zbMATH — the first resource for mathematics

A gluey particle model. (English) Zbl 1359.76112
Summary: We present here a new framework to handle the short-range interaction between rigid bodies in a viscous, incompressible fluid. This framework is built as the vanishing viscosity limit of a lubrication model. We restrict ourselves here to the case of a single particle and a rigid wall. Our approach is based on a standard first-order approximation for the lubrication force between two rigid bodies, where a small parameter \(\varepsilon\) plays the role of the underlying fluid viscosity. We establish convergence when \(\varepsilon\) goes to 0 of a subsequence of trajectories towards a solution to a problem of the hybrid type: it relies on two distinct states, unglued and glued, the latter being described by a new variable \(\gamma \) which expresses in a way the asymptotic smallness of the distance, and which plays the role of an adhesion potential. The limit problem has a surprising property: although it is well-posed in many situations, uniqueness does not generally hold as soon as left-hand clusters of contact times are allowed. Some prospective extensions of this model (other types of singularities, roughness of surfaces, macroscopic version) are proposed.

76D99 Incompressible viscous fluids
76D08 Lubrication theory
76M25 Other numerical methods (fluid mechanics) (MSC2010)
Full Text: DOI