Droplet spreading: intermediate scaling law by PDE methods
Lorenzo Giacomelli and Felix Otto
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Submission date: 07. Nov. 2000
published in: Communications on pure and applied mathematics, 55 (2002) 2, p. 217-254
DOI number (of the published article): 10.1002/cpa.10017
MSC-Numbers: 35B99, 35K30, 35K55, 35K65, 76D08
Keywords and phrases: intermediate scaling laws, qualitative behavior, higher order nonlinear degenerate parabolic equations, thin film, slip condition, lubrication theory
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We consider the spreading of a thin droplet of viscous liquid on a plane surface driven by capillarity. The standard lubrication approximation leads to an evolution equation for the film height h - which is ill-posed when the spreading is limited by the no-slip boundary condition at the liquid-solid interface, due to a singularity at the moving contact line. The most common relaxation of the no-slip boundary condition removes this singularity but introduces a new physical length scale: the slippage length b. It is believed that this microscopic length scale only enters logarithmically in the effective (that is, macroscopic) spreading behavior.
In this paper, we rigorously show that the naively expected spreading rate is indeed only altered by a logarithmic term involving b. More precisely, we prove a scaling law for the diameter of the apparent (that is, macroscopic) support of the droplet in time. This is an intermediate scaling law: It takes an initial layer to "forget" the initial droplet shape - whereas after a long time, the droplet is so thin that its spreading is governed by the physics on the scale b.
Our proof works by deriving suitable estimates for physically relevant integral quantities: the free energy, the length of the apparent support and their respective rates of change. As opposed to matched asymptotic methods, this PDE approach closely mimics a simple heuristic argument based on the gradient flow structure.