Dissipative Mechanisms

Abstract

A compressible heat-conducting homogeneous Newtonian fluid is described by the following system of equations for the velocity v, density ρ, and temperature θ at position x and time t:

$$\begin{array}{*{20}{c}} {\rho \left( {x,t} \right){{v}_{t}}\left( {x,t} \right) = - \rho \left( {x,t} \right)v\left( {x,t} \right)\bullet \nabla v\left( {x,t} \right)} \\ { - \nabla \pi \left( {p\left( {x,t} \right),\theta \left( {x,t} \right)} \right)} \\ { + \nabla \bullet \left[ {\mu \left( {\rho \left( {x,t} \right),\theta \left( {x,t} \right)} \right)\nabla v\left( {x,t} \right)} \right]} \\ \end{array}$$

((1.1))

$${{\rho }_{{\text{t}}}}({\text{x,t) = - v(x,t)}} \bullet \nabla \rho ({\text{x}},{\text{t}}){\text{ - }}\rho ({\text{x}},{\text{t}})\nabla \bullet {\text{v(x,t)}}$$

((1.2))

$$\frac{\partial }{{\partial {\text{t}}}}\varepsilon (\rho ({\text{x}},{\text{t}}),{\mkern 1mu} \theta ({\text{x}},{\text{t}})){\text{ = }}\nabla \bullet [\kappa (\rho ({\text{x}},{\text{t}}),{\mkern 1mu} \theta ({\text{x}},{\text{t}})\nabla \theta ({\text{x}},{\text{t}})]$$

((1.3))

Here and below, we denote partial derivatives with subscripts. The functions [0,∞) × [0, ∞) ∋ (ρ, θ) → π (ρ, θ), μ (ρ, θ), ε(ρ, θ), κ(ρ, θ) are prescribed constitutive functions, π represents a modified pressure, μ the viscosity, ε the internal energy, and κ the thermal conductivity. These constitutive functions are usually required to satisfy

$${{\pi }_{\rho }} > 0,{\mkern 1mu} \mu \geqslant 0,{\mkern 1mu} {{\varepsilon }_{\theta }} > 0,{\mkern 1mu} \kappa > {\text{0}}$$

((1.4))

Equations (1.1), (1.2), (1.3) are respectively the Navier-Stokes equations, the continuity equation, and the heat equation. If ρ and θ are regarded as given positive-valued functions, then (1.1) is a semilinear parabolic system for ν when μ is everywhere positive.