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

TL;DR

This paper analyzes traveling wave solutions of a thin-film equation in the partial wetting regime, revealing how microscopic contact angles influence macroscopic behavior and extending the Cox-Voinov law with rigorous asymptotics.

Contribution

It provides a detailed asymptotic analysis of traveling waves in partial wetting, connecting microscopic and macroscopic contact angles and identifying resonance effects based on the mobility exponent.

Findings

Asymptotic behavior varies with the mobility exponent n.

The Cox-Voinov law is extended to include microscopic contact angle dependence.

Resonances occur for specific values of n, affecting the solution asymptotics.

Abstract

We consider the thin-film equation $\partial_t h + \partial_y \left(m(h) \partial_y^3 h\right) = 0$ in $\{h > 0\}$ with partial-wetting boundary conditions and inhomogeneous mobility of the form $m(h) = h^3+\lambda^{3-n}h^n$, where $h \ge 0$ is the film height, $\lambda > 0$ is the slip length, $y > 0$ denotes the lateral variable, and $n \in (0,3)$ is the mobility exponent parameterizing the nonlinear slip condition. The partial-wetting regime implies the boundary condition $\partial_y h = \mathrm{const.} > 0$ at the triple junction $\partial\{h > 0\}$ (nonzero microscopic contact angle). Existence and uniqueness of traveling-wave solutions to this problem under the constraint $\partial_y^2 h \to 0$ as $h \to \infty$ have been proved in previous work by Chiricotto and Giacomelli in [Commun. Appl. Ind. Math., 2(2):e-388, 16, 2011]. We are interested in the asymptotics as $h \downarrow 0$ and $h \to \infty$. By reformulating the problem as $h \downarrow 0$ 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 $h\downarrow0$ depending on $n$. Together with the asymptotics as $h\to\infty$ characterizing the Cox-Voinov law for the velocity-dependent macroscopic contact angle as found by Giacomelli, the first author of this work, and Otto in [Nonlinearity, 29(9):2497-2536, 2016], 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.