TWO-PHASE STEFAN PROBLEM FOR GENERALIZED HEAT EQUATION

Аннотация

The generalized heat equation is very important for modeling of the heat transfer in bodies with a variable cross section, when the radial component of the temperature gradient can be neglected in comparison with the axial component. Such models can be applied in the theory of the heat- and mass transfer in the electrical contacts. In particular, the temperature field in a liquid metal bridge appearing at the opening electrical contacts can be modelled by the above considered Stefan problem for the generalized heat equation. The exact solution in the case when the temperature field in a liquid bridge is modelled by the generalized heat equation, while for the temperature of the solid contact zone is modelled by the spherical model, can be represented in the form of series with radial heat polynomials and integral error functions. The recurrence formulas for the coefficients of these series are given in papers published earlier in “News of the National Academy of Sciences of the Republic of Kazakhstan, Physic-mathematical series”.
The two-phase Stefan problem for the generalized heat equation is considered in this paper for the case when one of the phases collapses into a point at the initial time. That creates a serious difficulty for the solution by the standard method of reduction of the problem to the integral equations because these equations are singular. Another method is used here in the case, when the functions appearing in the initial and boundary conditions are analytical and can be expanded into Taylor series. Then the solution of the problem can be represented in the form of series for special functions (Laguerre polynomials and the confluent hypergeometric function) with undetermined coefficients. These special functions have a close link with the heat polynomials introduced by P.C. Rosenbloom and D.V. Widder.