A variational approach to the moving contact line hydrodynamics

TL;DR

This paper derives a variational form of the generalized Navier boundary condition to model moving contact lines in two-phase flows, successfully matching molecular dynamics results and revealing a power-law slip regime.

Contribution

It provides a variational derivation of the GNBC based on energy dissipation principles, offering a new theoretical foundation for contact line hydrodynamics.

Findings

GNBC reproduces molecular dynamics slip velocity profiles.

Transition from complete to partial slip follows a power-law.

Hydrodynamic model captures mesoscopic slip behavior.

Abstract

In immiscible two-phase flows, contact line denotes the intersection of the fluid-fluid interface with the solid wall. When one fluid displaces the other, the contact line moves along the wall. A classical problem in continuum hydrodynamics is the incompatibility between the moving contact line and the no-slip boundary condition, as the latter leads to a non-integrable singularity. The recently discovered generalized Navier boundary condition (GNBC) offers an alternative to the no-slip boundary condition which can resolve the moving contact line conundrum. We present a variational derivation of the GNBC through the principle of minimum energy dissipation (entropy production), as formulated by Onsager for small perturbations away from the equilibrium. Through numerical implementation of a continuum hydrodynamic model, it is demonstrated that the GNBC can quantitatively reproduce the moving contact line slip velocity profiles obtained from molecular dynamics simulations. In particular, the transition from complete slip at the moving contact line to near-zero slip far away is shown to be governed by a power-law partial slip regime, extending to mesoscopic length scales. The sharp (fluid-fluid) interface limit of the hydrodynamic model, together with some general implications of slip versus no-slip, are discussed.