TY - JOUR

T1 - Two-Timing Hypothesis, Distinguished Limits, Drifts, and Pseudo-Diffusion for Oscillating Flows

AU - Vladimirov, V. A.

N1 - Funding Information:
The author is grateful to Prof. K.I. Ilin for the checking of calculations. The author would like to express special gratitude to Profs. A.D.D. Craik and H.K. Moffatt for reading this manuscript and making useful critical remarks. Also many thanks to Profs. M.E. McIntyre, A.B. Morgulis, T.J. Pedley, M.R.E. Proctor, and D.W. Hughes for helpful discussions. This research is supported by the grant IG/SCI/DOMS/16/13 from Sultan Qaboos University, Oman.
Publisher Copyright:
© 2016 Wiley Periodicals, Inc., A Wiley Company

PY - 2017/4/1

Y1 - 2017/4/1

N2 - 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.

AB - 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.

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U2 - 10.1111/sapm.12152

DO - 10.1111/sapm.12152

M3 - Article

AN - SCOPUS:85002776514

SN - 0022-2526

VL - 138

SP - 269

EP - 293

JO - Studies in Applied Mathematics

JF - Studies in Applied Mathematics

IS - 3

ER -