Rigorous asymptotics of traveling-wave solutions to the thin-film equation and Tanner's law

Lorenzo Giacomelli, Manuel Gnann, and Felix Otto

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Submission date: 12. Aug. 2015 (revised version: May 2016)
Pages: 44
published in: Nonlinearity, 29 (2016) 9, p. 2497-2536 
DOI number (of the published article): 10.1088/0951-7715/29/9/2497
Bibtex
MSC-Numbers: 35K65, 34B16, 34B40, 34C45, 34E10, 76A20
Keywords and phrases: degenerate parabolic equations, Singular nonlinear boundary value problems, Boundary value problems on infinite intervals, Invariant manifolds, Perturbations, asymptotics, thin fluid films
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Abstract:
We are interested in traveling-wave solutions to the thin-film equation with zero microscopic contact angle (in the sense of complete wetting without precursor) and inhomogeneous mobility h³ + λ³⁻ⁿhⁿ, where h, λ, and n ∈ denote film height, slip length, and mobility exponent, respectively. Existence and uniqueness have been established by Maria Chiricotto and the first of the authors in previous work under the assumption of sub-quadratic growth as h →∞.

In the present note we investigate the asymptotics of solutions as h ↘ 0 (the contact-line region) and h →∞. As h ↘ 0 we observe, to leading order, the same asymptotics as for traveling waves or source-type self-similar solutions to the thin-film equation with homogeneous mobility hⁿ and we additionally characterize corrections to this law. Moreover, as h →∞ we identify, to leading order, the logarithmic Tanner profile - i.e., the solution to the corresponding unperturbed problem with λ = 0 - that determines the apparent macroscopic contact angle. Besides higher-order terms, corrections turn out to affect the asymptotic law as h →∞ only by setting the length scale in the logarithmic Tanner profile. Moreover, we prove that both the correction and the length scale depend smoothly on n. Hence, in line with the common philosophy, the precise modeling of liquid-solid interactions (within our model, the mobility exponent) does not affect the qualitative macroscopic properties of the film.