arXiv subjects

Tomáš Los

Publications and source records attributed to Tomáš Los.

4 recordsLinked to original sources

arXiv2026

On three-dimensional flows of thermo-viscoelastic fluids of Giesekus type

Viscoelastic rate-type fluid models constitute a fundamental framework for the mathematical description of complex materials exhibiting coupled elastic and viscous effects, with a wide range of applications in engineering, biomaterials, and medicine. In realistic regimes, thermal effects are essential and lead to strongly coupled systems in which heat conduction and temperature-dependent constitutive laws play a decisive role. In this paper, we develop a thermodynamically consistent model for heat-conducting viscoelastic rate-type fluids. We establish the existence of a global weak solution in the full three-dimensional setting. In contrast to the existing literature, no smallness, regularity, or structural restrictions on the initial data are imposed beyond natural energy and entropy bounds, and no additional regularising mechanisms such as artificial stress diffusion are required. The analysis is based on a weak-strong framework combining energy and entropy balances with compactness tools, allowing us to treat the full nonlinear coupling between the fluid velocity, the temperature, and the elastic stress.

math.AP↗

arXiv2024

On three dimensional flows of viscoelastic fluids of Giesekus type

Viscoelastic rate-type fluids are popular models of choice in many applications involving flows of fluid-like materials with complex micro-structure. A well-developed mathematical theory for the most of these classical fluid models is however missing. The main purpose of this study is to provide a complete proof of long-time and large-data existence of weak solutions to unsteady internal three-dimensional flows of Giesekus fluids subject to a no-slip boundary condition. As a new auxiliary tool, we provide the identification of certain biting limits in the parabolic setting, presented here within the framework of evolutionary Stokes problems. We also generalize the long-time and large-data existence result to higher dimensions, to viscoelastic models with multiple relaxation mechanisms and to viscoelastic models with different type of dissipation.

math.AP↗

arXiv2020

On unsteady flows of pore pressure-activated granular materials

We investigate mathematical properties of the system of nonlinear partial differential equations that describe, under certain simplifying assumptions, evolutionary processes in water-saturated granular materials. The unconsolidated solid matrix behaves as an ideal plastic material before the activation takes place and then it starts to flow as a Newtonian or a generalized Newtonian fluid. The plastic yield stress is non-constant and depends on the difference between the given lithostatic pressure and the pressure of the fluid in a pore space. We study unsteady three-dimensional flows in an impermeable container, subject to stick-slip boundary conditions. Under realistic assumptions on the data, we establish long-time and large-data existence theory.

math.AP↗

arXiv2019

On three-dimensional flows of pore pressure activated Bingham fluids

We are concerned with a system of partial differential equations describing internal flows of homogeneous incompressible fluids of Bingham type in which the value of activation (the so-called yield) stress depends on the internal pore pressure governed by an advection-diffusion equation. After providing the physical background of the considered model, paying attention to the assumptions involved in its derivation, we focus on the PDE analysis of the initial and boundary value problems. We give several equivalent descriptions for the considered class of fluids of Bingham type. In particular, we exploit the possibility to write such a response as an implicit tensorial constitutive equation, involving the pore pressure, the deviatoric part of the Cauchy stress and the velocity gradient. Interestingly, this tensorial response can be characterized by two scalar constraints. We employ a similar approach to treat stick-slip boundary conditions. Within such a setting we prove long time and large data existence of weak solutions to the evolutionary problem in three dimensions.

math.AP↗