A Nonlinear Two-phase Stefan Problem with Melting Point Gradient: a Constructive Approach

We consider a one-dimensional two-phase Stefan problem, modeling a layer of solid material floating on liquid. The model includes internal heat sources, variable total mass (resulting e.g. from sedimentation or erosion), and a pressuredependent melting point. The problem is reduced to a set of nonlinear integral equations, which provides the basis for an existence and uniqueness proof and a new numerical method. Numerical results are presented. Introduction In this paper, we study the temperature distribution in a thick layer of dense matter, floating on a deeper, denser layer. The materials of the two regions are polymorphs of each other. The problem of the equilibrium of the column of matter due to a disequilibrium of both the buoyancy and the thermal state is called a Stefan problem. The most familiar example of the Stefan problem is that of the history of a cake of ice floating in water after the surface of the ice has received a fresh snowfall. A geophysically derived example concerns the possibility that the Mohorovicic discontinuity is an isochemical phase transformation boundary between a denser mantle phase below and a lighter crustal phase above; the desequilibrium may arise, for example, from erosion or sedimentation at the surface or by a change in the thermal state in the mantle. The two examples differ especially in the sign of the slope of the pressure-temperature curve for the phase transformation. In this paper we assume that the pressure-temperature curve for the phase boundary interface is linear but make no specification of the sign. We shall assume that the total mass in the two regions is a continuously differentiable function of time. The elevations of the phase boundary and the free surface are to be determined as functions of time. Concerning the large literature on the Stefan problem, we mention only some papers which we consider particularly relevant for our purpose, namely those of Friedman [2], Rubinstein [6], MacDonald and Ness [3], O’Connell and Wasserburg [5], and Mori [4]. Rubinstein provides a broadly based review of the history of the Stefan problem. Papers [3] and [5] elucidate the geophysical importance of the problem: in them, the set of partial differential equations has been integrated numerically. Mori [4] treats the numerical solution of a similar problem in great