Abstract
This paper presents an extension of the non-field analytical method—known as the method of Kulish—to solving heat transfer problems in domains with a moving boundary. This is an important type of problems with various applications in different areas of science. Among these are heat transfer due to chemical reactions, ignition and explosions, combustion, and many others. The general form of the non-field solution has been obtained for the case of an arbitrarily moving boundary. After that some particular cases of the solution are considered. Among them are such cases as the boundary speed changing linearly, parabolically, exponentially, and polynomially. Whenever possible, the solutions thus obtained have been compared with known solutions. The final part of the paper is devoted to determination of the front propagation law in Stefan-type problems at large times. Asymptotic solutions have been found for several important cases of the front propagation.
Highlights
This paper presents an extension of the non-field analytical method—known as the method of Kulish— to solving heat transfer problems in domains with a moving boundary
Heat transfer in domains with a moving boundary is an important problem with various applications in different areas of science
Asymptotic solutions have been found for several important cases of the front propagation
Summary
In the coordinate system related to the boundary, the heat transfer problem becomes identical to Eq (1) with α = const , β = β(t) , γ = 0. The latter problem is of great importance while modelling processes of propagation of the phase transition fronts, chemical reactions, or combustion. To find the corresponding alternative form of the fractional operator (12) in the case of α = const , β = β(t) , γ = 0 , substitute t β2 t β2. For instance, the decreasing function β(t) is given in the form of a series with respect to exponential functions β = βne−nct , c, βn = const,.
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