María Higuera

Linear oscillations of weakly dissipative axisymmetric liquid bridges

Linear oscillations of axisymmetric capillary bridges are analyzed for large values of the modified Reynolds number C−1. There are two kinds of oscillating modes. For nearly inviscid modes (the flow being potential, except in boundary layers), it is seen that the damping rate −ΩR and the frequency ΩI are of the form ΩR=ω1C1/2+ω2C+O(C3/2) and ΩI=ω0+ω1C1/2+O(C3/2), where the coefficients ω0≳0, ω1<0, and ω2<0 depend on the aspect ratio of the bridge and the mode being excited. This result compares well with numerical results if C≲0.01, while the leading term in the expansion of the damping rate (that was already known) gives a bad approximation, except for unrealistically large values of the modified Reynolds number (C≲10−6). Viscous modes (involving a nonvanishing vorticity distribution everywhere in the liquid bridge), providing damping rates of the order of C, are also considered.

Modal description of internal optimal streaks

This paper deals with the definition and description of optimal streaky (S) perturbations in a Blasius boundary layer. First, the asymptotic behaviours of S-perturbations near the free stream and the leading edge are studied to conclude that the former is slaved to the solution inside the boundary layer. Based on these results, a quite precise numerical scheme is constructed that allows concluding that S-perturbations produced inside the boundary layer, near the leading edge, can be defined in terms of just one streamwise-evolving solution of the linearized equations, associated with the first eigenmode of an eigenvalue problem first formulated by Luchini ( J. Fluid Mech ., vol. 327, 1996, p. 101). Such solution may be seen as an internal unstable streaky mode of the boundary layer, similar to eigenmodes of linearized stability problems. The remaining modes decay streamwise. Thus, the definition of streaks in terms of an optimization problem that is used nowadays is not necessary.

Linear nonaxisymmetric oscillations of nearly inviscid liquid bridges

Linear nonaxisymmetric oscillations of liquid bridges are analyzed in the limit of large capillary Reynolds number C−1. A boundary layer analysis is used that yields a second approximation of the damping rate and frequency in terms of the capillary Reynolds number, the slenderness of the bridge and the axial and azimuthal wave numbers of the mode being excited. Very good agreement with previous numerical results is obtained for C≲0.01, while the first correction of the damping rate gives a poor approximation, except for unrealistically small values of C (C≲10−5).

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 considered.

The role of fluid dynamics on compressed/expanded surfactant monolayers

A typical experiment to measure monolayer surface rheological properties consists of two parallel, slightly immersed, moving solid barriers that compress and expand a shallow liquid layer that contains the surfactant monolayer in its free surface. The area between the barriers controls the surfactant concentration, which is frequently assumed as spatially constant. In order to minimize the fluid dynamics and other non-equilibrium effects, the barriers motion is very slow. Nevertheless, the surfactant concentration dynamics exhibit some unexpected features such as irreversibility, suggesting that the motion is not slow enough. We present a long wave theory that takes into account the fluid dynamics in the bulk phase coupled to the free surface elevation. In addition, apparent irreversibility is also discussed that may result from artifacts associated with the menisci dynamics when surface tension is measured using a Wilhelmy plate. Instead, additional, purely chemical, non-equilibrium effects are ignored. ...

Nonlinear dynamics of confined liquid systems with interfaces subject to forced vibrations.

A review is presented of the dynamic behavior of confined fluid systems with interfaces under monochromatic mechanical forcing, emphasizing the associated spatio-temporal structure of the fluid response. At low viscosity, vibrations significantly affect dynamics and always produce viscous mean flows, which are coupled to the primary oscillating flow and evolve on a very slow timescale. Thus, unlike the primary oscillating flow, mean flows may easily interact with the surface rheology, which generates dynamics that usually exhibit a much slower timescale than that of typical gravity-capillary waves. The review is made with an eye to the typical experimental devices used to measure surface properties, which usually consist of periodically forced, symmetric fluid systems with interfaces. The current theoretical description of these systems ignores the fluid mechanics, which could play a larger role than presently assumed.

Interaction of Nearly-Inviscid, Multi-mode Faraday Waves and Mean Flows

Faraday waves [1] are gravity-capillary waves that are excited on the surface of a fluid when its container is vibrated vertically and the vertical acceleration exceeds a threshold value. These waves have received much attention in the literature both as a basic fluid dynamical problem and as a paradigm of a pattern-forming system [2],[3],[4]. Unfortunately, in the low viscosity limit, there are several basic issues that remain unresolved, particularly in connection with the generation of mean flows in the bulk. The viscous part of these flows (also called streaming flow or acoustic streaming) is driven by the oscillatory boundary layers attached to the solid walls and the free surface by well-known mechanisms first uncovered by Schlichting [5] and Longuet-Higgins [6]. This mean flow has been shown recently to affect the dynamics of the primary waves at leading order in a related, laterally vibrated system [7]. This is somewhat similar to the effect of an internal circulation on surface wave dynamics in drops [8].

Faraday waves, streaming flow, and relaxation oscillations in nearly circular containers.

Faraday waves near onset in an elliptical container are described by a third-order system of ordinary differential equations with characteristic slow-fast structure. These equations describe the interaction of standing waves with a weakly damped streaming flow driven by Reynolds stresses in boundary layers at the free surface and the rigid walls, and capture the proliferation with decreasing damping of periodic and nonperiodic relaxation oscillations observed near onset in previous simulations. These structures are the result of slow drift through symmetry-related Hopf bifurcations.

Nearly Inviscid Faraday Waves in Slightly Rectangular Containers

In the weakly inviscid regime parametrically driven surface gravity-capillary waves generate oscillatory viscous boundary layers along the container walls and the free surface. Through nonlinear rectification these generate Reynolds stresses which drive a streaming flow in the nominally inviscid bulk; this flow in turn advects the waves responsible for the boundary layers. The resulting system is described by amplitude equations coupled to a Navier-Stokes-like equation for the bulk streaming flow, with boundary conditions obtained by matching to the boundary layers, and represents a novel type of pattern-forming system. The coupling to the streaming flow is responsible for new types of secondary instabilities of standing waves leading to chaotic dynamics, and in appropriate regimes can lead to the presence of relaxations oscillations. These are present because in the nearly inviscid regime the streaming flow decays much more slowly than the waves, and resemble a class of oscillations discovered by Simonelli and Gollub [J. Fluid Mech. 199 (1989), 349] in a domain with an almost square cross-section.

NEAR-RESONANT, STEADY MODE INTERACTION: PERIODIC, QUASI-PERIODIC AND LOCALIZED PATTERNS ∗

Motivated by the rich variety of complex periodic and quasi-periodic patterns found in systems such as two-frequency forced Faraday waves, we study the interaction of two spatially periodic modes that are nearly resonant. Within the framework of two coupled one-dimensional Ginzburg--Landau equations we investigate analytically the stability of the periodic solutions to general perturbations, including perturbations that do not respect the periodicity of the pattern, and which may lead to quasi-periodic solutions. We study the impact of the deviation from exact resonance on the destabilizing modes and on the final states. In regimes in which the mode interaction leads to the existence of traveling waves our numerical simulations reveal localized waves in which the wavenumbers are resonant and which drift through a steady background pattern that has an off-resonant wavenumber ratio.