An approximate analytic method for calculating the temperature field for a phase transition in an embankment in a permafrost region

摘要

In this study, we applied conformal mapping and irregular perturbation methods to develop an analytic method to calculate the temperature field for a phase transition in the irregular domain of an embankment in a permafrost region. We then used this method to derive an approximate formula for the unstable temperature field of the phase transition in a homogeneous embankment over an irregular domain. First, we used conformal mapping to project a two-dimensional homogeneous embankment with irregular boundaries onto a one-dimensional semi-infinite area to derive the thermal conduction equations for unstable phase transitions and a continuity equation for the inter-facial heat flux in the mapping coordinates. The irregularity of the boundary of the solution domain meant that only the mapping in the spatial domain could be used to inexpensively impose the boundary conditions analytically onto the regular boundary, thereby enabling the formulation of an analytical solution for the temperature field for a phase transition over an irregular domain. A small parameter perturbation and the orthogonal polynomial approximation were then used to analytically compute the thermal conduction equation for a phase transition in the mapping coordinate system.