Weak solutions to lubrication systems describing the evolution of bilayer thin films

Abstract

The existence of global nonnegative weak solutions is proved for coupled one-dimensional lubrication systems that describe the evolution of nanoscopic bilayer thin polymer films that take account of Navier-slip or no-slip conditions at both liquid-liquid and liquid-solid interfaces. In addition, in the presence of attractive van der Waals and repulsive Born intermolecular interactions existence of positive smooth solutions is shown.

Modelling Pattern Formation in Dip-Coating Experiments

“…However, for two-field long-wave hydrodynamic equations this had to our knowledge only been shown before for dewetting two-layer films [55,56] but neither for surfactant-covered films nor for films of solutions or suspensions. Recently, the gradient dynamics form in the case of two-layer films has been extended to include the case with slip and has been employed in deeper mathematical analyses of such models [32,34,33]. We expect that the here reviewed gradient dynamics form will also facilitate similar analyses for films of suspensions, solutions and mixtures.…”

Existence of weak solutions for particle-laden flow with surface tension

“…The system (2) is closely related to the equations governing transport of insoluble surfactant on the fluid surface [11], for which the concentration satisfies an equation with a non-degenerate diffusion term. Existence and positivity of weak solutions was established in [13,1] using a finite element approach and studied for more general systems in later work by [8,14,10]. The techniques employed there are almost applicable to (2), but must be modified to account for a few key differences in the structure of the equations.…”

“…[existence] Let (9)-(13) hold. Assume that the initial data (h 0 , ψ 0 ) satisfy (8) and (14). Then, for any time T > 0, there exists a weak solution (h, ψ) of the problem (4)-(7) in the sense of Definition 3.1.…”

Weak solutions to a two-phase thin film model with insoluble surfactant driven by capillary effects

“…The study of weak solutions for a two-phase thin film system without surfactant has been addressed in [12] (n = 2) and [8,12] (n = 3) 1 . Results regarding the existence of global non-negative weak solutions to a system describing the dynamics of a one-phase thin film with insoluble surfactant are subject in [4,6,9,15].…”