An explicit solution for a two-phase Stefan problem with a similarity exponential heat sources

Abstract

A two-phase Stefan problem with heat source terms in both liquid and solid phases for a semi-in¯nite phase-change material is considered. The internal heat source functions are given by g j (x; t) = (¡1) j+1 ½l t exp ³ ¡( x 2a j p t + d j ) 2´( j = 1 solid phase; j = 2 liquid phase), ½ is the mass density, l is the fusion latent heat by unit of mass; a 2 j is the di®usion coe±cient, x is spatial variable, t is the temporal variable and d j 2 R. A similarity solution is obtained for any data when a temperature boundary condition is imposed at the¯xed face x = 0; when a°ux condition of the type ¡q 0 = p t (q 0 > 0) is imposed on x = 0 then there exists a similarity solution if and only if a restriction on q 0 is satis¯ed.