Dense Suspension Flows: a Mathematical Model Consistent with Thermodynamics

We address the flows of dense suspensions of particles within the framework of two-velocity continuum. Thermodynamics of such a continuum is developed by the method suggested in the papers of L. D. Landau and I. M. Khalatnikov. As an application, we consider the convective settling problem. We capture the Boycott effect and prove that the enhanced sedimentation occurs in a 10 tilted vessel due to vortices. We do not call on additional interphase forces like the Stokes drag, the virtual mass force, the Archimedes force, the Basset-Boussinesq force and etc. Instead, we apply a generalized Fick's law for the particle mass concentration flux vector.

Open Mathematical Aspects of Continuum Thermodynamics: Hyperbolicity, Boundaries and Nonlinearities

Thermodynamics is continuously spreading in the engineering practice, which is especially true for non-equilibrium models in continuum problems. Although there are concepts and approaches beyond the classical knowledge, which are known for decades, their mathematical properties, and consequences of the generalizations are less-known and are still of high interest in current researches. Therefore, we found it essential to collect the most important and still open mathematical questions that are related to different continuum thermodynamic approaches. First, we start with the example of Classical Irreversible Thermodynamics (CIT) in order to provide the basis for the more general and complex frameworks, such as the Non-Equilibrium Thermodynamics with Internal Variables (NET-IV) and Rational Extended Thermodynamics (RET). Here, we aim to present that each approach has its specific problems, such as how the initial and boundary conditions can be formulated, how the coefficients in the partial differential equations are connected to each other, and how it affects the appearance of nonlinearities. We present these properties and comparing the approach of NET-IV and RET to each other from these points of view. In the present work, we restrict ourselves on non-relativistic models.