The Coxâ€"Voinov law for traveling waves in the partial wetting regime.

We consider the thin-film equation ∂ t h + ∂ y m (h) ∂ y 3 h = 0 in { h > 0} with partial-wetting boundary conditions and inhomogeneous mobility of the form m (h) = h 3 + λ 3âˆ' n h n , where h â©ľ 0 is the film height, λ > 0 is the slip length, y > 0 denotes the lateral variable, and n ∈ (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 ∂ y 2 h â†' 0 as h â†' ∞ 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 â†" 0 and h â†' ∞. By reformulating the problem as h â†" 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 â†" 0 depending on n. Together with the asymptotics as h â†' ∞ characterizing the Coxâ€"Voinov law for the velocity-dependent macroscopic contact angle as found by Giacomelli, the first author of this work, and Otto (2016 Nonlinearity 29 2497â€"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 Coxâ€"Voinov law for the macroscopic contact angle depends continuously differentiably on the microscopic contact angle. [ABSTRACT FROM AUTHOR]