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 are the correct boundary conditions 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, for which a slip condition, associated with a small slip parameter, λ, serves to alleviate. In this paper, we discuss what occurs as the slip parameter, λ, 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 λ|). The crucial result is that in the case where time is held constant, the λ → 0 limit converges to the slip-free equation, and contact line slippage occurs as a regular perturbative effect. However, if λ → 0 and t → ∞, then contact line slippage is a leading-order singular effect.

Droplet breakup in a stagnation-point flow

“…), this criterion corresponds to the limit in which the minimum droplet thickness is of the order of micrometres and must be governed by van der Waals forces. In addition, we note that the second spatial derivative of the numerical solution to (3.5) has a logarithmic singularity at the contact line, as previously discussed by Ren, Trinh & Weinan (2015) and similarly by Constantin et al. (1993).…”

“…εL ∼ O(0.1 mm)), this criterion corresponds to the limit in which the minimum droplet thickness is of the order of micrometres and must be governed by van der Waals forces. In addition, we note that the second spatial derivative of the numerical solution to (3.5) has a logarithmic singularity at the contact line, as previously discussed by Ren, Trinh & Weinan (2015) and similarly by Constantin et al (1993). Despite the singularity in the curvature, Ren et al (2015) pointed out that their numerical solutions of h and h x remain bounded near the contact line, while Constantin et al (1993) argued that their computations with the Crank-Nicolson scheme converge to the steady weak solution for h (x, t).…”

“…Despite the singularity in the curvature, Ren et al. (2015) pointed out that their numerical solutions of and remain bounded near the contact line, while Constantin et al. (1993) argued that their computations with the Crank–Nicolson scheme converge to the steady weak solution for .…”

Asymptotic analysis of the evaporation dynamics of partially wetting droplets

“…On the other hand, an asymptotically consistent expansion forṙ in which O(1/ ln 2 λ) terms are retained requires, in principle, the inclusion of an O(1/ ln λ) correction to r(0) to be used as an initial condition to (3.20), as compared to the value of r(0) for the full problem. This correction would arise from a (mostly numerical) treatment of the capillary action stage which occurs for t = O(1) (see also Ren et al 2015;Oliver et al 2015;Saxton et al 2016). Yet, although our simulations repeatedly show that the capillary action phase and the initial droplet profile are unimportant for the subsequent dynamics, one can take the r(0) correction to be a priori negligibly small by considering initial drop shapes which are already sufficiently close to the quasistatic profiles, (3.3).…”

Effective boundary conditions for dynamic contact angle hysteresis on chemically inhomogeneous surfaces

“…The relation between the (macroscopic) apparent contact angle and the contact line velocity (characterized by the capillary number) is derived by matched expansions and asymptotic analysis. The model was further validated and developed in Sui & Spelt (2013b) and Sibley, Nold & Kalliadasis (2015a) and generalized in Ren, Trinh & E (2015) and Zhang & Ren (2019) for distinguished limits in different time regimes. In real simulations, one can use the macroscopic model directly and there is no need to resolve the microscopic slip region in the vicinity of the contact line.…”