Boundary conditions for dynamic wetting - A mathematical analysis
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https://doi.org/10.1140/epjst/e2020-900249-7
THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS
Regular Article
Boundary conditions for dynamic wetting A mathematical analysis Mathis Frickea and Dieter Bothe Department of Mathematics, Technische Universit¨ at Darmstadt, Darmstadt, Germany Received 31 October 2019 / Accepted 6 July 2020 Published online 14 September 2020 Abstract. The moving contact line paradox discussed in the famous paper by Huh and Scriven has lead to an extensive scientific discussion about singularities in continuum mechanical models of dynamic wetting in the framework of the two-phase Navier–Stokes equations. Since the no-slip condition introduces a non-integrable and therefore unphysical singularity into the model, various models to relax the singularity have been proposed. Many of the relaxation mechanisms still retain a weak (integrable) singularity, while other approaches look for completely regular solutions with finite curvature and pressure at the moving contact line. In particular, the model introduced recently in [A.V. Lukyanov, T. Pryer, Langmuir 33, 8582 (2017)] aims for regular solutions through modified boundary conditions. The present work applies the mathematical tool of compatibility analysis to continuum models of dynamic wetting. The basic idea is that the boundary conditions have to be compatible at the contact line in order to allow for regular solutions. Remarkably, the method allows to compute explicit expressions for the pressure and the curvature locally at the moving contact line for regular solutions to the model of Lukyanov and Pryer. It is found that solutions may still be singular for the latter model.
1 Introduction Starting with the work by Huh and Scriven [1], the scientific discussion about singularities became central for the continuum mechanical modeling of dynamic wetting (see, e.g., [2–8]). While it is generally accepted that the non-integrable singularity in the viscous dissipation introduced by the no-slip condition (see [1]) is unphysical for a viscous fluid, there are different approaches to relax the singularity. See [4–9] and references therein for an overview of existing models and the field of dynamic wetting in general. In particular, the articles and controversial discussion notes contained in [6] provide a comprehensive overview about methods and open questions. For the sake of brevity, we only consider two particular approaches in this paper. Note, however, that the method of compatibility analysis is general in nature and is applicable to other modeling approaches as well. a
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The European Physical Journal Special Topics
One of the most prominent choices is the Navier slip law which allows for tangential slip at the solid boundary according to hv, ni = 0, −λ(v − w)k = 2η(Dn)k ,
(1) (2)
where λ > 0 is a friction coefficient, w is the velocity of the solid boundary, n is the unit outer normal and D = 21 (∇v + ∇v T ) is the rate-of-deformation tensor. The parameter L = η/λ is called slip-length and control
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