Two-Timing Hypothesis, Distinguished Limits, Drifts, and Pseudo-Diffusion for Oscillating Flows

The aim of this paper is: using the two-timing method to study and classify the multiplicity of distinguished limits and asymptotic solutions for the advection equation with a general oscillating velocity field. Our results are: (i)the dimensionless advection equation that contains two independent small parameters, which represent the ratio of two characteristic time-scales and the spatial amplitude of oscillations; the related scaling of the variables and parameters uses the Strouhal number; (ii)an infinite sequence of distinguished limits has been identified; this sequence corresponds to the successive degenerations of a drift velocity; (iii)we have derived the averaged equations and the oscillatory equations for the first four distinguished limits; derivations are performed up to the fourth orders in small parameters; (v)we have shown, that each distinguished limit generates an infinite number of parametric solutions; these solutions differ from each other by the slow time-scale and the amplitude of the prescribed velocity; (vi)we have discovered the inevitable presence of pseudo-diffusion terms in the averaged equations, pseudo-diffusion appears as a Lie derivative of the averaged tensor of quadratic displacements; we have analyzed the matrix of pseudo-diffusion coefficients and have established its degenerated form and hyperbolic character; however, for one-dimensional cases, the pseudo-diffusion can appear as ordinary diffusion; (vii)the averaged equations for four different types of oscillating velocity fields have been considered as the examples of different drifts and pseudo-diffusion; (viii)our main methodological result is the introduction of a logical order into the area and classification of an infinite number of asymptotic solutions; we hope that it can help in the study of the similar problems for more complex systems; (ix)our study can be used as a test for the validity of the two-timing hypothesis, because in our calculations we do not employ any additional assumptions.