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

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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: 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; an infinite sequence of distinguished limits has been identified; this sequence corresponds to the successive degenerations of a drift velocity; 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; 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; 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; the averaged equations for four different types of oscillating velocity fields have been considered as the examples of different drifts and pseudo‐diffusion; 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; 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.
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Document Type: Research Article

Publication date: April 1, 2017

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