## 2 Theory
### Solution Strategy
#### 2.3.1 Velocity for the Liquid Phase  
Replacing the liquid phase velocity, current, and force in Equations (3)-(5) results in Equation (13), a unique equation in which the kinematic (\(v_{l}\)) viscosity, the liquid volume fraction (\(\phi_{l}\)), the fluid density (\(\rho_{l}\)), and the magnetic flux density \(B_{0}\) appear as constants.  
\[\frac{\partial w_{l}(r,t)}{\partial t}=v_{l}\frac{\partial^{2}w_{l}(r,t)}{ \partial z^{2}}+v_{l}\frac{1}{r}\frac{\partial w_{l}(r,t)}{\partial r}-\frac{ \sigma}{\phi_{l}\rho_{l}}B_{0}{}^{2}w_{l}(r,t) \tag{13}\]  
Expressing the liquid phase velocity as \(w_{l}(r,t)=F(r)G(t)\) and computing its partial derivatives to be introduced in Equation (13), \(\partial w_{l}/\partial t=F\hat{G}\), \(\partial w_{l}/\partial r=GF\) and \(\partial^{2}w_{l}/\partial r^{2}=GF^{\prime\prime}\). This process leads to Equation (14).  
\[F\hat{G}=v_{l}GF^{\prime\prime}+v_{l}\frac{1}{r}GF^{\prime}-\frac{\sigma}{ \phi_{l}\rho_{l}}B_{0}{}^{2}FG \tag{14}\]  
All terms in Equation (14) can be multiplied by \(1/(FG)\). In addition, the proportionality factor \(\eta\) is introduced. Thus, Equation (15) can be solved by separation of variables.  
\[\frac{\hat{G}}{G}=v_{l}\frac{F^{\prime\prime}}{F}+v_{l}\frac{1}{r}\frac{F^{ \prime}}{F}-\frac{\sigma}{\phi_{l}\rho_{l}}B_{0}{}^{2}=-\eta^{2} \tag{15}\]  
The solution for the time-dependent function \(G\) is direct, resulting in Equation (16), where \(A\) is a real constant.  
\[G(t)=Ae^{-\eta^{2}t} \tag{16}\]  
Rearranging Equation (15), one obtains Equation (17).  
\[r^{2}F^{\prime\prime}+rF^{\prime}+\left(\frac{\eta^{2}}{v_{l}}-\frac{\sigma}{ \phi_{l}\mu_{l}}B_{0}{}^{2}\right)\!r^{2}F=0 \tag{17}\]  
Defining \(\beta^{2}=\left(\eta^{2}/v_{l}-\sigma/(\phi_{l}\mu_{l})B_{0}{}^{2}\right)\), the solution to Equation (17) can be found in terms of the Bessel functions \(J_{0}\) and \(Y_{0}\), where \(J_{0}\) is the Bessel function of the first kind, and \(Y_{0}\) is the Bessel function of second kind, with \(C_{1}\) and \(C_{2}\) as constants to be found (Equation (18)).  
\[F(r)=C_{1}J_{0}(\beta\,r)+C_{2}Y_{0}(\beta\,r) \tag{18}\]  
Considering \(J_{0}(\beta\,r)\) and \(Y_{0}(\beta\,r)\) when \(r\to 0\), the constant accompanying \(Y_{0}\) is chosen to be zero as \(Y_{0}\) diverges, leading to Equation (19), where \(C_{1}=C\) is a real constant. The \(z\)-component of the velocity is expressed in Equation (20).  
\[F(r)=C_{1}J_{0}(\beta\,r) \tag{19}\]  
\[w_{l}(r,t)=CJ_{0}(\beta\,r)e^{-\eta^{2}t} \tag{20}\]Setting the boundary condition \(w(R,t)=0\) leads to \(J_{0}(\beta\,R)=0\), meaning that \(\beta_{k}\ R=\gamma_{k}\), where the \(\gamma_{k}\) values are the zeros of \(J_{0}\) for \(k=1,\ 2,\ 3\ldots\infty\). It follows that \(\beta_{k}{}^{2}\ R^{2}=\gamma_{k}{}^{2}\); thus, \(R^{2}\Big{(}\,\eta^{2}{}_{k}/v-\sigma{B_{0}}^{2}/(\phi_{l}\mu)\Big{)}=\gamma_{k }{}^{2}\). It is noticed that \(\beta_{k}{}^{2}\) is always positive as \(\gamma_{k}{}^{2}>0\) holds. Solving for \(\eta^{2}{}_{k}\), one obtains Equation (21), where the terms at the right side of the equation are associated to viscous and magnetic damping, respectively.  
\[\eta^{2}{}_{k}=\frac{\
u_{l}}{R^{2}}\Bigg{(}\gamma_{k}{}^{2}+\frac{R^{2} \sigma{B_{0}}^{2}}{\phi_{l}\mu}\Bigg{)}=\
u_{l}\Bigg{(}\beta_{k}{}^{2}+\frac{ \sigma{B_{0}}^{2}}{\phi_{l}\mu}\Bigg{)} \tag{21}\]  
Setting the initial condition \(w(r,0)=W_{0}\) leads to Equation (22), which is based on the orthogonality properties of Bessel functions [36; 37].  
\[w_{l}(r,t)=2W_{0}\sum_{k=1}^{\infty}\frac{J_{0}(\beta_{k}\ r)}{\beta_{k}RJ_{1} (\beta_{k}R)}e^{-\eta_{k}{}^{2}t} \tag{22}\]