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Two‐Timing Hypothesis, Distinguished Limits, Drifts, and Pseudo‐Diffusion for Oscillating Flows
Author(s) -
Vladimirov V. A.
Publication year - 2017
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12152
Subject(s) - mathematical analysis , mathematics , dimensionless quantity , amplitude , advection , scaling , diffusion , strouhal number , limit (mathematics) , quadratic equation , physics , geometry , mechanics , reynolds number , quantum mechanics , turbulence , thermodynamics
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.