A multiple-timing analysis of temporal ratcheting

TL;DR

This paper develops a two-timing perturbation method to analyze temporal ratchets in a particle-fluid system under dual-frequency vibrations, predicting net particle velocities for specific frequency ratios and providing formulas for these velocities.

Contribution

It introduces a novel two-timing perturbation approach to predict and quantify temporal ratchets in a fluid-particle system with dual-frequency excitation.

Findings

Temporal ratchets occur at specific frequency ratios like 2, 3/2, 4/3.

Closed-form formulas for net velocities are derived.

First- and third-order schemes predict the existence of ratchets.

Abstract

We develop a two-timing perturbation analysis to investigate the existence of temporal ratchets in an exemplary system of a particle moving in a tank of fluid in response to an external vibration of the tank. We consider two-mode vibrations with angular frequencies $\omega$ and $\alpha\omega$, where $\alpha$ is a rational number. It has been established, in other physical systems, that if $\alpha$ is a ratio of odd and even integers (e.g., $\tfrac{2}{1}$, $\tfrac{3}{2}$, $\tfrac{4}{3}$), the system yields a net response: here, a nonzero time-average particle velocity. Our first-order and third-order two-timing perturbation schemes predict the existence of temporal ratchets for, respectively, $\alpha=2$ and $\alpha=\tfrac{3}{2}$, $\tfrac{4}{3}$. More importantly, we find closed form formula for the magnitude and direction of the induced net velocities for these $\alpha$ values.