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## Über dieses Buch

This contributed volume is based on talks given at the August 2016 summer school “Fluids Under Pressure,” held in Prague as part of the “Prague-Sum” series. Written by experts in their respective fields, chapters explore the complex role that pressure plays in physics, mathematical modeling, and fluid flow analysis. Specific topics covered include:Oceanic and atmospheric dynamicsIncompressible flowsViscous compressible flowsWell-posedness of the Navier-Stokes equationsWeak solutions to the Navier-Stokes equationsFluids Under Pressure will be a valuable resource for graduate students and researchers studying fluid flow dynamics.

## Inhaltsverzeichnis

### Chapter 1. An Approach to the Primitive Equations for Oceanic and Atmospheric Dynamics by Evolution Equations

Abstract
The primitive equations for oceanic and atmospheric dynamics are a fundamental model for many geophysical flows. In this chapter we present a summary of an approach to these equations based on the theory of evolution equations. In particular, we discuss the hydrostatic Stokes operator, well-posedness results in critical spaces within the L p(L q)-scale, within the scaling invariant space L (L 1) for Neumann boundary conditions, and within the L (L p) space for mixed boundary conditions and p > 3. In addition, we investigate real analyticity of solutions, convergence of the scaled Navier-Stokes equations to the primitive equations, and the existence of periodic solutions for large forces.
Matthias Hieber, Amru Hussein

### Chapter 2. Viscous Compressible Flows Under Pressure

Abstract
This chapter deals with the role of pressure in the theory of viscous compressible flows. The pressure state laws and viscosities are described. Special attention is devoted to non-monotone pressure laws and pressure dependent viscosities. The global existence proofs are discussed for approximate systems. Some relevant physical applications are described, including among others the anelastic Euler equations, shallow water model, granular media, or mixture problems.
Didier Bresch, Pierre-Emmanuel Jabin

### Chapter 3. Global Well-Posedness for Incompressible–Incompressible Two-Phase Problem

Abstract
This chapter is devoted to some mathematical analysis of the two-phase problem for the viscous incompressible–incompressible capillary flows separated by sharp interface, this problem being called two-phase problem for short, and the Stokes equations with transmission conditions on the sharp interface which is arised from the two-phase problem.
The maximal regularity is a character of the system of equations of parabolic type, and it is a very powerful tool in solving quasilinear equations of parabolic type. The authors of this lecture note have developed a systematic method to derive the maximal regularity theorem for the initial-boundary value problem for the Stokes equations with non-homogeneous boundary conditions, which is based on the $$\mathcal {R}$$ bounded solution operators theory and L. Weis’ operator valued Fourier multiplier theorem. The notion of $$\mathcal {R}$$ boundedness plays an essential role in the Weis’ theory, which takes the place of boundedness in the standard Fourier multiplier theorem of Marcinkiewicz-Mikhilin-Hölmander type.
In this lecture note, we explain how to use the $$\mathcal {R}$$-bounded solution operators to derive the maximal regularity theorem for the Stokes equations with transmission conditions, and as an application of our maximal regularity theorem, we prove the local well-posedness of the two-phase problem, where the solutions are obtained in the L p in time and L q in space maximal regularity class. So far, this framework gives us the best possible regularity class of parabolic quasilinear equations.
Moreover, we prove the global well-posedness for the two-phase problem both in the bounded domain case and the unbounded domain case. A key tool is the decay property of the C 0 analytic semigroup associated with the Stokes equations with transmission conditions. In the bounded domain case, the decay properties are obtained essentially from the analysis of zero eigenvalue. As a result we prove the exponential stability of our C 0 analytic semigroup in some quotient space, which, together with the conservation of momentum and angular momentum and the maximal regularity theorem, yields the global well-posedness in the case of small initial data and the ball-like reference domain.
On the other hand, in the unbounded domain case, the zero is a continuous spectrum for the Stokes equations with transmission conditions, and so we can prove the polynomial decay properties for the C 0 analytic semigroup only, which, combined with L p-L q maximal regularity theorem with suitable choices of p and q, yields the L p time summability of the L q space norm of solutions to the nonlinear problem. From this we prove the global well-posedness for the small initial data in the unbounded domain case. Notice that the L p summability yields L 1 summability in view of the Hölder inequality, which is enough to handle the kinetic equations.
What we want to emphasize here is that our method is based on the construction of $$\mathcal {R}$$ bounded solution operators and spectral analysis of the zero eigenvalue or generalized eigenvalue of the generalized resolvent problem for the Stokes equations with transmission conditions. The spectral analysis here can be used widely to study the other parabolic linear and quasilinear equations arising from the mathematical study of viscous fluid flows as well as other models in mathematical physics like MHD, multicomponent flows, namatic crystal flows, and so on.
Yoshihiro Shibata, Hirokazu Saito

