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

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.