A Generalized Stefan Problem in a Diffusion Model with Evaporation

Abstract

A model for species diffusion is presented, with evaporation at a moving free boundary. The resulting problem resembles a one-phase Stefan problem with superheating, but the usual Stefan condition at the moving boundary is replaced by a version which, in the classical setting, would violate conservation of energy. In the fast evaporation limit, however, the problem reduces to a classical nonlinear Stefan problem with negative latent heat.

Spin coating of an evaporating polymer solution

“…(22)), while the solution of the leading order inner problem (24) for max x j/(x,t)j in this time regime is given by squares; it continues to agree well with the solution of (3) also in the long-time regime. The thin line with the solid diamonds corresponds to the solution for h(t) to the leading order asymptotic problem (28), (30) in the long time regime. The two vertical dotted lines indicate the times t ¼ =d 2 ¼ 28:9 and t ¼ À1=2 ¼ 1:69 Â 10 3 , respectively.…”

“…19 The first theoretical treatment of spin coating an evaporating solution is due to Meyerhofer 20 and was later extended by Sukanek, 21 Bornside et al, 22 and Reisfeld et al 23,24 Additional effects if a volatile component is added, such as variable viscosity and diffusion coefficients during the thinning of the film and its effects on the stability of the film, have also been intensely studied asymptotically and numerically during the past decade. [25][26][27][28][29][30][31] One important feature that occurs due to the evaporation of the volatile component is the phenomenon of "skin" formation. This has first been studied by Lawrence, 32,33 see also de Gennes 34 and Okuzono et al 35 for further discussions on this aspect.…”

“…In the long time regime, the solution to Eqs. (28) and (30) is indicated by a thin line with solid diamonds for h(t). The two vertical dotted lines correspond to the times t ¼ =d 2 ¼ 2:89 Â 10 À2 and t ¼ À1=2 ¼ 1:69 Â 10 3 , respectively.…”

Well-posedness of a parabolic moving-boundary problem in the setting of Wasserstein gradient flows

“…Some of our results are not new: existence and uniqueness for classical solutions of a very similar one-dimensional problem has been proved by Van de Fliert and Van der Hout [11], and general methods for Stefan-like problems can also be applied [12]. The restriction to one dimension also allows for a formulation in terms of the variable U (x) := x −∞ ρ, which is very similar to a classical Stefan problem [28].…”

Local weak solvability of a moving boundary problem describing swelling along a halfline

“…The question (Q1) motivates us to choose for a free boundary modeling strategy of the swelling process, while the answer to (Q2) turns out to be able to clarify a concept of global existence of solutions to the evolution problem behind (Q1). It is worth noting that from the mathematical standpoint, our free boundary problem resembles remotely the classical one phase Stefan problem and its variations for handling unsaturated flow through capillary fringes, superheating, phase transitions, or evaporation; compare, for instance, [6,8,9,12] and references cited therein. Our contribution is in line with the existing mathematical modeling and analysis work of swelling by Fasano and collaborators (see [4,5], e.g.…”