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

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

Chaotic oscillations in a nearly inviscid, axisymmetric capillary bridge at 2:1 parametric resonance

We consider the 2:1 internal resonances (such that Ω1>0 and Ω2 ≃ 2Ω1 are natural frequencies) that appear in a nearly inviscid, axisymmetric capillary bridge when the slenderness Λ is such that 0<Λ<π (to avoid the Rayleigh instability) and only the first eight capillary modes are considered. A normal form is derived that gives the slow evolution (in the viscous time scale) of the complex amplitudes of the eigenmodes associated with Ω1 and Ω2, and consists of two complex ODEs that are balances of terms accounting for inertia, damping, detuning from resonance, quadratic nonlinearity, and forcing. In order to obtain quantitatively good results, a two-term approximation is used for the damping rate. The coefficients of quadratic terms are seen to be nonzero if and only if the eigenmode associated with Ω2 is even. In that case the quadratic normal form possesses steady states (which correspond to mono- or bichromatic oscillations of the liquid bridge) and more complex periodic or chaotic attractors (corresponding to periodically or chaotically modulated oscillations). For illustration, several bifurcation diagrams are analyzed in some detail for an internal resonance that appears at Λ ≃ 2.23 and involves the fifth and eighth eigenmodes. If, instead, the eigenmode associated with Ω2 is odd, and only one of the eigenmodes associated with Ω1 and Ω2 is directly excited, then quadratic terms are absent in the normal form and the associated dynamics is seen to be fairly simple

Propioceptores articulares y musculares

La función de los mecanorreceptores de las articulaciones y músculos se considera asociada a la propiocepción. Sin embargo, existen evidencias de que la propiocepción no sólo depende del morfotipo de mecanorreceptor presente en dichos tejidos sino también de las propiedades de las neuronas sensitivas primarias y las fibras sensitivas asociadas a ellos, así como de su proyección sobre el asta posterior de la médula espinal. Este artículo resume las bases morfológicas de la propiocepción a nivel del sistema nervioso periférico, analizando: a) las neuronas sensitivas primarias propioceptivas; b) los tipos de fibras nerviosas sensitivas que llegan a los propioceptores; c) la inervación sensitiva de articulaciones y músculos; d) los morfotipos de mecanorreceptores asociados a la propiocepción; e) los datos recientes obtenidos a partir de animales deficientes en factores clave para el desarrollo del sistema propioceptor.Peer Reviewe

Weakly Nonlinear Oscillations of Nearly Inviscid Axisymmetric Liquid Briges

A weakly nonlinear analysis is presented of the small oscillations of nearly inviscid liquid bridges subjected to almost resonant axial vibrations of the disks. An amplitude equation is derived for the evolution of the complex amplitude of the oscillations that exhibits hysteresis and period doublings. We also analyse the steady streaming in the bulk forced by the oscillatory boundary layers near the disks; the boundary layer near the free surface produces no forcing terms. In particular some experimentally observed patterns are explained, and some new, non-observed ones are predicted. We conclude that the structure of this steady flow is not the appropriate one to counterbalance steady thermocapillary convection, but our results indicate how to get the desired counterbalancing effect

CORRIGENDUM: Surface wave damping in a brimful circular cylinder

As pointed out to us by Mr T. Heath, the following printing errors can be quite misleading when using the formulas in the paper to obtain eigenfrequencies and damping rates to compare with experiments: in (A 13) 1 should read −1 on the right-hand side; in (A 22) and (A 26) Ω20 should read Ω−20; in (A 25) the factor Ω40 must be omitted on the right-hand side. When revising again the printed version of the paper, we discovered several additional misprints: A factor C was omitted in the first two integrals in the expression for J2, immediately following equation (2.9). The sign of the second expression for I1 in (2.23) should be changed. The expression (W0Wz +3WW0z)z=0 should read 2(W0Wz +WW0z)z=0 in equation (2.24). The expression W0(1, z)W0z(1, z) in (2.26) should read W0(r, 0)W0z(r, 0). None of the misprints above affect the results of the paper, which were obtained with the correct expressions

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

The interaction of thermocapillary convection and low-frequency vibration in nearly-inviscid liquid bridges

The combined effect of thermocapillary stress and steady forcing due to vibrations of the disks in a model-half-zone axisymmetric liquid bridge is considered for low-viscosity liquids (i.e., with a large capillary Reynolds number), and low nondimensional vibration frequencies (i.e., small as compared to the capillary Reynolds number). An asymptotic model is derived for the slowly-varying streaming flow in the bulk (outside the oscillatory boundary layers) resulting from both effects that includes also buoyancy and other thermal expansion effects. This model is used to first analyze the steady streaming flow patterns in isothermal conditions and then to show that mechanical vibrations can annihilate almost completely thermocapillary flows of fairly large Reynolds numbers provided that: (i) the Prandtl number is appropriately small, (ii) both disks are vibrated, and (iii) the vibrating amplitudes, frequency and phases are appropriate (the counterbalancing effect depends crucially on the difference of the vibrating phases of both disks