Dynamics of nearly unstable axisymmetric liquid bridges

The dynamics of a noncylindrical, axisymmetric, marginally unstable liquid bridge between two equal disks is analyzed in the inviscid limit. The resulting model allows for the weakly nonlinear description of both the (first stage of) breakage for unstable configurations and the (slow) dynamics for stable configurations. The analysis is made for both slender and short liquid brides. In the former range, the dynamics breaks reflection symmetry on the midplane between the supporting disks and can be described by a standard Duffing equation, while for short bridges reflection symmetry is preserved and the equation is still Duffing-like but exhibiting a quadratic nonlinearity. The asymptotic results compare well with existing experiments

Linear oscillations of axisymmetric viscous liquid bridges

Small amplitude free oscillations of axisymmetric capillary bridges are considered for varying values of the capillary Reynolds number C-1 and the slenderness of the bridge Λ . A semi-analytical method is presented that provides cheap and accurate results for arbitrary values of C-1 and Λ ; several asymptotic limits (namely, C>> 1, C>>1, Λ >> 1 \ {and} \ |π -Λ |>> 1 ) are considered in some detail, and the associated approximate results are checked. A fairly complete picture of the (fairly complex) spectrum of the linear problem is obtained for varying values of C and Λ . Two kinds of normal modes, called capillary and hydrodynamic respectively, are almost always clearly identified, the former being associated with free surface deformation and the latter, only with the internal flow field; when C is small the damping rate associated with both kind of modes is comparable, and the hydrodynamic ones explain the appearance of secondary (steady or slowly-varying) streaming flow

A note on the effect of surface contamination in water wave damping

Asymptotic formulas are derived for the effect of contamination on surface wave damping in a brimful circular cylinder; viscosity is assumed to be small and contamination is modelled through Marangoni elasticity with insoluble surfactant. It is seen that an appropriately chosen finite Marangoni elasticity provides an explanation for a significant amount of the unexplained additional damping rate in a well-known experiment by Henderson & Miles (1994); discrepancies are within 15%, significantly lower than those encountered by Henderson & Miles (1994) under the assumption of inextensible film

La Razonable Utilidad de las Matemáticas: Una Visión Personal

Este artículo es una modificación de la Lección Magistral que impartí el 10 de diciembre de 2008 a la Promoción de Ingenieros Aeronáuticos que se graduó ese día. En atención al auditorio, el tono de la lección era relativamente elemental, casi de divulgación. Aunque gran parte de los lectores de esta revista son ingenieros aeronáuticos, me ha parecido conveniente elevar solamente un poco ese tono en el artículo, por dos razones. Primero, porque otros lectores de la revista pueden agradecerlo. Y, en segundo lugar, porque el tono ayuda a recordar las ideas básicas, relativamente sencillas, que hay detrás de muchos problemas complicados

Singular Langmuir-Hinshelwood Reaction-Diffusion Problems: Strongly Nonisothermal Conditions

The steady state reaction-diffusion problem for a permeable catalytic particle is considered, when the reaction rate is of the Langmuir-Hinshelwood type and the activation energy is large. It is shown that there are multiple solutions for the Damkohler number belonging to a certain interval. An arbitrarily large number of solutions appear for symmetric particles: (a) in two dimensions if the adsorption effects are sufficiently important and the reaction order is negative, and (b) in three dimensions. An asymptotic analysis provides approximate analytical expressions for the response curves and for the multiplicity bounds

Singular Langmuir-Hinshelwood reaction-diffusion problems. strong absorption under quasi-isothermal conditions

The steady state reaction-diffusion problem is considered for a permeable catalytic particle with Langmuir-Hinshelwood kinetics under isothermal and quasi-isothermal conditions. It is known that there may be multiple solutions due to either strong adsorption or external thermal effects; in the first case, an arbitrarily large number of solutions may appear for symmetric pellets in two and three dimensions. An asymptotic analysis provides analytical expressions for the response curve of the particle and for the multiplicity bounds. The approximate results compare quite well with those computed numerically, even in cases in which the gauge functions of the approximation scheme are of the logarithmic type

Viscous effects in parametrically excited water waves.

The effect of viscosity is considered in the capillary.gravity waves that are parametrically excited by vertical vibrations in a horizontal fluid layer. special attention latory boundary layer attached to the solid walls and the free surface. It is wxplained that this secondary mean flow affects the dynamics of the primary waves themselves. Several specific limiting cases of practical interest are considered to illustrate the consequences of this coupled evolution

Large activation energy analysis of the ignition of self-heating porous bodies

A large activation energy analysis of the problem of thermal ignition of self-heating porous bodies is carried out by means of a regular perturbation method. A correction to the well-known Frank-Kamenetskii estimate of the ignition limit is calculated, for symmetric bodies, by using similarity properties of the equations giving higher order terms in an expansion in powers of \/E (E = activation energy). Our estimate compares well with numerical results, and differs from others in the literature, which are not better than Frank-Kamenetskii's one from an asymptotic point of view. Dirichlet and Robin type of boundary conditions are considered. A brief analysis of the extinction problem for no reactant consumption is also presented

Weakly nonlinear nonaxisymmetric oscillations of capillary bridges at small viscosity

Weakly nonlinear nonaxisymmetric oscillations of a capillary bridge are considered in the limit of small viscosity. The supporting disks of the liquid bridge are subjected to small amplitude mechanical vibrations with a frequency that is close to a natural frequency. A set of equations is derived for accounting the slow dynamics of the capillary bridge. These equations describe the coupled evolution of two counter-rotating capillary waves and an associated streaming flow. Our derivation shows that the effect of the streaming flow on the capillary waves cannot be a priori ignored because it arises at the same order as the leading (cubic) nonlinearity. The system obtained is simplified, then analyzed both analytically and numerically to provide qualitative predictions of both the relevant large time dynamics and the role of the streaming flow. The case of parametric forcing at a frequency near twice a natural frequency is also considere

On the steady streaming flow due to high-frequency vibration in nearly inviscid liquid bridges

The steady streaming flow due to vibration in capillary bridges is considered in the limiting case when both the capillary Reynolds number and the non-dimensional vibration frequency (based on the capillary time) are large. An asymptotic model is obtained that provides the streaming flow in the bulk, outside the thin oscillatory boundary layers near the disks and the interface. Numerical integration of this model shows that several symmetric and non-symmetric streaming flow patterns are obtained for varying values of the vibration parameters. As a by-product, the quantitative response of the liquid bridge to high-frequency axial vibrations of the disks is also obtained