The motion of a compressible viscous and heat conducting fluid occupying a domain is described by a triple of functions – the density , the velocity , the temperature . These functions satisfy the Navier-Stokes-Fourier system of equations
where is a viscous stress tensor, the heat flux obeys Fourier’s law
is the entropy production, is the pressure, is the specific entropy, is the specific internal energy, Ma and Fr denote the Mach and Froude numbers.
Let and are average quantities
Setting Ma=, Fr=, where is a small parameter, the triple of unknown functions is represented by
It is proved that the limits
satisfy to the Oberbeck-Boussinesq system
where the viscosity coefficient , the specific heat at constant pressure , the heat conductivity coefficient and the coefficient of thermal expansion are evaluated at .
It is shown that the oscillations of the sound waves are effectively damped by the presence of a “wavy bottom” of physical domain.