Explicit solutions for a two-phase unidimensional Lamé-Clapeyron-Stefan problem with source terms in both phases

A. C. BRIOZZO - M. F. NATALE - D. A. TARZIA

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

Elsevier

Año: 2007 vol. 329 p. 145 - 162

0022-247X

A two-phase Stefan problem with heat source terms of a general similarity type in both liquid and solid phases for a semi-infinite phase-change material is studied. We assume the initial temperature is a negative constant and we consider two different boundary conditions at the fixed face $x=0$, a constant temperature or a heat flux of the form $-q_{0}/sqrt{t}% ;(q_{0}>0).$ The internal heat source functions are given by $g_{j}(x,t)=% dfrac{ ho l}{t}eta _{j}left( rac{x}{2a_{j}sqrt{t}} ight) ;$($j=1$ solid phase; $j=2;$liquid phase) where $eta _{j}=eta _{j}left( eta ight) ;$are functions with appropriate regularity properties, $ ho $ is the mass density,$;l$ is the fusion latent heat by unit of mass$% ,;a_{j}^{2};$is the diffusion coefficient,$;x$ is spatial variable and $% t $is the temporal variable. We obtain for both problems explicit solutions with a restriction for data only for the second boundary conditions on $x=0.$ Moreover, the equivalence of the two free boundary problems is also proved. We generalize the solution obtained in J. L. Menaldi - D. A. Tarzia, Comput. Appl. Math., 12 (1993), 123-142 for the one-phase Stefan problem. Finally, a particular case where $eta _{j};$($j=1,2$) are of exponential type given by $;eta _{j}left( x ight) =exp left( -(x+d_{j})^{2} ight) $ with $% ;x$ and $d_{j}in Bbb{R}$ is also studied in details for both boundary temperature conditions at $x=0.$ This type of heat source terms is important through the use of microwave energy following E. P. Scott, J. of Heat Transfer, 116 (1994), 686-693. We obtain a unique solution of the similarity type for any data when a temperature boundary condition at the fixed face $%x=0$ is considered; a similar result is obtained for a heat flux condition imposed on $x=0$ if an inequality for parameter $q_{0}$ is satisfied.