A dynamical systems approach for the contact-line singularity in thin-film flows. (October 2016)

• Record Type: Journal Article
• Authors: Belgacem, Fethi Ben; Gnann, Manuel V.; Kuehn, Christian
• Is Part Of: Nonlinear analysis. Volume 144(2016)
• Page Start: 204
• Page End: 235
• Publication Date: 2016-10
• Subjects: 76A20 -- 37N10 -- 35K25 -- 35K65 -- 34B16; Thin film equation -- Self-similar solution -- Contact line -- Center manifolds -- Resonances -- Boundary-value problem; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.06.010
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We are interested in a complete characterization of the contact-line singularity of thin-film flows for zero and nonzero contact angles. By treating the model problem of source-type self-similar solutions, we demonstrate that this singularity can be understood by the study of invariant manifolds of a suitable dynamical system. In particular, we prove regularity results for singular expansions near the contact line for a wide class of mobility exponents and for zero and nonzero dynamic contact angles. Key points are the reduction to center manifolds and identifying resonance conditions at equilibrium points. The results are extended to radially-symmetric source-type solutions in higher dimensions. Furthermore, we give dynamical systems proofs for the existence and uniqueness of self-similar droplet solutions in the nonzero dynamic contact-angle case.