### Chapter 4. The Role of Pressure in the Theory of Weak Solutions to the Navier-Stokes Equations

Abstract
Sections 4.1 and 4.2 contain an introduction, notation and definitions and basic properties of used function spaces and operators. A pressure, associated with a weak solution to the Navier-Stokes equations for incompressible fluid, is constructed in Sect. 4.3. The interior regularity of the pressure in regions, where the velocity satisfies Serrin’s integrability conditions, is studied in Sect. 4.4. Finally, Sect. 4.5 is devoted to criteria of regularity for weak solutions to the Navier-Stokes equations, formulated in terms of the pressure.
Jiří Neustupa

### Chapter 5. Flows of Fluids with Pressure Dependent Material Coefficients

Abstract
It has been well documented in many studies that the material parameters of a fluid may essentially depend on the pressure and that they can vary by several orders of magnitude. The material parameters, for which this dependence is observed, are mainly the viscosity (due to the internal forces in the fluid) and the friction (due to fluid–(rigid) solid interaction). In addition, these large variations with respect to the pressure in the material parameters occur although the variations of the density are almost negligible (in comparison with other parameters). Therefore it is still reasonable to describe the above mentioned phenomena in many fluids by incompressible models. Likewise, the viscosity and the drag of many fluids vary with the shear rate and such shear (rate)-dependent viscosity and friction are extensively used, ranging from geophysics, chemical engineering, and bio-material science up to the food industry, enhanced oil recovery, carbon dioxide sequestration, or extraction of unconventional oil deposits, etc.
The aim of this study is to present an overview of available results for models with very complicated rheological laws used in engineering praxes. As particular examples that fit into the class of models studied here, we refer to the Darcy model, to the Brinkman models, and to the Bingham models. Nevertheless, the aim of this study is much more ambitious and we go much beyond these standard models and present a kind of unifying theory, which is based on the use of the so-called maximal monotone graphs, which seems to be very appropriate from the point of view of mathematical analysis of the problem.
Miroslav Bulíček

### Chapter 6. Finite Element Pressure Stabilizations for Incompressible Flow Problems

Abstract
The simulation of incompressible flow problems with pairs of velocity-pressure finite element spaces that do not satisfy the discrete inf-sup condition requires a so-called pressure stabilization. This chapter provides a survey of available methods which are presented for the Stokes problem to concentrate on the main ideas and to avoid additional difficulties originating from more complicated models. The methods can be divided into residual-based stabilizations and stabilizations that utilize only the pressure. For the first class, a comprehensive numerical analysis is presented, whereas for the second class, the presentation is more concise except for a detailed analysis of a local projection stabilization method. Connections of various pressure stabilizations to inf-sup stable discretizations with velocity spaces enriched by bubble functions are also discussed. Numerical studies compare several of the available pressure stabilizations.
Volker John, Petr Knobloch, Ulrich Wilbrandt

### Chapter 7. Finite-Volume Methods for Navier-Stokes Equations

Abstract
Flows of engineering interest can only be predicted by using numerical solution methods, because the governing equations cannot be solved analytically. Among many possible approaches, finite-volume methods have become popular in this field and are the basis for most commercial and public computer codes used to simulate fluid flow. One of the most widely used methods, which is applicable to complex geometries and arbitrary polyhedral computational grids, is described in this chapter. The solution method was originally developed for incompressible flows and later extended to compressible flows at any speed. It is also described how to deal with moving grids which follow the motion of solid bodies. The applicability of the method to various flow types is demonstrated by simulating flow around sphere at various Reynolds numbers, ranging from a creeping flow at Re = 5 to a supersonic flow at Re = 5,000,000. The final example involves a sphere oscillating in water at 6000 Hz with a very small amplitude, which requires that compressibility of water is taken into account. The fact that the same method can be applied to such a wide class of flows, including a wide range of turbulence-modeling approaches, is the main reason why it is used in general-purpose commercial codes.
Milovan Perić
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