Finite-time singularities in the dynamical evolution of contact lines

2013/02/05 by Dmitry E. Pelinovsky, D. E. Pelinovsky, Pelinovsky, D. E. +3
Engineering · Materials Science · Mathematics · Physics and Astronomy · #Adhesion, Friction, and Surface Interactions #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Fluid Dynamics and Thin Films #Surface Modification and Superhydrophobicity #math.AP #physics.flu-dyn

paper · pdf · doi:10.48550/arxiv.1302.1218

9 pages, 2 figures

arxiv created 2013/02/05 · openalex publication_date 2013/02/05 · arxiv updated 2013/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study finite-time singularities in the linear advection-diffusion equation with a variable speed on a semi-infinite line. The variable speed is determined by an additional condition at the boundary, which models the dynamics of a contact line of a hydrodynamic flow at a 180 contact angle. Using apriori energy estimates, we derive conditions on variable speed that guarantee that a sufficiently smooth solution of the linear advection--diffusion equation blows up in a finite time. Using the class of self-similar solutions to the linear advection-diffusion equation, we find the blow-up rate of singularity formation. This blow-up rate does not agree with previous numerical simulations of the model problem.

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