The elastic Landau–Levich problem

Abstract

We study the classical Landau–Levich dip-coating problem in the case where the interface has significant elasticity. One aim of this work is to unravel the effect of surface-adsorbed hydrophobic particles on Landau–Levich flow. Motivated by recent findings (Vella, Aussillous & Mahadevan, Europhys. Lett., vol. 68, 2004, pp. 212–218) that a jammed monolayer of adsorbed particles on a fluid interface makes it respond akin to an elastic solid, we use the Helfrich elasticity model to study the effect of interfacial elasticity on Landau–Levich flow. We define an elasticity number, $\mathit{El}$, which represents the relative strength of viscous forces to elasticity. The main assumptions of the theory are that $\mathit{El}$ be small, and that surface tension effects are negligible. The shape of the free surface is formulated as a nonlinear boundary value problem: we develop the solution as an asymptotic expansion in the small parameter ${\mathit{El}}^{1/ 7} $ and use the method of matched asymptotic expansions to determine the film thickness as a function of $\mathit{El}$. The solution to the shape of the static meniscus is not as straightforward as in the classical Landau–Levich problem, as evaluation of higher-order effects is necessary in order to close the problem. A remarkable aspect of the problem is the occurrence of multiple solutions, and five of these are found numerically. In any event, the film thickness varies as ${\mathit{El}}^{4/ 7} $ in qualitative agreement with the experiments of Ouriemi & Homsy (Phys. Fluids, 2013, in press).

The elastocapillary Landau–Levich problem

“…Equation (1.2) is the key result in Dixit & Homsy (2013). A remarkable feature of this result is that unlike the classical Landau-Levich result,h ∞ is not unique.…”

“…In a previous paper, Dixit & Homsy (2013), we studied the effect of interfacial elasticity on the classical Landau-Levich dip-coating flow in the absence of surface tension. The results obtained in that analysis have motivated a detailed investigation of the relative roles of surface tension and elasticity in the Landau-Levich problem.…”

“…The subscript c inh ∞,c denotes that the above result is valid for a clean interface possessing a constant surface tension. If the fluid interface is replaced with an elastic membrane, then (1.1) is replaced with the following result, valid in the absence of surface tension (Dixit & Homsy 2013):…”

A pinned or free-floating rigid plate on a thin viscous film

“…Table 1 shows a selection of previous work on both the pinned and free-floating problems for both rigid and elastic plates, and illustrates that the much of the previous work has ignored the effect of surface tension, which is a key ingredient of the present work. The Free Surfaces Surface Tension Elasticity Conway & Richman (1983) 0 no yes Moriarty & Terrill (1996) 2 yes no Kamiyama & Kohnsari (2000) 0 no yes Hosoi & Mahadevan (2004) 0 no yes Giacomin et al (2012) 1 no yes 0 no no Dixit & Homsy (2013) 0 no yes This work (Parts 1 and 2) 1 or 2 yes no and yes Table 1. A selection of previous work on both the pinned and free-floating problems.…”

“…Pranckh & Scriven 1990, Iliopoulos & Scriven 2005, and Giacomin et al 2012, and the elastic drag-out problem (e.g. Dixit & Homsy 2013). Also relevant here is the recent work by and on the so-called "washboard" instability which can be created by moving a pivoted rigid plate over a layer of fluid or granular material.…”