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Teoreticheskaya i Matematicheskaya Fizika, 2020, Volume 203, Number 1, Pages 91–105
DOI: https://doi.org/10.4213/tmf9821
(Mi tmf9821)
 

Boundary layer collapses described by the two-dimensional intermediate long-wave equation

J. O. Olooab, V. I. Shriraa

a School of Computing and Mathematics, Keele University, Staffordshire, United Kingdom
b The Catholic University of Eastern Africa, Nairobi, Kenya
References:
Abstract: We study the nonlinear dynamics of localized perturbations of a confined generic boundary-layer shear flow in the framework of the essentially two-dimensional generalization of the intermediate long-wave (2d-ILW) equation. The 2d-ILW equation was originally derived to describe nonlinear evolution of boundary layer perturbations in a fluid confined between two parallel planes. The distance between the planes is characterized by a dimensionless parameter $D$. In the limits of large and small $D$, the 2d-ILW equation respectively tends to the 2d Benjamin–Ono and 2d Zakharov–Kuznetsov equations. We show that localized initial perturbations of any given shape collapse, i.e., blow up in a finite time and form a point singularity, if the Hamiltonian is negative, which occurs if the perturbation amplitude exceeds a certain threshold specific for each particular shape of the initial perturbation. For axisymmetric Gaussian and Lorentzian initial perturbations of amplitude $a$ and width $\sigma$, we derive explicit nonlinear neutral stability curves that separate the domains of perturbation collapse and decay on the plane $(a,\sigma)$ for various values of $D$. The amplitude threshold $a$ increases as $D$ and $\sigma$ decrease and tends to infinity at $D\to0$. The 2d-ILW equation also admits steady axisymmetric solitary wave solutions whose Hamiltonian is always negative; they collapse for all $D$ except $D=0$. But the equation itself has not been proved for small $D$. Direct numerical simulations of the 2d-ILW equation with Gaussian and Lorentzian initial conditions show that initial perturbations with an amplitude exceeding the found threshold collapse in a self-similar manner, while perturbations with a below-threshold amplitude decay.
Keywords: boundary layer instability, nonlinear evolution equation, collapse, singularity formation, laminar–turbulent transition.
Funding agency Grant number
Natural Environment Research Council NE/M016269/1
European Union's Seventh Framework Programme 612610
Commonwealth Scholarship Commission in the UK KECA-2016-30
This research was supported in part by the UK NERC (Grant No. NE/M016269/1) and the EU (Grant No. FP7 612610). The research of J. Oloo was supported by the Commonwealth Scholarship Commission (Grant No. KECA-2016-30), without which this work would not have happened.
Received: 17.09.2019
Revised: 04.11.2019
English version:
Theoretical and Mathematical Physics, 2020, