Comparing Extremes: Mixing of Fluids, Mixing of Solids

Abstract

Mixing’s reach is remarkably wide; examples appear in the liquid-liquid, solid-liquid, and solid-solid areas. Traditional engineering approaches tackle all of these problems on a case-by-case basis, no description being expected to cover all possible situations. Much can be gained, however, by juxtaposing extremes; fluid-fluid mixing and solid-solid mixing provide such bounds. For fluids mixing, basic understanding at a continuum level is firmly established: Navier-Stokes equations provide a first principles description valid on macroscopic scales for most problems. Somewhat paradoxically, basic understanding for solids mixing is much less developed despite the fact that a first principles description — particle dynamics — is arguably better for solids than for fluids. Shortcomings of continuum descriptions manifest themselves on macroscopic scales, particle segregation being an instance where physical meso-scale processes are not well understood. Slow mixing of powders and slow mixing of fluids can be described by maps; in powders the map is a succession of distinct avalanches; in fluids the repetition of stretching and folding. In both cases rather simple pictures can be extended to the point that non-trivial conclusions can be obtained. In mixing of fluids repeated stretching and folding leads to chaos, the disorder often being accompanied by symmetries and regularity; in powders by unmixed cores and systems that mix or do not mix depending upon the sense of rotation.