Stability criteria for fluid flows

In this volume, the main methods, techniques and tricks used to derive sufficient conditions for fluid flow stability are discussed. In general, nonlinear and linear cases require different treatments, thus we have to dfferentiate between linear and nonlinear criteria. With a few exceptions, the treatment is analytical, but connections with the geometric viewpoint of dynamical systems are also outlined. Inequalities and their use are crucial for finding stability criteria. That is why particular attention is paid to classical or generalized analytical inequalities, especially to those relating integrals of functions and their derivatives. The best constants involved into the last ones can be viewed as extrema of some associated functionals. If the extrema are with constraints, the corresponding inequalities are the so-called isoperimetric inequalities. Further, in order to solve the associated variational problems, direct methods, based on expansions in Fourier series upon total sets of functions, turned out to be among the most efficient. The Fourier series can be introduced directly into the functional or into the corresponding Euler equations, which, in the isoperimetric case, are eigenvalue problems. Moreover, the expansion functions may be chosen to satisfy all boundary conditions of the problem or part of them (especially when even and odd derivatives occur in equations or/and boundary conditions). Finally, in looking for variational principles natural conditions may occur. Several variational aspects related to functional inequalities used to prove stability criteria emerge, to justify the insertion of an entire chapter(3) devoted to variational problems. Algebraic and differential inequalities are summarized in Appendix 1 together with some formulae of tensor analysis. A great amount of hydrodynamic and hydromagnetic stability criteria exist, and we do not intend to present them all, having chosen to limit ourselves to the founder's criteria, our own results, and a few other results of the Italian and Romanian schools in the field, for mixtures and in the Benard magnetic case, for free or rigid walls. In addition, we are concerned mainly with convection problems (including temperature; concentration; magnetic, Soret, Dufour, Hall, ion-slip, dielectrophoretic effects in horizontal layers) for viscous fluids or fluid mixtures. Only in a few cases, horizontal convection and other effects are considered. In most cases, we use variational methods, methods of Hilbert spaces theory, methods based on inequalities (isoperimetric or not), the Fourier series method and a direct method based on the characteristic equation. This is why, with a few exceptions, in the linear cases treated by us, the ordinary differential equations (ode's) have constant coeffcients but a higher order. In the nonlinear case, for the sake of simplicity, in order to have a symmetrizable linearized part, we preferred basic equilibria or steady flows. Consequently, this book considers fluid flows whose stability properties do not depend on local phenomena. Nowadays, hydrodynamic stability theory is involved in important ecological and industrial problems, requiring a lot of effects, and characteristics of fluids congurations, other than traditional ones, being taken into account. We treat a few of these complex problems in a didactic way, in order to be useful to other similar topics. We do not repeat classical and, by now, simple results of hydrodynamic and hydromagnetic stability theory. They can be found in the basic monographs on the topic. We go further instead with more complex but still basic subjects, e.g. linearization principle, universal stability criteria, stability spectrum estimates, variational principles, improved energy methods, treatment of problems with intricate boundary conditions. Moreover, for all worked examples, detailed computations are given. Throughout the book we try as much as possible to treat the most realistic cases. Firstly, this is related especially to the boundary conditions which \spoil the symmetry" of the mathematical problem, the given problem being reformulated so that elegant functional analytic methods apply to the new setting. Secondly, we bring into actuality some other powerful approaches (e.g. Budiansky-DiPrima (BD) method, backward integration method) frequently used several decades ago, and almost forgotten by now by applied mathematicians. Thirdly, we call the attention of mathematicians interested in fluid flow stability towards powerful methods (e.g. Joseph's differentiation of parameters approach, B-D method, the direct method). Minimal prerequisites are: fundamentals of classical calculus of variations (Eulerequations, isoperimetric problems, variational principles), linear and nonlinear functional analysis.