TY - JOUR

T1 - Coupling the Navier–Stokes–Fourier equations with the Johnson–Segalman stress-diffusive viscoelastic model: Global-in-time and large-data analysis

AU - Bathory, Michal

AU - Bulíček, Miroslav

AU - Málek, Josef

PY - 2024/1/19

Y1 - 2024/1/19

N2 - We prove that there exists a large-data and global-in-time weak solution to a system of partial differential equations describing the 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 principal 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.

AB - We prove that there exists a large-data and global-in-time weak solution to a system of partial differential equations describing the 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 principal 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.

KW - Johnson–Segalman

KW - weak solution

KW - Viscoelastic heat-conducting fluids

KW - Johnson-Segalman

UR - http://www.scopus.com/inward/record.url?scp=85183532393&partnerID=8YFLogxK

U2 - 10.1142/S0218202524500064

DO - 10.1142/S0218202524500064

M3 - Article

VL - 34

SP - 417

EP - 476

JO - Mathematical Models and Methods in Applied Sciences

JF - Mathematical Models and Methods in Applied Sciences

SN - 0218-2025

IS - 3

ER -