On non-Newtonian Fluids with Energy Transfer

Abstract

The model combining incompressible Navier-Stokes' equation in a non-Newtonian p-power-law modification and the nonlinear heat equation is considered. Existence of its (very) weak solutions is proved for p > 11/5 under mild assumptions of the temperature-dependent stress tensor by careful successive limit passage in a Galerkin approximation. (2000). 35Q35, 76A05, 80A20. Mathematics Subject Classification

Existence of Proper Weak Solutions to the Navier-Stokes-Fourier System

“…Note that µ(θ ε )|Du ε | 2 → µ(θ ν )|Du ν | 2 holds in L 1 (Q T ) (compare to (25)) and it is not required the strong convergence of the pressure ∇p ε → ∇p ν in L 2 (Q T ) (compare to (24)) since (28) holds for φ = p ε . Then, the initial value problem (26)-(27) passes to the limit as ε tends to zero (ν fixed), concluding the below regularized problem.…”

“…Indeed, in the N-S-F system, the viscosity is a temperature dependent function and the temperature is a solution of a parabolic equation with L 1 -data due to the existence of the Joule effect. By these reasons, in the work [6,25] (and some references therein) the N-S-F system is constituted by momentum and total energy equations in order to the L 1 dissipative term in the internal energy equation is obtained by approximative methods using the weakly sequential lower semicontinuity of the norm. On the other hand, in the N-S system the study of the class of suitable weak solutions is being of interest since [7,20,26,27], if homogeneous Dirichlet boundary conditions are enforced (see for instance [15,16]).…”