Unsteady coupled high flux-mass and momentum transfer with a moving boundary

Abstract

The simultaneous transient developing mass and momentum boundary layers in a moving-wall (Stefan) problem resulting from the flow of a fluid through a channel with dissolving walls is determined. Two effects which influence the course of the problem are considered: a flux induced velocity, referred to as the Stefan-Nusselt effect, characterized by a parameter called the potential ratio and a velocity due to the density difference between the phases, superimposed on the Stefan-Nusselt velocity, and characterized by a parameter called the density ratio. The changing geometric confines of the problem requires the introduction of a mobile coordinate system in lieu of the conventional Eulerian coordinate system. No known solution by previous authors have considered both these effects in a mobile coordinate system. The boundary equations, considering these effects, are developed in vector form for a wall which is skewed to the coordinates of a conventional x, y coordinate system. The general diffusion equation in terms of moving coordinates is derived from basic principles and shown to contain additional terms which are not obtainable by a chain-rule conversion. The additional terms cancel out in differential form but must be included in finite difference form to ensure conservation. The problem is then solved by finite differences using the vorticity-stream function method. A non-dimensional flux is defined and plots showing the variation of the non-dimensional flux, mass concentration, vorticity and velocity as well as the shape of the channel wall with the governing parameters— Re, Sc, the potential ratio, and the ratio of solid to fluid density are shown as a function of time and distance. The influence of the parameters are discussed. A wave-wall effect, which occurred in some of the problems, is described. The mathematical techniques developed are applicable to solving a class of problems which have heretofore received scanty attention, namely problems in which the important consideration is the shape of the domain for the desired or optimum potential distribution. Moving-wall, separation flow, explosion propagation and free surface problems are in this class.