Transport of interfaces with surface tension by 2D viscous flows
Summary: We consider the problem of finding a global weak solution for two-dimensional, incompressible viscous flow on a torus, containing a surface-tension bearing curve transported by the flow. This is the simplest case of a class of two-phase flows considered by \textit{P. I. Plotnikov} [Sib. Math. J. 34, No.~4, 704--716 (1993); translation from Sib. Mat. Zh. 34, No.~4, 127--141 (1993; Zbl 0814.76007)] and \textit{H. Abels} [Interfaces Free Bound. 9, No.~1, 31--65 (2007; Zbl 1124.35060)]. Our work complements Abels' analysis by examining this special case in detail. We construct a family of approximations and show that the limit of these approximations satisfies, globally in time, an incomplete set of equations in the weak sense. In addition, we examine criteria for closure of the limit system, we find conditions which imply nontrivial dependence of the limiting solution on the surface tension parameter, and we obtain a new system of evolution equations which models our flow-interface problem, in a form that may be useful for further analysis and for numerical simulations.
scientific article; zbMATH DE number 5690536