# Numerical Analysis: Multiscale methods in fluid dynamics

Multiscale problems appear in several areas of fluid dynamics. Complex fluids is one example, which abound in nature, with examples such as polymeric solutions, milk foam, paper pulp, blood and emulsions (mixtures of two immiscible liquid substances). Simple Newtonian liquids with immersed elastic or solid particles form suspensions that can display very complex and non-Newtonian dynamics. See for instance the example of sedimentation of point particles in Stoeks flow above. In general, flows of these complex fluids exhibit a strong multiscale coupling where the micro-scale dynamics of the particles affects the macro-dynamics of the flow and vice versa. Traditionally complex fluids have been simulated using macroscopic mathematical models that take into account the micro-scale influence on the macro-scale flow via constitutive relations that are more complex than for Newtonian flow. There are, however, many regimes where these models are insufficient.

In recent years there has been an intense activity and interest in the numerical treatment of general multiscale problems. Numerical frameworks like the “heterogeneous multiscale methods” (HMM), “superparameterization” and “equation free methods” have been developed and analyzed. The main idea of these frameworks is to bypass the precise macroscopic modeling and instead directly couple micro- and macroscale simulations in a systematic way. The microscopic model is never simulated fully, but only locally in time and space, to appropriately drive a solver for the macroscopic model. In this way, the computational cost will be much smaller than for direct simulation of the full fine scale model, while scale separation, or other scale structure assumptions, will ensure the accuracy of the approach.

The main objective is to develop methods based on the HMM framework for multiscale problems in fluid dynamics, with a special focus on particulate flows, i.e. suspensions with particles (solid or elastic of different shapes) immersed in a fluid. One aim would be to couple direct simulations of individual particles in Stokes flow with a model for the resulting macroscopic flow. In the initial stage we study elliptic PDEs with rapidly varying coefficients as a model problem.