Two-dimensional Stefan problem using level set method

Abstract

Cancer cell invasion across the surrounding tissues is driven by finger-like protrusions known as invadopodia. Plasma membrane of a cancer cell has the elastic characteristic that gives the idea to employ the knowledge of Stefan problem to solve the problem in cancer cell invasion. For this purpose, plasma membrane is treated as a free boundary interface. In fact, Stefan problem is a well-known free boundary problem that requires the solution of moving interface(s) as time proceeds along with the change in the temperature distribution. Hence, this study investigated twodimensional free boundary Stefan problem that focused on the solidification process. A mathematical model of solidification is first introduced by Chen et al. (1997) considering the jump of the temperature distribution fromthe liquid to the solid regions. The velocity concerning the movement of the interface is calculated by taking the first derivative of the temperature with respect to space. The model is solved numerically by using finite-difference of level set method [13]. Two algorithms of the level set method are presented in modelling and discretization parts, to handle the topology changes of Stefan problem, systematically. Results of the interface movement and temperature distribution are presented graphically and discussed in order to visualize the solidification process.