Real materials are neither ideal solids nor ideal liquids. Lava, glaciers, toothpaste, tomato sauce, hair gel, non-drip paint and many more gel-like fluids are usually plastic solids, but they begin to flow like a viscous liquid under sufficiently large stresses. This phenomenon is called viscoplasticity.
From the mathematical viewpoint, the Bingham, Casson and Herschel-Bulkley models for viscoplastic fluids lead to challenging convex optimisation problems in function spaces, coupled with the Navier-Stokes and possibly further transport equations. I am developing fast numerical algorithms to solve such problems
- in their genuinely nonsmooth form without regularisation
- with an optimal worst-case convergence rate
- without any heuristic algorithm parameters.