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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 \(\Omega\subset \mathbb{R}^3\) 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 \[ \begin{aligned} \frac{\partial \rho}{\partial t}&+\text{div}\,(\rho u)=0,\\ \frac{\partial }{\partial t}(\rho u)+\text{div}\,(\rho u\otimes u)&+ \frac{1}{\text{Ma}^2}\nabla p(\rho,\vartheta)=\text{div}\,S+\frac{1}{\text{Fr}^2}\rho\nabla F,\\ \frac{\partial }{\partial t}(\rho s(\rho,\vartheta))&+\text{div}\,(\rho s(\rho,\vartheta)u)+\text{div}\,\left(\frac{q}{\vartheta}\right)=\sigma,\\ \frac{d}{dt}\int_\Omega&\left(\frac{\text{Ma}^2}{2}\rho | u| ^2 +\rho e(\rho,\vartheta)-\frac{\text{Ma}^2}{\text{Fr}^2}\rho F \right)dx=0, \end{aligned}\tag{1} \] 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 \[ \bar{\vartheta}=\frac{1}{| \Omega| }\int_\Omega\vartheta\,dx,\quad \bar{\rho}=\frac{1}{| \Omega| }\int_\Omega\rho\,dx. \] 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 \[ \begin{aligned} \text{div}\,u=0&,\\ \bar{\rho}\big(\frac{\partial u}{\partial t}+\text{div}\,(u\otimes u)\big)+\nabla P&=\bar{\mu}\Delta u-r\nabla F,\\ \bar{\rho}\bar{c}_p\big(\frac{\partial \theta}{\partial t}+\text{div}\,(\theta u)\big)-& \text{div}\,(\bar{k}\nabla \theta)=0,\\ r+\bar{\rho}\bar{\alpha}(\theta-\bar{\theta})&=0, \end{aligned} \] 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.

MSC:

35Q30 Navier-Stokes equations
35B35 Stability in context of PDEs
76N15 Gas dynamics (general theory)
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