Basic a priori bounds are derived. We give a complete proof of existence of weak solutions to the Navier—Stokes—Fourier system.
Feireisl / Novotný | Singular Limits in Thermodynamics of Viscous Fluids
The proof is very technical and rather involved combining various techniques of nonlinear analysis and the theory of partial differential equations. Our goal was to provide a concise but at the same time self—contained treatment of the problem without any restriction on the size of the initial data and the length of the existence interval. The extreme generality of the full Navier-Stokes-Fourier system whereby the equations describe the entire spectrum of possible motions—ranging from sound waves, cyclone waves in the atmosphere, to models of gaseous stars in astrophysics—constitutes a serious defect of the equations from the point of view of applications.
Eliminating unwanted or unimportant modes of motion, and building in the essential balances between flow fields, allow the investigator to better focus on a particular class of phenomena and to potentially achieve a deeper understanding of the problem.
Scaling and asymptotic analysis play an important role in this approach. By scaling the equations, meaning by choosing appropriately the system of the reference units, the parameters determining the behavior of the system become explicit.
Asymptotic analysis provides a useful tool in the situations when certain of these parameters called characteristic numbers vanish or become infinite. We develop the general ideas concerning singular limits in fluid mechanics focusing on the incompressible low Mach number limit with spatially homogeneous constant density profile.
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We show that solutions of the Navier-Stokes-Fourier system approach in the regime the Oberbeck- Boussinesq approximation. We develop methods to handle the case of strongly stratified fluids. In particular, both thermal and caloric equations of state modify their form reflecting substantial changes of the material properties of the fluid.
One of the most delicate issues in the analysis of singular limits for the Navier-Stokes-Fourier system in the low Mach number regime is the influence of acoustic waves.
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If the physical domain is bounded and the complete slip boundary conditions imposed, the acoustic waves, being reflected by the boundary, inevitably develop high frequency oscillations resulting in the weak convergence of the velocity field, in particular, its gradient part converges to zero only in the sense of integral means. This rather unpleasant phenomenon creates additional problems when handling the convective term in the momentum equation.
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Here, we focus on the mechanisms by which the acoustic energy may be dissipated, and the ways how the dissipation may be used in order to show strong pointwise convergence of the velocities in the incompressible limit. Many theoretical problems in continuum fluid mechanics are formulated on unbounded physical domains, most frequently on the whole Euclidean space. Although, arguably, any physical but also numerical space is necessarily bounded, the concept of unbounded domain offers a useful approximation in the situations when the influence of the boundary or at least its part on the behavior of the system can be neglected.
We examine the incompressible limit of the Navier—Stokes—Fourier System in the situation when the spatial domain is large with respect to the characteristic speed of sound in the fluid. We interpret certain results on the singular limits of the Navier-Stokes-Fourier system in terms of the acoustic analogies.
An acoustic analogy is represented by a non-homogeneous wave equation supplemented with source terms obtained simply by regrouping the original primitive system.
Singular limits in thermodynamics of viscous fluids
In the low Mach number regime, the source terms may be evaluated on the basis of the limit incompressible system. This is the principal idea of the so-called hybrid method used in numerical analysis. Nowadays classical statements are appended with the relevant reference material, while complete proofs are provided in the cases when a compilation of several different techniques is necessary. A significant part of the theory presented below is related to general problems in mathematical fluid mechanics and may be of independent interest.
We refer to the classical monographs by Batchelor [ 20 ] or Lamb [ ] for the full account on the mathematical theory of continuum fluid mechanics.
It can be represented by the Navier-Stokes-Fourier-system that combines Newton's rheological law for the viscous stress and Fourier's law of heat conduction for the internal energy flux. As a summary, this book studies singular limits of weak solutions to the system governing the flow of thermally conducting compressible viscous fluids. Skip to main content Skip to table of contents.
Advertisement Hide. Singular Limits in Thermodynamics of Viscous Fluids. Front Matter Pages i-xxxvi.
Fluid Flow Modeling. Pages Weak Solutions, A Priori Estimates.
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Existence Theory. Asymptotic Analysis — An Introduction. Singular Limits — Low Stratification.