# Low stratification of a heat-conducting fluid in time-dependent domain

## TL;DR

This paper investigates the low Mach number limit of the Navier-Stokes-Fourier system in a moving domain with low stratification, revealing an additional term in the temperature equation due to domain motion.

## Contribution

It extends the analysis of low Mach number limits to moving domains, identifying a new term in the temperature equation linked to boundary motion.

## Key findings

- Limit system is the Oberback-Boussinesq system with an extra temperature term.
- Properties of Helmholtz decomposition on moving domains are crucial.
- Eigenvalues of Neumann Laplace operator depend on time.

## Abstract

We study the low Mach number limit of the full Navier-Stokes-Fourier system in the case of low stratification with ill-prepared initial data for the problem stated on moving domain with prescribed motion of the boundary. Similarly as in the case of a fixed domain we recover as a limit the Oberback-Boussinesq system, however we identify one additional term in the temperature equation of the limit system which is related to the motion of the domain and which is not present in the case of a fixed domain. One of the main ingredients in the proof are the properties of the Helmholtz decomposition on moving domains and the dependence of eigenvalues and eigenspaces of the Neumann Laplace operator on time.