Mathematical models for the study of granular fluids

Abstract

This Ph.D. thesis aims to obtain and to develop some mathematical models to understand some aspects of the dynamics of heterogeneous granular fluids. More precisely, the expected result is to develop three models, one where the dynamics of the granular material is modeled using a mixture theory approach, and the other two, where we consider the granular fluid is modeled using a multiphase approach involving rigid structures and fluids. More precisely: In the first model, we obtained a set of equations based on the mixture theory using homogenization tools and a thermodynamic procedure. These equations reflect two essential properties of granular fluids: the viscous nature of the intersticial fluid and a Coulomb-type of behavior of the granular component. With our equations, we study the problem of a dense granular heterogeneous flow, composed by a Newtonian fluid and a solid component in the setting of the Couette flow between two infinite cylinders. In the second model, we consider the motion of a rigid body in a viscoplastic material. The 3D Bingham equations model this material, and the Newton laws govern the displacement of the rigid body. Our main result is the existence of a weak solution for the corresponding system. The weak formulation is an inequality (due to the plasticity of the fluid), and it involves a free boundary (due to the motion of the rigid body). The proof is achieved using an approximated problem and passing it to the limit. The approximated problems consider the regularization of the convex terms in the Bingham fluid and by using a penalty method to take into account the presence of the rigid body. In the third model, we consider the motion of a perfect heat conductor rigid body in a heat conducting Newtonian fluid. The 3D Fourier-Navier-Stokes equations model the fluid, and the Newton laws and the balance of internal energy model the rigid body. Our main result is the existence of a weak solution for the corresponding system. The weak formulation is composed by the balance of momentum and the balance of total energy equation which includes the pressure of the fluid, and it involves a free boundary (due to the motion of the rigid body). To obtain an integrable pressure, we consider a Navier slip boundary condition for the outer boundary and the mutual interface. As in the second problem, the proof is achieved using an approximated problem and passing it to the limit. The approximated problems consider a regularization of the convective term and a penalty method to take into account the presence of the rigid body.