Singular Limits in Thermodynamics of Viscous Fluids

The fundamental laws of continuum mechanics interpreted as infinite families of integral identities introduced in Chapter 1, rather than systems of partial differential equations, give rise to the concept of weak (or variational) solutions that can be vastly extended to extremely divers physical systems of various sorts. The main stumbling block of this approach when applied to the field equations of fluid mechanics is the fact that the available a priori estimates are not strong enough in order to control the flux of the total energy and/or the dissipation rate of the kinetic energy. This difficulty has been known since the seminal work of Leray [132] on the incompressible Navier-Stokes system, where the validity of the so-called energy equality remains an open problem, even in the class of suitable weak solutions introduced by Caffarelli et al. [37]. The question is whether or not the rate of decay of the kinetic energy equals the dissipation rate due to viscosity as predicted by formula (1.39). It seems worth noting that certain weak solutions to hyperbolic conservation laws indeed dissipate the kinetic energy whereas classical solutions of the same problem, provided they exist, do not. On the other hand, however, we are still very far from complete understanding of possible singularities, if any, that may be developed by solutions to dissipative systems studied in fluid mechanics. The problem seems even more complex in the framework of compressible fluids, where Hoff [113] showed that singularities survive in the course of evolution provided they were present in the initial data. However, it is still not known if the density may develop “blow up” (gravitational collapse) or vanish (vacuum state) in a finite time. Quite recently, Brenner [28] proposed a daring new approach to fluid mechanics, where at least some of the above mentioned difficulties are likely to be eliminated.

The informal notion of a well-posed problem captures many of the desired features of what we mean by solving a system of partial differential equations. Usually a given problem is well posed if

This chapter develops the general ideas discussed in Section 4.2 focusing on the singular limits characterized by the spatially homogeneous (constant) distribution of the limit density. We start with the scaled Navier-Stokes-Fourier system introduced in Section 4.1 as a primitive system, where we take the Mach number Ma proportional to a small parameter ɛ, .

We expand the methods developed in the previous chapter in order to handle the strongly stratified systems discussed briefly in Section 4.3. In comparison with the previous considerations, a new aspect arises, namely the thermodynamic state variables ϱ and ϑ undergo a scaling procedure similar to that of kinematic quantities like velocity, length, and time. In particular, both thermal and caloric equations of state modify their form reflecting substantial changes of the material properties of the fluid.

As we have seen in the previous chapters, 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 the acoustic waves. If the physical domain is bounded, the acoustic waves, being reflected by the boundary, inevitably develop high frequency oscillations resulting in the weak convergence of the velocity field. This rather unpleasant phenomenon creates additional problems when handling the convective term in the momentum equation (cf. Sections 5.4.7, 6.6.3 above). In this chapter, we focus on the mechanisms so-far neglected by which the acoustic energy is dissipated into heat, and the ways in which the dissipation may be used in order to show strong (pointwise) convergence of the velocity.

Many theoretical problems in continuum fluid mechanics are formulated on unbounded physical domains, most frequently on the whole Euclidean space ℝ³. Although, arguably, any physical but also numerical space is necessarily bounded, the concept of an unbounded domain offers a useful approximation in situations when the influence of the boundary on the behavior of the system can be neglected. The acoustic waves examined in the previous chapters are often ignored in meteorological models, where the underlying ambient space is large when compared with the characteristic speed of the fluid as well as the speed of sound. However, as we have seen in Chapter 5, the way the acoustic waves “disappear” in the asymptotic limit may include fast oscillations in the time variable that may produce undesirable numerical instabilities. In this chapter, 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. Remarkably, although very large, our physical space is still bounded exactly in the spirit of the leading idea of this book that the notions of “large” and “small” depend on the chosen scale.

We interpret our previous results on the singular limits of the Navier-Stokes-Fourier system in terms of the acoustic analogies discussed briefly in Chapters 4, 5. Let us recall that an acoustic analogy is represented by a non-homogeneous wave equation supplemented with source terms obtained simply by regrouping the original (primitive) system. 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. Our goal is to discuss the advantages as well as limitations of this approach in light of the exact mathematical results obtained so far.

For readers’ convenience, a number of standard results used in the preceding text is summarized in this chapter. 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.

The material collected in Chapter 1 is standard. We refer to the classical monographs by Batchelor [18] or Lamb [131] for a full account of the mathematical theory of continuum fluid mechanics. A more recent treatment may be found in Truesdell and Noll [192] or Truesdell and Rajagopal [193]. An excellent introduction to the mathematical theory of waves in fluids is contained in Lighthill’ s book [135].