BACK-DIFFUSION IN A FINITE MEDIUM WITH A MOVING BOUNDARY
C. K. Meadley · The Quarterly Journal of Mechanics and Applied Mathematics · 1971
A mathematical model is formulated to describe the distribution of solute concentration in a layer of a chemical solution when a free surface recedes by continuous evaporation of the solvent. It is considered that the evaporation activates a back-diffusion of solute away from the receding free surface and into the remaining liquid, resulting in an unsteady one-dimensional concentration field. Formulation of the problem is on the basis of Fick‘s laws of diffusion and is shown to be equivalent to solving a fixed-point boundary, initial-value problem for a general, second-order, linear parabolio equation. If a(t) = l(t)/D, where l(t) is the liquid depth variation in time and D is the diffusion coefficient, a solution of the boundary-value problem is required in the form of a functional on a(t). It is found that the problem has a unique solution within an interval 0 0, when a(t) is analytic and non-vanishing. A method for constructing an approximate mathematical solution, when 0 < /da/dt/ ≤4, is described. It is further shown that an exact solution is available when the form of the depth variation l(t) corresponds to one member of a particular one-parameter family of curves.