Temperature field in a solid surrounding a migrating heat source

The present research is related to contact melting by a moving heat source of arbitrary shape. Heat conduction in the melting material is governed by a 3D differential equation, where the thermal conductivity of the surrounding material is assumed to be strongly temperature dependent. By using the Green's formula, the boundary value problem is converted to the boundary integral equation. This nonlinear equation is solved numerically by iterations utilizing the boundary element method. Different shapes of heat sources are investigated. Since the obtained integral equation is the Fredholm-type equation of the first kind and belongs to the family of so-called ill-posed problems, supplementary computations that verify the stability of numerical algorithm are provided. For the special cases associated with thermodrilling technology, some analytical estimations and solutions are obtained. Particularly, if the melting velocity is high (ePe>10), asymptotic solutions are found. In this case the integral equation is significantly reduced, simplifying the computations. Numerical results are in good agreement with the closed-form solutions available for the elliptical shape of a solid-liquid interface.