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On the incompressible limit for the Navier-Stokes-Fourier system in domains with wavy bottoms. (English) Zbl 1158.35072
The motion of a compressible viscous and heat conducting fluid occupying a domain $\Omega\subset \bbfR^3$ is described by a triple of functions -- the density $\rho(x,t)$, the velocity $u(x,t)$, the temperature $\vartheta(x,t)$. These functions satisfy the Navier-Stokes-Fourier system of equations $$\aligned \frac{\partial \rho}{\partial t}&+\text{div}\,(\rho u)=0,\\ \frac{\partial }{\partial t}(\rho u)+\text{div}\,(\rho u\otimes u)&+ \frac{1}{\text{Ma}^2}\nabla p(\rho,\vartheta)=\text{div}\,S+\frac{1}{\text{Fr}^2}\rho\nabla F,\\ \frac{\partial }{\partial t}(\rho s(\rho,\vartheta))&+\text{div}\,(\rho s(\rho,\vartheta)u)+\text{div}\,\left(\frac{q}{\vartheta}\right)=\sigma,\\ \frac{d}{dt}\int_\Omega&\left(\frac{\text{Ma}^2}{2}\rho \vert u\vert ^2 +\rho e(\rho,\vartheta)-\frac{\text{Ma}^2}{\text{Fr}^2}\rho F \right)dx=0, \endaligned\tag1$$ where $S$ is a viscous stress tensor, the heat flux $q$ obeys Fourier’s law $$q=-k(\vartheta)\nabla\vartheta,$$ $\sigma$ is the entropy production, $p$ is the pressure, $s$ is the specific entropy, $e$ is the specific internal energy, Ma and Fr denote the Mach and Froude numbers. Let $\bar{\vartheta}$ and $\bar{\rho}$ are average quantities $$\bar{\vartheta}=\frac{1}{\vert \Omega\vert }\int_\Omega\vartheta\,dx,\quad \bar{\rho}=\frac{1}{\vert \Omega\vert }\int_\Omega\rho\,dx.$$ Setting Ma=$\varepsilon$, Fr=$\sqrt{\varepsilon}$, where $\varepsilon$ is a small parameter, the triple of unknown functions is represented by $$\rho=\rho_\varepsilon=\bar{\rho}+\varepsilon r_\varepsilon,\quad u=u_\varepsilon,\quad \vartheta=\vartheta_\varepsilon=\bar{\vartheta}+\varepsilon \theta_\varepsilon.$$ It is proved that the limits $$r_\varepsilon\rightarrow r,\quad u_\varepsilon\rightarrow u,\quad \theta_\varepsilon\rightarrow \theta \quad \text{as}\ \varepsilon\rightarrow 0$$ satisfy to the Oberbeck-Boussinesq system $$\aligned \text{div}\,u=0&,\\ \bar{\rho}\big(\frac{\partial u}{\partial t}+\text{div}\,(u\otimes u)\big)+\nabla P&=\bar{\mu}\Delta u-r\nabla F,\\ \bar{\rho}\bar{c}_p\big(\frac{\partial \theta}{\partial t}+\text{div}\,(\theta u)\big)-& \text{div}\,(\bar{k}\nabla \theta)=0,\\ r+\bar{\rho}\bar{\alpha}(\theta-\bar{\theta})&=0, \endaligned$$ where the viscosity coefficient $\bar{\mu}$, the specific heat at constant pressure $\bar{c}_p$, the heat conductivity coefficient $\bar{k}$ and the coefficient of thermal expansion $\bar{\alpha}$ are evaluated at $\bar{\rho},\bar{\vartheta}$. It is shown that the oscillations of the sound waves are effectively damped by the presence of a “wavy bottom” of physical domain.

35Q30Stokes and Navier-Stokes equations
35B35Stability of solutions of PDE
76N15Gas dynamics, general
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