Mathematical Fluid Mechanics

This contribution is a review of mathematical results concerning the Navier—Stokes equations. The reader should be aware that it is far from being complete and that it has been organized according to the taste and field of interest of the writer.

Compared with the needs of physicists, engineers and even people who design large numerical methods, the contribution of mathematics seems to be very limited. Therefore, I try to show that even in this limitation, mathematics contributes seriously to the understanding of the problem.

Eventually both for physical reason and to reach a better understanding, it seems adequate to insert the Navier—Stokes equations in a large hierarchy of equations. This hierarchy starts from the Hamiltonian systems with large number of particles and ends with models of turbulence used in engineering. Emphasis is put on this hierarchy and the relations existing between its various steps.

A relatively simple iterative scheme for steady compressible viscous flow was given in a recent paper of Heywood and Padula [5]. It was intended as a basis for both the existence theory and for numerical methods. Among the assumptions they made in introducing the scheme, one was that the force should be small. On the other hand, by another method, Novotnÿ and Pileckas [7] recently extended the existence theory for such flow to the case of forces that are small perturbations of large potential forces. The purpose of the present paper is to incorporate the ideas of Novotnÿ and Pileckas, for treating such forces, into the simpler and more constructive scheme of Heywood and Padula. This is one step in our larger objective, which is to base the existence theory for compressible viscous flow on constructive methods that may be suggestive of new numerical methods.

We present a mathematical rigorous analysis of the linear Rayleigh—Taylor instability for some simple model flows which occur in connection with inertial confinement fusion. We consider the effect of a smooth density profile and of a convection velocity on the growth rate of the instability. We use an asymptotic analysis for studying the growth of large wavelength perturbations.

We review some recent results on a priori estimates, compactness, global existence, and the long-time behaviour of solutions to the Navier-Stokes equations of a compressible fluid flow.

In modern technologies one often meets the necessity to solve compressible flow (i. e. flow of gases) with a complicated structure. Let us mention, e. g., aviation design, turbomachinery, automobile, food and chemical industry, etc. There is a number of simplified models used for the simulation of gas flow, as inviscid irrotational subsonic flow, small disturbance transonic flow and full potential transonic flow. Modern computers allow us to use more and more complex models. Here we shall be concerned with the models of gas flow described by the Euler equations (inviscid flow) and the Navier—Stokes equations (viscous flow). We shall formulate initial-boundary value problems of gas dynamics and discuss numerical methods for their solution.

This article is devoted to recent developments and open questions concerning instabilities in ideal fluid flows. It is argued that in some appropriate sense almost all steady flows of an ideal incompressible fluid are unstable. However there are different kinds of instability. Many of the instabilities that are described could be termed “slow” and technically they are associated with the Jordan cell structure of the governing operator as opposed to the “fast” instabilities associated with isolated unstable eigenvalues. Numerous examples are given to stress the importance for the existence of instabilities of the norm in which the growth of disturbance is measured.

This contribution deals with the modern finite volume schemes solving the Euler and Navier-Stokes equations for transonic flow problems. We will mention the TVD theory for first order and higher order schemes and some numerical examples obtained by 2D central and upwind schemes for 2D transonic flows in the GAMM channel or through the SE 1050 turbine cascade of Skoda Plzeñ.

In the next part two new 2D finite volume schemes are presented. Explicit composite scheme on a structured triangular mesh and implicit scheme realized on a general unstructured mesh. Both schemes are used for the solution of inviscid transonic flows in the GAMM channel and the implicit scheme also for the flows through the SE 1050 turbine cascade using both triangular and quadrilateral meshes. For the case of the flows through the SE 1050 turbine we compare the numerical results with the experiment.

The TVD MacCormack method as well as a finite volume composite scheme are extended to a 3D method for solving flows through channels and turbine cascades.

This paper continues the efforts of Xie [4], [5], [6], [7], [8] and the present author [2] to prove an inequality of the form sup₀< for solutions of the Stokes equations in an arbitrary three-dimensional domain 1^-2, with a constantcindependent of the domain. Here the norms are L²-norms over S2, and A denotes the Stokes operator. The function u is assumed to be solenoidal, and to vanish on the boundary and at spatial infinity. Xie [6] proved the inequality modulo one missing point that he left as a conjecture. Recently, in [22], we showed that the desired inequality would also follow from another conjecture that seems to us more approachable. The present paper offers some partial results and observations from our efforts to prove this new conjecture.

We discuss uniqueness, regularity and stability of weak and strong solutions to the Navier-Stokes equations. We first introduce the classL ^s (0,T;L ^r (ℝ ⁿ ))of Serrin and give a brief survey of well—posedness. Then we devote ourselves to various estimates in the BMO—Hardy spaces and to the Sobolev embedding in the Besov space in the critical case. Making use of the technique in the harmonic analysis, we prove a criterion on regularity and break-down of the solutions. Finally, we show the stability of smooth solutions with definite convergence rates.

We formulate sufficient conditions for regularity of a suitable weak solution(v; p ) in a sub—domainD of the time—space cylinderQT in Section 3. The conditions are anisotropic in the sense that the assumptions about v_i v₂ (the first two components of velocity) differ from the assumptions about the third component of velocity v₃. The question what types of deformations of infinitely small volumes of the fluid support regularity and what types contribute to a blow—up is studied in Section 4. Finally, we mention some open problems in Section 5.