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

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 ${\partial }_{y}h=const.>0$ 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 $h\downarrow 0$ 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.

24 pages, 3 figures, revised version; changed title, extended introduction

References in corpus (3)

• Non-negative Martingale Solutions to the Stochastic Thin-Film Equation with Nonlinear Gradient Noise
• Relaxation to equilibrium in the one-dimensional thin-film equation with partial wetting
• Martingale solutions to the stochastic thin-film equation in two dimensions

Cited by in corpus (1)

• Droplet motion with contact-line friction: long-time asymptotics in complete wetting