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

Nearly inviscid Faraday waves in containers with broken symmetry

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 various types of drift instabilities of standing waves, 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. Two model systems, obtained by projection of the Navier-Stokes-like equation onto the slowest mode of the domain, are examined to clarify the origin of this behavior. In the first the domain is an elliptically distorted cylinder while in the second it is an almost square rectangle. In both cases the forced symmetry breaking results in a nonlinear competition between two nearly degenerate oscillatory modes. This interaction destabilizes standing waves at small amplitudes and amplifies the role played by the streaming flow. In both systems the coupling to the streaming flow triggered by these instabilities leads to slow drifts along slow manifolds of fixed points or periodic orbits of the fast system, and generates behavior that resembles bursting in excitable systems. The results are compared to experiments

Low-dimensional modeling of streaks in a wedge flow boundary layer

This paper is concerned with the low dimensional structure of optimal streaks in a wedge flow boundary layer, which have been recently shown to consist of a unique (up to a constant factor) three-dimensional streamwise evolving mode, known as the most unstable streaky mode. Optimal streaks exhibit a still unexplored/unexploited approximate self-similarity (not associated with the boundary layer self-similarity), namely the streamwise velocity re-scaled with their maximum remains almost independent of both the spanwise wavenumber and the streamwise coordinate; the remaining two velocity components instead do not satisfy this property. The approximate self-similar behavior is analyzed here and exploited to further simplify the description of optimal streaks. In particular, it is shown that streaks can be approximately described in terms of the streamwise evolution of the scalar amplitudes of just three one-dimensional modes, providing the wall normal profiles of the streamwise velocity and two combinations of the cross flow velocity components; the scalar amplitudes obey a singular system of three ordinary differential equations (involving only two degrees of freedom), which approximates well the streamwise evolution of the general streaks

Low-Dimensional Modelling for optimal streaks in the blasius boundary layer

This paper is concerned with the low dimensional structure of optimal streaks in the Blasius boundary layer. Optimal streaks are well known to exhibit an approximate self-similarity, namely the streamwise velocity re-scaled with their maximum remains almost independent of both the spanwise wavenumber and the streamwise coordinate. However, the reason of this self-similar behavior is still unexplained as well as unexploited. After revisiting the structure of the streaks near the leading edge singularity, two additional approximately self-similar relations involving the velocity components and their wall normal derivatives are identified. Based on these properties, we derive a low dimensional model with two degrees of freedom. The comparison with the results obtained from the linearized boundary layer equations shows that this model is consistent and provide good approximations

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 bedefined in terms of just one stream wise-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 stream wise. Thus, the definition of streaks in terms of an optimization problem that is used nowadays is not necessary

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

Influence of the Surface Viscosity on the Breakup of a Surfactant-Laden Drop

We examine both theoretically and experimentally the breakup of a pendant drop loaded with an insoluble surfactant. The experiments show that a significant amount of surfactant is trapped in the resulting satellite droplet. This result contradicts previous theoretical predictions, where the effects of surface tension variation were limited to solutocapillarity and Marangoni stresses.We solve numerically the hydrodynamic equations, including not only those effects but also those of surface shear and dilatational viscosities. We show that surface viscosities play a critical role to explain the accumulation of surfactant in the satellite droplet.Ministerio de Economía y Competitividad DPI2013-46485-C3-1-R, TRA2013- 45808-RJunta de Extremadura GR1004

Relaxation oscillations in a nearly inviscid Faraday system

In the nearly inviscid regime parametrically driven surface gravity-capillary waves couple to a streaming flow driven in oscillatory viscous boundary layers. In an elliptical container of small eccentricity this coupling can lead to relaxation oscillations

Coupled Amplitude-Streaming Flow Equations for Nearly Inviscid Faraday Waves in Small Aspect Ratio Containers

We derive a set of asymptotically exact coupled amplitude-streaming flow ({CASF}) equations governing the evolution of weakly nonlinear nearly inviscid multimode Faraday waves and the associated streaming flow in finite geometries. The streaming flow is found to play a particularly important role near mode interactions. Such interactions come about either through a suitable choice of parameters or through breaking of degeneracy among modes related by symmetry. An example of the first case is provided by the interaction of two nonaxisymmetric modes in a circular container with different azimuthal wavenumbers. The second case arises when the shape of the container is changed from square to slightly rectangular, or from circular to slightly noncircular but with a plane of symmetry. The generation of streaming flow in each of these cases is discussed in detail and the properties of the resulting CASF equations are described. A preliminary analysis suggests that these equations can resolve discrepancies between existing theory and experimental results in the first two of the above cases

Nonlinear Dynamics in experimental devices with compressed/expanded surfactant monolayers

A theory is provided for a common experimental set up that is used to measure surface properties in surfactant monolayers. The set up consists of a surfactant monolayer (over a shallow liquid layer) that is compressed/expanded in a periodic fashion by moving in counter-phase two parallel, slightly immersed solid barriers, which vary the free surface area and thus the surfactant concentration. The simplest theory ignores the fluid dynamics in the bulk fluid, assuming spatially uniform surfactant concentration, which requires quite small forcing frequencies and provides reversible dynamics in the compression/expansion cycles. In this paper, we present a long-wave theory for not so slow oscillations that assumes local equilibrium but takes the fluid dynamics into account. This simple theory uncovers the physical mechanisms involved in the surfactant behavior and allows for extracting more information from each experimental run. The conclusion is that the fluid dynamics cannot be ignored, and that some irreversible dynamics could well have a fluid dynamic origi