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When a droplet spreads on a solid substrate, it is unclear what are thecorrect boundary conditions to impose at the moving contact line. The classicalno-slip condition is generally acknowledged to lead to a non-integrablesingularity at the moving contact line, for which a slip condition, associatedwith a small slip parameter, $\lambda$, serves to alleviate. In this paper, wediscuss what occurs as the slip parameter, $\lambda$, tends to zero. Inparticular, we explain how the zero-slip limit should be discussed inconsideration of two distinguished:: one where time is held constant $t =O(1)$, and one where time tends to infinity at the rate $t = O(|\log\lambda|)$. The crucial result is that in the case where time is held constant,the $\lambda \to 0$ limit converges to the slip-free equation, and contact lineslippage occurs as a regular perturbative effect. However, if $\lambda \to 0$and $t \to \infty$, then contact line slippage is a leading-order singulareffect.

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