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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 Ω 3 is described by a triple of functions – the density ρ(x,t), the velocity u(x,t), the temperature ϑ(x,t). These functions satisfy the Navier-Stokes-Fourier system of equations

ρ t+div(ρu)=0, t(ρu)+div(ρuu)+1 Ma 2 p(ρ,ϑ)=divS+1 Fr 2 ρF, t(ρs(ρ,ϑ))+div(ρs(ρ,ϑ)u)+divq ϑ=σ,d dt Ω Ma 2 2ρ|u| 2 +ρe(ρ,ϑ)-Ma 2 Fr 2 ρFdx=0,(1)

where S is a viscous stress tensor, the heat flux q obeys Fourier’s law

q=-k(ϑ)ϑ,

σ 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 ϑ ¯ and ρ ¯ are average quantities

ϑ ¯=1 |Ω| Ω ϑdx,ρ ¯=1 |Ω| Ω ρdx·

Setting Ma=ε, Fr=ε, where ε is a small parameter, the triple of unknown functions is represented by

ρ=ρ ε =ρ ¯+εr ε ,u=u ε ,ϑ=ϑ ε =ϑ ¯+εθ ε ·

It is proved that the limits

r ε r,u ε u,θ ε θasε0

satisfy to the Oberbeck-Boussinesq system

divu=0,ρ ¯u t + div (uu)+P=μ ¯Δu-rF,ρ ¯c ¯ p θ t + div (θu)-div(k ¯θ)=0,r+ρ ¯α ¯(θ-θ ¯)=0,

where the viscosity coefficient μ ¯, the specific heat at constant pressure c ¯ p , the heat conductivity coefficient k ¯ and the coefficient of thermal expansion α ¯ are evaluated at ρ ¯,ϑ ¯.

It is shown that the oscillations of the sound waves are effectively damped by the presence of a “wavy bottom” of physical domain.

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