## 7 Applications to polar-symmetric problems
### Melting of iced-water cylinder  
We study the melting of an iced water cylinder, whose cross-section is shown in Figure 21. The cylinder is initially at the melting temperature and an essential boundary condition \(T\big{(}R\big{)}=T_{1}>T_{\text{\tiny{tun}}}\) is imposed to the external surface so that the initially solid cylinder progressively melts. The problem has cylindrical symmetric and can be described by the spatial coordinate \(r\). The energy balance equation in cylindrical coordinates and for isotropic heat conduction is  
\[\rho\frac{\partial}{\partial t}\big{(}cT\big{)}=k\Bigg{(}\frac{\partial^{2} T}{\partial r^{2}}+\frac{1}{r}\frac{\partial T}{\partial r}\Bigg{)}, \tag{65}\]  
and the position of the phase interface \(r_{\text{\tiny{s}}}\big{(}t\big{)}\) is governed by Stefan law,  
\[\rho L\frac{dr_{\text{\tiny{s}}}}{dt}=-\Bigg{(}k_{\text{\tiny{ia}}}\frac{ \partial T_{\text{\tiny{ia}}}}{\partial r}-k_{\text{\tiny{sol}}}\frac{ \partial T_{\text{\tiny{sol}}}}{\partial r}\Bigg{)}_{r_{\text{\tiny{ren}}}}. \tag{66}\]  
The phase-interface temperature condition is  
Figure 20: Temperature field at different time instants. \[T\left(r_{*},t\right)=T_{\text{tran}}\,. \tag{67}\]  
The boundary conditions are  
\[ \begin{split} T\left(R,t\right)=T_{\text{t}}>T_{\text{tran}}\,,\\ q\left(0,t\right)=-k\left.\frac{\partial T}{\partial r}\right|_{r =0}=0,\end{split}  \tag{68}\]  
where the latter is due to the symmetry of the problem and expresses the natural boundary condition of absence of heat flow in the axis of the cylinder. The initial conditions are  
\[ \begin{split} T\left(r,0\right)=T_{\text{tran}}\,,\quad 0\leq r<R,\\ r_{*}\left(0\right)=a.\end{split}  \tag{69}\]  
We develop the PSS approximation to compare the numerical results. Under the PSS hypotheses, Eq. (65) becomes  
\[\frac{\partial^{2}T_{\text{PSS}}}{\partial r^{2}}+\frac{1}{r}\frac{\partial T _{\text{PSS}}}{\partial r}=0\,. \tag{70}\]  
By integrating and applying the boundary conditions of Eqs (68), the approximated temperature field is obtained as  
\[T_{\text{PSS}}\left(r,t\right)=\left(1-\frac{\log\frac{r}{R}}{R}\right)\left( T_{\text{t}}-T_{\text{tran}}\right)+T_{\text{tran}},\quad r\in\Omega_{\text{ }_{\text{lidq}}}\,. \tag{71}\]  
By substituting in Eq. (66) and integrating it with the initial condition in Eqs (69), an implicit relation between the interface position and the time is obtained,  
\[\frac{k_{\text{lig}}\left(T_{\text{t}}-T_{\text{tran}}\right)}{\rho L}_{t}= \frac{\left(r_{*}^{\text{PSS}}\right)^{2}}{2}\left(\log\frac{r_{*}^{\text{PSS }}}{R}-\frac{1}{2}\right)-\frac{a^{2}}{2}\left(\log\frac{a}{R}-\frac{1}{2} \right). \tag{72}\]  
Figure 21: Cross-section of the melting ice cylinder. For the numerical analysis we consider the data in Table 1, from which a relatively high value of Stefan number is achieved,  
\[\alpha=\frac{L}{c_{\text{lio}}\left(T_{1}-T_{\text{tran}}\right)} \simeq 80\;, \tag{73}\]  
so that the PSS approximation represents an adequate reference for comparing the numerical results. Figure 22 shows the interface position calculated through the PSS approximation and the XFEM simulations with two different spatial discretisations, with 10 and 40 equally-sized finite elements. The increase in the spatial discretisation improves the quality of the numerical solution especially in the last stages of the melting process, when the phase-interface is close to the cylinder axis. This is mainly due to the evaluation of the integrals in Eqs (37). In particular, the accuracy of a finite element is higher if the areas of the cylindrical surfaces constructed with the radii at the two nodes of the considered element are closer. The temperature field at some time instants, \(t=2400\) s, \(4800\) s,..., \(12000\) s, is represented in Figure 23. Temperature fields evaluated through PSS approximation and XFEM simulation differ, if they are evaluated at the same time instant, because of the error in the phase-interface position.