Francisco J. Mancebo

On the linearization of some singular, nonlinear elliptic problems and applications☆

This paper deals with the spectrum of a linear, weighted eigenvalue problem associated with a singular, second order, elliptic operator in a bounded domain, with Dirichlet boundary data. In particular, we analyze the existence and uniqueness of principal eigenvalues. As an application, we extend the usual concepts of linearization and Frechet derivability, and the method of sub and supersolutions to some semilinear, singular elliptic problems.

Faraday instability threshold in large-aspect-ratio containers

We consider the Floquet linear problem giving the threshold acceleration for the appearance of Faraday waves in large-aspect-ratio containers, without further restrictions on the values of the parameters. We classify all distinguished limits for varying values of the various parameters and simplify the exact problem in each limit. The resulting simplified problems either admit closed-form solutions or are solved numerically by the well-known method introduced by Kumar & Tuckerman (1994). Some comparisons are made with ( a ) the numerical solution of the original exact problem, ( b ) some ad hoc approximations in the literature, and ( c ) some experimental results.

Weakly nonuniform thermal effects in a porous catalyst: asymptotic models and local nonlinear stability of the steady states

This paper considers a first-order, irreversible exothermic reaction in a bounded porous catalyst, with smooth boundary, in one, two, and three space dimensions. It is assumed that the characteristic reaction time is sufficiently small for the chemical reaction to be confined to a thin layer near the boundary of the catalyst, and that the thermal diffusivity is large enough for the temperature to be uniform in the reaction layer, but that it is not so large as to avoid significant thermal gradients inside the catalyst. For appropriate realistic limiting values of the several nondimensional parameters of the problem, several time-dependent asymptotic models are derived that account for the chemical reaction at the boundary (that becomes essentially impervious to the reactant), heat conduction inside the catalyst, and exchange of heat and reactant with the surrounding unreacted fluid. These models possess asymmetrical steady states for symmetric shapes of the catalyst, and some of them exhibit a rich dynami...

Weakly-nonlinear analysis of the Rayleigh–Taylor instability in a vertically vibrated, large aspect ratio container

We consider a horizontal liquid layer supported by air in a wide (as compared to depth) container, which is vertically vibrated with an appropriately large frequency, intending to counterbalance the Rayleigh-Taylor instability of the fiat, rigid-body vibrating state. We apply a long-wave, weakly-nonlinear analysis that yields a generalized Cahn-Hilliard equation for the evolution of the fluid interface, with appropriate boundary conditions obtained by a boundary layer analysis. This equation shows that the stabilizing effect of vibration is like that of surface tension, and is used to analyze the linear stability of the fiat state, and the local bifurcation at the instability threshold.