An inequality for the coefficient 𝜎 of the free boundary 𝑠(𝑡)=2𝜎√𝑡 of the Neumann solution for the two-phase Stefan problem

Abstract

Abstract. We consider a semi-infinite body (e.g. ice), represented by (0, -I-oo), with an initial temperature -c < 0 having a heat flux h(t) = -h0/yjt (h0 > 0) in the fixed face x = 0. If h0 > cki/yjna! there exists a solution, of Neumann type, for the resulting twophase Stefan problem. If we connect it with the Neumann problem (on x = 0 the body has a temperature b > 0) we obtain the inequality erf(<r/a2) < (k2 bajki ca2) for the coefficient a of the free boundary s(f) = 2ajt, where kt and af are respectively the thermal conductivity and thermal diffusivity coefficients of the corresponding i phase (i = 1: solid phase, i = 2: liquid phase). If h0 < ck1/y/na1 there is no solution of the initial problem and if h0 = ckjjnai the problem has no physical meaning and corresponds to the case where the latent heat of fusion L tends to infinity. position of the solid-liquid interface (free boundary) at time t > 0 0(x, t) temperature defined for x > 0, t > 0 02 (x, t) water temperature defined for 0 < x < s(t), t > 0 0i (x, t) ice temperature defined for x > s(t), t > 0 c2 specific heat of water C! specific heat of ice I latent heat of fusion p mass density k2 thermal conductivity of water kx thermal conductivity of ice -c<0 initial temperature b > 0 temperature in the fixed face x = 0 h(t) heat flux in the fixed face x = 0 * Received October 1, 1980.

Bounds for the subsistence of a problem of heat conduction

“…This paper was motivated by [2] and [3](see also [7,9,10]). In [8] the author find an exact solution for a particular type of heat flux q. In the paper by TarziaTurner (1992) the authors have presented a similar problem with temperature and convective boundary conditions and they proved that certain conditions on the data are necessary and sufficient in order to obtain a change-phase in the material.…”

Exact solutions for drying with coupled phase-change in a porous medium with a heat flux condition on the surface

“…A semi-infinite porous medium is dried by maintaining a heat flux condition at x = 0 of the type −q 0 / √ t, with q 0 > 0, which was firstly considered in [21]. Initially, the whole body is at uniform temperature t 0 and uniform moisture potential u 0 .…”

Untitled

“…Using methods that mainly consist of comparisonscarried out by means of the maximum principles for the heat equation-between the solution of the problem in question with the exact solution of a similar problem, the authors determine necessary or sufficient conditions on initial and boundary data so as to have one of the three possibilities: /* = 0, 0 < t* < +00, or t* = +oo. Initial papers related to this subject are [6] and [7], where an explicit solution for a problem in a semi-infinite slab is considered for which conditions on data to obtain an instantaneous change of phase (i.e., t* = 0) are established.…”