Lagrangian Transport through Surfaces in Compressible Flows

Abstract

A material-based, i.e., Lagrangian, methodology for exact integration of flux by volume-preserving flows through a surface has been developed recently in [Karrasch, SIAM J. Appl. Math., 76 (2016), pp. 1178-1190]. In the present paper, we first generalize this framework to general compressible flows, thereby solving the donating region problem in full generality. Second, we demonstrate the efficacy of this approach on a slightly idealized version of a classic two-dimensional mixing problem: transport in a cross-channel micromixer, as considered recently in 1 We use the term surface as the short form of hypersurface, i.e., a codimension-one submanifold of space or space-time. In three dimensions, this refers to classic surfaces, in two dimensions to curves. arXiv:1707.00518v2 [physics.flu-dyn]