Mixing and Mixtures in Geo- and Biophysical Flows: A Focus on Mathematical Theory and Numerical Methods

Activated fluids: continuum description, analysis and computational results

Josef Málek

Charles University in Prague


In the first part, we provide a systematic classification of incompressible fluid-like materials, ranging from the Euler fluid to rigid-body solids, paying particular attention to the concept of viscosity, fluidity and their generalizations (leading to shear-thickening, shear-thinning, stress-thinning and stress-thickening fluids) and involving also responses that are due to an activation criterion (while Bingham fluid can serve as a standard example, the Euler-fluid, which, after activation, responds as a Navier-Stokes fluid, serves as an interesting new class). In addition, we provide a similar systematic study for both activated and non-activated boundary conditions, ranging from slip to no-slip. Implicit constitutive theory provides an elegant framework for expressing such responses involving activation criteria in a compact form that is more suitable for further mathematical analysis. The presentation will also include an introduction to a general thermodynamic approach suitable for the development of constitutive relations for stress, energy flux, etc. In the second part, we explore mathematical properties of unsteady internal three-dimensional flows in bounded smooth domains for activated incompressible fluids and activated boundary conditions. We study the global-in-time existence of weak solutions in the sense of Leray. After reformulation of the problem in the setting of maximal monotone graphs, we first explain both the easy steps and the difficulties in establishing the stability of the problem under consideration with respect to weakly converging sequences. Finally, we study in detail one interesting nontrivial case, which concerns Bingham fluids with stick-slip boundary conditions.