We consider the stability of a chemical equilibrium of a thermally conducting two component reactive viscous mixture which is situated in a horizontal layer heated from below and experiencing a catalyzed chemical reaction at the bottom plate. We provide a method for an easy derivation of the nonlinear stability bound, derived explicitely in terms of the involved physical parameters. It enables us to derive a linearization principle in a large sense, i.e. to prove that the linear and nonlinear stability bounds are equal.

In this paper we study the nonlinear Lyapunov stability of the mechanical equilibrium for a fluid mixture in a plane layer, in presence of linear skewsymmetric effects, such as the Coriolis term in the rotating Bénard problem. If we reformulate the nonlinear stability problem, projecting the initial perturbation evolution equations on some suitable orthogonal subspaces, we preserve the contribution of the skewsymmetric term, and jointly all the nonlinear terms vanish, indipendently from the boundary conditions. In this way we recover some sufficient conditions of nonlinear stability that are necessary and sufficient conditions of linear stability too. Because of this we can not require from the beginning that some functionals occurring in the energy relation must be definite positive. After studying the problem we can select only the results where such functionals satisfy the previous requisites, by using the linear instability analysis.

In this paper we study the nonlinear Lyapunov stability of the conduction-diffusion solution in a layer of a rotating Newtonian fluid, heated and salted from below. If we reformulate the nonlinear stability problem, projecting the initial perturbation evolution equations on some suitable orthogonal subspaces, we preserve the contribution of the Coriolis term, and jointly all the nonlinear terms vanish. We prove that, if the principle of exchange of stabilities holds, the linear and nonlinear stability bounds are equal. We find that the nonlinear stability bound is nothing else but the critical Rayleigh number obtained solving the linear instability problem

In this work we consider the non linear stability of a chemical equilibrium of a thermally conducting two component reactive viscous mixture which is situated in a horizontal layer heated from below and experiencing a catalyzed chemical reaction at the bottom plate. The evolution equation for the perturbation energy is deduced with an approach which generalizes the Joseph’s parametric differentiation method. Moreover, the nonlinear stability bound for the chemical equilibrium of the fluid mixture is derived in terms of thermal and concentrational non dimensional numbers.

Si studiano la stabilità lineare e non lineare dell'equilibrio termodiffusivo nel problema di Bénard, e si dimostra la coincidenza del limiti di stabilità lineare e non lineare, nel caso in cui l'instabilità insorga come convezione stazionaria.

Si studia la stabilità dell'equilibrio chimico di una miscela viscosa termoconduttrice, in uno strato piano. Nell'ipotesi in cui sussista il principio di scambio delle stabilità si determina la regione dello spazio dei parametri in cui i limiti di stabilità lineare e non lineare coincidono.

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 r

In this paper we study the nonlinear Lyapunov stability of the conduction-diffusion solution of a rotating couple-stress fluid, in a layer heated and salted from below. After reformulating the perturbation evolution equations in a suitable equivalent form, we derive the appropriate Lyapunov function and we prove that, if the principle of exchange of stabilities holds, the linear and nonlinear stability bounds are equal. The nonlinear stability bound is exactly the critical Rayleigh number obtained solving the linear instability of the conduction-diffusion solution