On the distinguished limits of the Navier slip model of the moving contact line problem

Abstract

When a droplet spreads on a solid substrate, it is unclear what the correct boundary conditions are to impose at the moving contact line. The classical no-slip condition is generally acknowledged to lead to a non-integrable singularity at the moving contact line, which a slip condition, associated with a small slip parameter, ${\it\lambda}$ , serves to alleviate. In this paper, we discuss what occurs as the slip parameter, ${\it\lambda}$ , tends to zero. In particular, we explain how the zero-slip limit should be discussed in consideration of two distinguished limits: one where time is held constant, $t=O(1)$ , and one where time tends to infinity at the rate $t=O(|\!\log {\it\lambda}|)$ . The crucial result is that in the case where time is held constant, the ${\it\lambda}\rightarrow 0$ limit converges to the slip-free equation, and contact line slippage occurs as a regular perturbative effect. However, if ${\it\lambda}\rightarrow 0$ and $t\rightarrow \infty$ , then contact line slippage is a leading-order singular effect.