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.

在Huh和Scriven的著名论文中讨论的移动接触线悖论引起了关于两相Navier-Stokes方程框架中动态润湿的连续力学模型中的奇点的广泛科学讨论。由于防滑条件在模型中引入了不可积分的奇异性，因此提出了各种松弛奇异性的模型。许多松弛机制仍然保持弱（可积分）奇异性，而其他方法则寻求在运动接触线上具有有限曲率和压力的完全规则的解。特别是，该模型最近在[AV Lukyanov，T. Pryer，Langmuir 33，8582（2017）]的目标是通过修改边界条件进行常规求解。本工作将相容性分析的数学工具应用于动态润湿的连续模型。基本思想是，边界条件必须在接触线上兼容，以便进行常规求解。值得注意的是，该方法允许为移动接触线局部计算压力和曲率的显式表达式，以对Lukyanov和Pryer模型进行常规求解。发现对于后一种模型，解决方案可能仍然是奇异的。