The Cox-Voinov law for traveling waves in the partial wetting regime

Abstract

We consider the thin-film equation ∂th+∂ym(h)∂y3h=0 in {h > 0} with partial-wetting boundary conditions and inhomogeneous mobility of the form m(h) = h 3 + λ 3-n h n , where h ∼ 0 is the film height, λ > 0 is the slip length, y > 0 denotes the lateral variable, and n ϵ (0, 3) is the mobility exponent parameterizing the nonlinear slip condition. The partial-wetting regime implies the boundary condition ∂ y h = const. > 0 at the triple junction ∂{h > 0} (nonzero microscopic contact angle). Existence and uniqueness of traveling-wave solutions to this problem under the constraint ∂y2h→0 as h → ∞ have been proved in previous work by Chiricotto and Giacomelli (2011 Commun. Appl. Ind. Math. 2 e-388, 16). We are interested in the asymptotics as h ↓ 0 and h → ∞. By reformulating the problem as h ↓ 0 as a dynamical system for the difference between the solution and the microscopic contact angle, values for n are found for which linear as well as nonlinear resonances occur. These resonances lead to a different asymptotic behavior of the solution as h ↓ 0 depending on n. Together with the asymptotics as h → ∞ characterizing the Cox-Voinov law for the velocity-dependent macroscopic contact angle as found by Giacomelli, the first author of this work, and Otto (2016 Nonlinearity 29 2497-536), the rigorous asymptotics of traveling-wave solutions to the thin-film equation in partial wetting can be characterized. Furthermore, our approach enables us to analyze the relation between the microscopic and macroscopic contact angle. It is found that the Cox-Voinov law for the macroscopic contact angle depends continuously differentiably on the microscopic contact angle.