Critical Forcing Amplitude Research Articles

Drop Ejection from an Oscillating Rod

The dynamics of a liquid drop which is supported on a solid rod that is forced to undergo large-amplitude, time-periodic oscillations along its axis is studied using a computational approach based on the Galerkin/finite element method and an adaptive mesh generation technique which enables discretization of overturning interfaces and analysis of drop breakup. When the forcing amplitude is small, the drop deformations are small and the drop remains intact as it undergoes shape oscillations. Larger forcing amplitudes result in the formation of a liquid thread, or neck, which connects two fluid masses: the fluid adjacent to the rod and a nearly globular fluid mass. If the drop deforms such that its length is sufficiently large, the thread ruptures and the globular fluid mass—a so-called primary drop—is ejected from the fluid remaining on the rod. The critical forcing amplitude Ac necessary to attain this length and hence drop breakup, the interface shape at breakup, and the volume of the ejected primary drop are determined computationally as functions of the Reynolds number, forcing frequency, and drop size. Over a wide range of values of the forcing amplitude above Ac, ejection occurs as the drop recedes from its maximum length during its second period of oscillation. These results show that Ac increases as Reynolds number and/or drop size decreases. The maximum length the drop reaches prior to ejection and the position and velocity of the rod for times approaching breakup are shown to profoundly affect the dynamics of drop breakup and the resulting interface shapes. These results show that the forcing amplitude and/or frequency can be chosen so as to prevent the formation of long liquid necks, which typically favor the formation of satellite droplets after breakup. Because drop ejection does not rely on external forces other than those due to rod motion, this method of drop formation holds promise for microgravity applications as well as terrestrial drop-on-demand technologies in which gravitational force is negligible compared to surface tension force.

Parameter dependence of stochastic resonance in the stochastic Hodgkin-Huxley neuron.

Recently, the phenomena of stochastic resonance (SR) have attracted much attention in the studies of the excitable systems under inherent noise, in particular, nervous systems. We study SR in a stochastic Hodgkin-Huxley neuron under Ornstein-Uhlenbeck noise and periodic stimulus, focusing on the dependence of properties of SR on stimulus parameters. We find that the dependence of the critical forcing amplitude on the frequency of the periodic stimulus shows a bell-shaped structure with a minimum at the stimulus frequency, which is quite different from the monotonous dependence observed in the bistable system at a small frequency range. The frequency dependence of the critical forcing amplitude is explained in connection with the firing onset bifurcation curve of the Hodgkin-Huxley neuron in the deterministic situation. The optimal noise intensity for maximal amplification is also found to show a similar structure.

Fractal basin boundaries and homoclinic orbits for periodic motion in a two-well potential.

A fractal-looking basin boundary for forced periodic motions of a particle in a two-well potential is observed in numerical simulation. The fractal structure seems to be correlated with the appearance of homoclinic orbits in the Poincar\'e map as calculated by Holmes using the method of Melnikov. Below this critical forcing amplitude the basin boundary appears to be smooth and nonfractal. This example raises questions about predictability in nonchaotic dynamics of nonlinear systems.

A Further Study of the Annual Cycle of the Zonal Mean Circulation in the Middle Atmosphere

The influence on the stratospheric mean circulation of planetary wavenumber 2 disturbances excited by steady forcing at the 100 mb level is investigated using a global semi-spectral primitive equation model. There exists a critical forcing amplitude below which the waves have little effect on the mean flow, while above which the waves produce subseasonal time scale vacillations in the winter hemisphere, including both major and minor warmings. Wave transience induced by the evolving mean flow distribution is responsible for generating the vacillations, while thermal damping serves to reduce the wave-driven changes in the mean flow.

Parametric instability of internal gravity waves

A continuously stratified fluid, when subjected to a weak periodic horizontal acceleration, is shown to be susceptible to a form of parametric instability whose time dependence is described, in its simplest form, by the Mathieu equation. Such an acceleration could be imposed by a large-scale internal wave field. The growth rates of small-scale unstable modes may readily be determined as functions of the forcing-acceleration amplitude and frequency. If any such mode has a natural frequency near to half the forcing frequency, the forcing amplitude required for instability may be limited in smallness only by internal viscous dissipation. Greater amplitudes are required when boundaries constrain the form of the modes, but for a given bounding geometry the most unstable mode and its critical forcing amplitude can be defined.An experiment designed to isolate the instability precisely confirms theoretical predictions, and evidence is given from previous experiments which suggest that its appearance can be the penultimate stage before the traumatic distortion of continuous stratifications under internal wave action.A preliminary calculation, using the Garrett & Munk (197%) oceanic internal wave spectrum, indicates that parametric instability could occur in the ocean at scales down to that of the finest observed microstructure, and may therefore have a significant role to play in its formation.