Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Coupling the Navier-Stokes-Fourier equations with the Johnson-Segalman stress-diffusive viscoelastic model: Global-in-time and large-data analysis

Michal Bathory, Miroslav Bulíček, Josef Málek

TL;DR

This work develops a mathematically rigorous framework for a coupled Navier–Stokes–Fourier system describing an incompressible heat-conducting viscoelastic fluid with stress diffusion, driven by a Johnson–Segalman-type elastic response and temperature-dependent material properties. By deriving the model from fundamental balance laws and thermodynamics, the authors ensure thermodynamic consistency and formulate a robust weak solution concept that incorporates temperature, entropy, and energy inequalities. The main contribution is a global-in-time existence theory for large data, proven via a sophisticated approximation scheme (Galerkin with cut-offs) and careful limits that preserve positivity and energy-entropy structure; in $d\le3$, a pressure field is obtained and local energy balance holds. The results provide a solid foundation for rigorous analysis of thermo-mechanically coupled viscoelastic flows with stress diffusion, with potential implications for polymeric and complex fluids where temperature effects are essential.

Abstract

We prove that there exists a~large-data and global-in-time weak solution to a~system of partial differential equations describing an unsteady flow of an incompressible heat-conducting rate-type viscoelastic stress-diffusive fluid filling up a~mechanically and thermally isolated container of any dimension. To overcome the~principle difficulties connected with ill-posedness of the~diffusive Oldroyd-B model in three dimensions, we assume that the~fluid admits a~strengthened dissipation mechanism, at least for excessive elastic deformations. All the~relevant material coefficients are allowed to depend continuously on the~temperature, whose evolution is captured by a~thermodynamically consistent equation. In fact, the~studied model is derived from scratch using only the~balance equations for linear momentum and energy, the~formulation of the~second law of thermodynamics and the~constitutive equation for the~internal energy. The~latter is assumed to be a~linear function of temperature, which simplifies the~model. The~concept of our weak solution incorporates both the~temperature and entropy inequalities, and also the~local balance of total energy provided that the~pressure function exists.

Coupling the Navier-Stokes-Fourier equations with the Johnson-Segalman stress-diffusive viscoelastic model: Global-in-time and large-data analysis

TL;DR

This work develops a mathematically rigorous framework for a coupled Navier–Stokes–Fourier system describing an incompressible heat-conducting viscoelastic fluid with stress diffusion, driven by a Johnson–Segalman-type elastic response and temperature-dependent material properties. By deriving the model from fundamental balance laws and thermodynamics, the authors ensure thermodynamic consistency and formulate a robust weak solution concept that incorporates temperature, entropy, and energy inequalities. The main contribution is a global-in-time existence theory for large data, proven via a sophisticated approximation scheme (Galerkin with cut-offs) and careful limits that preserve positivity and energy-entropy structure; in , a pressure field is obtained and local energy balance holds. The results provide a solid foundation for rigorous analysis of thermo-mechanically coupled viscoelastic flows with stress diffusion, with potential implications for polymeric and complex fluids where temperature effects are essential.

Abstract

We prove that there exists a~large-data and global-in-time weak solution to a~system of partial differential equations describing an unsteady flow of an incompressible heat-conducting rate-type viscoelastic stress-diffusive fluid filling up a~mechanically and thermally isolated container of any dimension. To overcome the~principle difficulties connected with ill-posedness of the~diffusive Oldroyd-B model in three dimensions, we assume that the~fluid admits a~strengthened dissipation mechanism, at least for excessive elastic deformations. All the~relevant material coefficients are allowed to depend continuously on the~temperature, whose evolution is captured by a~thermodynamically consistent equation. In fact, the~studied model is derived from scratch using only the~balance equations for linear momentum and energy, the~formulation of the~second law of thermodynamics and the~constitutive equation for the~internal energy. The~latter is assumed to be a~linear function of temperature, which simplifies the~model. The~concept of our weak solution incorporates both the~temperature and entropy inequalities, and also the~local balance of total energy provided that the~pressure function exists.
Paper Structure (7 sections, 4 theorems, 340 equations)

This paper contains 7 sections, 4 theorems, 340 equations.

Table of Contents

  1. Introduction
  2. Thermodynamical compatibility of the model
  3. Weak formulation & main result
  4. Existence of a weak solution: the approximative problem
  5. Existence of a weak solution: limits w, l
  6. Global energy equality for dle3
  7. Auxiliary results

Key Result

Theorem 3.1

Suppose that all the assumptions of Definition Dws are fulfilled. Then, there exists a weak solution of the system s1--s15ic in the sense of Definition Dws. In addition, if $d\leq 3$ and $\Omega\in\mathcal{C}^{1,1}$, then there is a pressure ${\rm p}\in L^{\frac{d+2}{d}}(Q;\mathbb{R})$ such that the and also Qv can be generalized to

Theorems & Definitions (8)

  • Definition 3.1
  • Theorem 3.1
  • Lemma A.1
  • proof
  • Lemma A.2
  • proof
  • Lemma A.3
  • proof