rm D}}\right|_{x_{\rm D}=1}= \left\{ \begin{array}{cc}0&0<z_{\rm D}<n_{1}\\ -\frac{m}{2(1-n_{1})}&n_{1}\leq z_{\rm D}<1\end{array} \right. \tag{7}\]  
\[\left.s_{\rm D}\right|_{x_{\rm D}=R_{\rm D}}=0\ \ \ \ (t_{\rm D}>0,\ 0\leq z_{\rm D}\leq 1) \tag{8}\]  
\[\left.\frac{\partial s_{\rm D}}{\partial z_{\rm D}}\right|_{z_{\rm D}=0}= \left.\frac{\partial s_{\rm D}}{\partial z_{\rm D}}\right|_{z_{\rm D}=1}=0\ \ \ \ (t_{\rm D}>0,\ 0\leq x_{\rm D}\leq R_{\rm D}) \tag{9}\]  
and  
\[\left.s_{\rm D}\right|_{t_{\rm D}=0}=0\ \ \ \ (0\leq x_{\rm D}\leq R_{\rm D},\ 0 \leq z_{\rm D}\leq 1) \tag{10}\]  
where subscript D represents a dimensionless symbol. Applying the Laplace transform to Equations (6)-(9) with the initial conditions of Equation (10) gives  
\[\left.\frac{\partial^{2}\widetilde{s}_{\rm D}}{\partial x_{\rm D}}^{2}+\frac{ m^{2}}{k}\frac{\partial^{2}\widetilde{s}_{\rm D}}{\partial z_{\rm D}}^{2}-p \widetilde{s}_{\rm D}=0\right. \tag{11}\]  
\[\left.\frac{\partial\widetilde{s}_{\rm D}}{\partial x_{\rm D}}\right|_{x_{\rm D }=1}=\left\{ \begin{array}{cc}0&0<z_{\rm D}<n_{1}\\ -\frac{m}{2(1-n_{1})p}&n_{1}\leq z_{\rm D}<1\end{array} \right. \tag{12}\]  
\[\left.\widetilde{s}_{\rm D}\right|_{x_{\rm D}=R_{\rm D}}=0 \tag{13}\]  
and  
\[\left.\frac{\partial\widetilde{s}_{\rm D}}{\partial z_{\rm D}}\right|_{z_{\rm D }=0}=\left.\frac{\partial\widetilde{s}_{\rm D}}{\partial z_{\rm D}}\right|_{z _{\rm D}=1}=0 \tag{14}\]  
where \(\widetilde{s}_{\rm D}(p,x_{\rm D},z_{\rm D})=\int_{0}^{\infty}s_{\rm D}(t_{ \rm D},x_{\rm D},z_{\rm D}){\rm e}^{-pt_{\rm D}}{\rm d}t_{\rm D}\), and \(p\) is the Laplace transform parameter. Applying the finite cosine transform to Equations (11)-(13) with the vertical boundary conditions of Equation (14) yields  
\[\left.\frac{\partial^{2}\widetilde{s}_{\rm D}}{\partial x_{\rm D}^{2}}-\left[ \frac{(n\pi m)^{2}}{k}+p\right]\widetilde{s}_{\rm D}=0\right. \tag{15}\]  
\[\left.\frac{\partial^{\widetilde{s}}_{\rm D}}{\partial x_{\rm D}}\right|_{x_{ \rm D}=1}=\int_{n_{1}}^{1}-\frac{m}{2(1-n_{1})p}\ \Theta(v_{n},z_{\rm D}){\rm d}z_{\rm D} \tag{16}\]and  
\[\widetilde{\mathfrak{s}}_{\rm D}\Big{|}_{x_{\rm D}=R_{\rm D}}=0 \tag{17}\]  
where \(\widetilde{\mathfrak{s}}_{\rm D}(p,x_{\rm D},n)=\int_{0}^{1}\widetilde{\mathfrak{ s}}_{\rm D}(p,x_{\rm D},z_{\rm D})\Theta(\
u_{n},z_{\rm D})dz_{\rm D}\); \(n\) is the finite cosine transform parameter; \(\Theta(\
u_{n},z_{\rm D})=\cos\
u_{n}z_{\rm D}\); and \(\
u_{n}\) is the root of the equation \(\sin\
u_{n}z_{\rm D}=0\), which can be expressed as \(\
u_{n}=n\pi\). According to the boundary conditions of Equations (16) and (17), the governing Equation (15) can be solved as  
\[\widetilde{\mathfrak{s}}_{\rm D}(p,x_{\rm D},n)=\int_{n_{1}}^{1}\frac{m}{2(1-n _{1})p}\Theta(\
u_{n},z_{\rm D})dz_{\rm D}\frac{\sinh[\lambda_{n}(p)(R_{\rm D }-x_{\rm D})]}{\lambda_{n}(p)\cosh[\lambda_{n}(p)(R_{\rm D}-1)]} \tag{18}\]  
where \(\lambda_{n}(p)=\sqrt{\frac{(m\pi\pi)^{2}}{k}+p}\).