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In this paper a linear theory for dielectric relaxation phenomena in polarizable reacting fluid mixtures is developed, in the frame of thermodynamics of irreversible processes with internal variables. The microscopic irreversible phenomena giving rise to dielectric relaxation are described splitting the total specific polarization in two irreversible parts and introducing one of these partial specific polarizations as internal variable in the thermodynamic state vector. The phenomenological equations for these fluid mixtures are derived and, in the linear case, a generalized Debye equation for dielectric relaxation phenomena is derived. Special cases are also treated. Linear theories for polarizable continuous media with dielectric relaxation phenomena were derived in the same frame of non-equilibrium thermodynamics with internal variables in previous papers by one of the authors (LR). A phenomenological theory for these phenomena was developed by Maugin for complex materials, using microscopic considerations and introducing particular partial polarizations per unit mass. The obtained results in this paper have applications in several fields of applied sciences, as, for instance, in medicine and biology, where complex fluids presenting dielectric relaxation, are constitued by different types of molecules, with own dielectric susceptibility and relaxation time.

In this paper we study the nonlinear Lyapunov stability of the conduction-diusion solution of a rotating couple-stress uid, in a layer heated and salted from below. After reformulating the perturbation evolution equations in a suit- able 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 num-ber obtained solving the linear instability problem of the conduction- diusion solution.

In this paper we study the nonlinear Lyapunov stability of the conduction-diffusion solution of the rotating Benard problem. We provide a method for a derivation of the optimum nonlinear stability bound. It allows us to derive a linearization principle in a larger sense, i.e. to prove that, if the principle of exchange of stabilities holds, the linear and nonlinear stability bounds are equal. After reformulating the perturbation evolution equations in a suitable equivalent form, we derive the appropriate Lyapunov function and for the first time we find that the nonlinear stability bound is nothing else but the critical Rayleigh number obtained solving the linear instability problem of the conduction-diffusion solution.

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

In this paper, by the Lyapunov direct method, we study the nonlinear Lyapunov stability of the conduction–diffusion solution of the anisotropic magnetic Benard problem, for a fully ionized fluid. We show that, if the conduction–diffusion solution is linearly stable, it is asymptotically nonlinearly stable.

Assuming the validity of the principle of exchange of stabilities, the linear stability of the mechanical equilibrium of a horizontal fluid layer subject to a vertical temperature gradient, a vertical magnetic field, in the presence of Hall and ion-slip effects to normal mode perturbations is studied. The Budiansky–DiPrima method is used to solve the governing boundary value problem for an 11th order system of ordinary differential equations containing four physical parameters. The exact neutral stability curve was obtained in the form of a series. For truncation to one and two terms the coupled action of Hall and ion-slip currents is found destabilizing. The closed form of the Rayleigh number as a function of the other parameters, defining the neutral curves up the two terms, was obtained. The given problem was split into an even and an odd problem. The computation shows that the odd problem corresponding to odd velocity and temperature perturbations with respect to the vertical coordinate is more appropriate than the even one.

NEUTRAL THERMAL HYDRODYNAMIC AND HYDROMAGNETIC STABILITY HYPERSURFACES FOR A MICROPOLAR FLUID LAYER

The global stability of some mechanical equilibrium of a horizontal layer of a binary mixture heated and salted from below in the presence of the thermal diffusion and thermal diffusive conductivity is studied. The complexity of the problem induced by the large number of the physical parameters was avoided by means of two choices of the involved constants (others than in the Joseph approach). By using the Joseph's parametric differentiation method, the explicit formula for the nonlinear stability bound is easily derived in terms of all involved physical parameters.

We prove that, in the linear stability of the thermodiffusive flows in a layer heated from below, Hall and ion-slip currents have stabilizing effect with respect to the hydrodynamic case, but they have instabilizing effect with respect to the isotropic M.H.D. case.

We prove that in anisotropc MHD, ion slip currents can have instabilizing as well as stabilizing effects, with respect to Hall currents, on the linear stability of the thermodiffusive equilibrium of a fluid in a plane layer.