The Stefan problem with arbitrary initial and boundary conditions

Abstract

Abstract.The paper is concerned with the free boundary problem of a semi-infinite body with an arbitrarily prescribed initial condition and an arbitrarily prescribed boundary condition at its face. An analytically exact solution of the problem is established, which is expressed in terms of some functions and polynomials of the similarity variable x/tin and time t. Convergence of the series solution is considered and proved. Hence the solution also serves as an existence proof. Some special initial and boundary conditions are discussed, which include the Neumann problem and the one-phase problem as special cases. Introduction.Diffusive processes with a change of phase of the materials occur in many scientific and engineering problems. They are known as Stefan problems or free boundary problems. Many examples of these problems can be given, e.g. the melting or freezing of an ice-water combination, the crystallization of a binary alloy or dissolution of a gas bubble in a liquid. To find the solutions to this class of problems has been the subject of investigations by many researchers. Because of the presence of a moving boundary between the two phases, the problem is nonlinear. Various mathematical methods and techniques have been used to study free boundary problems. They have been discussed and summarized in several books [1][2][3][4][5] and many survey papers [6][7][8][9].Free boundary problems have been studied since the nineteenth century by Lame and Clapeyron in 1831, Neumann in the 1860s and Stefan in 1889. However, the only known exact solutions are those of Neumann and some of their extensions and variations. These solutions are all expressible in a single similarity variable x/t1'2. This is to say that the partial differential equations of the problem are reducible to a set of ordinary differential equations. In the case of classical free boundary problems, the reductions are possible only when the body is semi-infinite and the boundary and initial conditions are of certain special forms. Neumann considered a semi-infinite body of a constant initial temperature which is suddenly in contact with a different temperature at its face. The exact solutions of the temperature of both phases are then found in the form of the similarity variable x/t1/2. And the solution of the interface location is proportional to t1/2. A generalization of the problem to cover arbitrary initial and boundary conditions is nontrivial, since the reduction to ordinary differential equations is no longer possible. No exact solutions have yet been found for the Neumann problem with arbitrary initial and boundary conditions.It is the purpose of this paper to study the Neumann problem with arbitrary initial and boundary conditions, i.e., the free boundary problem of a semi-infinite body with an

Analytic solution of the Stefan problem in finite mediums

“…The method of obtaining analytic solutions for phase change problems in semi-infinite mediums can be found in [6,7], and it can be seen that the solutions are obtained only for a certain class of problems in which the governing equations as well as the boundary conditions are invariant under a similarity variable which transforms the partial differential equations for the temperature distribution into ordinary differential equations. Tao solved the Stefan problem with arbitrary initial and boundary conditions [9] as well as with arbitrary heat flux and initial conditions [6]. In all these studies, the material has been considered semi-infinite, which can be a good approximation for many cases in which the material is so long that the boundary at one end remains unaffected for all practical purposes.…”

“…Although the Stefan problem with Dirichlet boundary conditions is solved in this paper, the proposed technique can be applied to the Stefan problem with arbitrary heat flux and initial conditions by expanding the boundary and initial conditions in appropriate power series as has been done in [9] and [6], respectively.…”

“…In the last section, we have obtained the solutions (18) and (19) of the proposed Stefan problem, and have given a method for determining the constants fm, gm, and am . We shall now establish the existence of these solutions by proving [13,9] convergence of each series that appears in the solutions. Combining (18) with (7a), (19) with (7b), and using the zeroth solutions, we obtain for t]x < tjQ , and im erfc(tj2) < im erfc(rj0) for rj2> t]0.…”

Approximate analytical solution of a Stefan’s problem in a finite domain

“…The notation G"(x) is after Tao [4], With two phases in Stefan's problem, we use Kt in place of t, where k is the thermal diffusivity of a phase. Substituting from (1.3), (1.4) transforms to 1 r°° i Gn(x) = -= (x-l)"e-xdl (1.5) n\sjn J-oo which integrates to an «th degree polynomial The serial type solution (1.2) has been used to solve Stefan's problem in a semiinfinite domain [1,2,3,4], We shall transform it to an integral type solution.…”

Temperature and moving boundary in two-phase freezing due to an axisymmetric cold spot

“…For one-dimensional class I problems, some of the important analytical techniques employed are (i) similarity solutions such as the one given in [9], (ii) series solutions by Tao [10], (iii) perturbation solutions [11], and (iv) approximate methods [12]. Several other references can be found in the references listed above.…”