Distinguished Limits of the Navier Slip Model for Moving Contact Lines in Stokes Flow

Abstract

The no-slip boundary condition in classical fluid mechanics is violated at a moving contact line, and it leads to an infinite rate of energy dissipation when combined with hydrodynamic equations. To overcome this difficulty, the Navier slip condition associated with a small parameter $\lambda$, named the slip length, has been proposed as an alternative boundary condition. In a recent work of Ren, Trinh, and E [J. Fluid Mech., 772 (2015), pp. 107--126], the distinguished limits of a spreading droplet when $\lambda$ tends to zero were studied using a thin film equation with the Navier slip condition. In this paper, we extend this analysis to the more general situation where the flow is modeled by the Stokes equation. In particular, we consider two distinguished limits as the slip length $\lambda$ tends to zero: one where time is held constant $t=O(1)$, and the other where time goes to infinity at the rate $t=O(|\ln\lambda|)$. It is found that when time is held constant, the contact line dynamics converges to the slip-free equation, and contact line slippage occurs as a regular perturbative effect. On the other hand, when time goes to infinity, significant contact line displacement occurs and the contact line slippage becomes a leading-order singular effect. In this latter case, we recover the earlier analysis, e.g., by Cox [J. Fluid Mech., 168 (1986), pp. 169--194], after rescaling time.