Two-timing Hypothesis, Distinguished Limits, Drifts, and Vibrodiffusion for Oscillating Flows

TL;DR

This paper develops a comprehensive two-timing method to analyze scalar advection in oscillating flows, revealing new averaged equations, distinguished limits, drift velocities, and the phenomenon of vibrodiffusion, with broad applicability.

Contribution

It extends the two-timing method to classify distinguished limits, derive generalized averaged equations, and introduce vibrodiffusion, without additional assumptions, advancing the theoretical understanding of oscillating flows.

Findings

Identification of two independent small parameters in advection.

Derivation of averaged equations for four distinguished limits.

Discovery of vibrodiffusion as a Lie derivative of quadratic displacements.

Abstract

In this paper we develop and use the two-timing method for a systematic study of a scalar advection caused by a general oscillating velocity field. Mathematically, we study and classify the multiplicity of distinguished limits and asymptotic solutions produced in the two-timing framework. Our calculations go far beyond the usual ones, performed by the two-timing method. We do not use any additional assumptions, hence our study can be seen as a test for the validity and sufficiency of the two-timing hypothesis. Physically, we derive the averaged equations in their maximum generality (and up to high orders in small parameters) and obtain qualitatively new results. Our results are: (i) the dimensionless advection equation contains \emph{two independent dimensionless small parameters}: the ratio of two time-scales and the spatial amplitudes of oscillations; (ii) we identify a sequence of \emph{distinguished limit solutions} which correspond to the successive degenerations of a \emph{drift velocity}; (iii) for a general oscillating velocity field we derive the averaged equations for the first \emph{four distinguished limit solutions}; (iv) we show, that \emph{each distinguish limit solution} produces an infinite number of \emph{parametric solutions} with a Strouhal number as the only large parameter; those solutions differ from each other by the slow time-scale and the velocity amplitude; (v) the striking outcome of our calculations is the inevitable appearance of \emph{vibrodiffusion}, which represents a Lie derivative of the averaged tensor of quadratic displacements; (vi) our main methodological result is the introduction of a logical order/classification of the solutions; we hope that it opens the gate for applications of the same ideas to more complex systems; (vii) five types of oscillating flows are presented as examples of different drifts and vibrodiffusion.