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

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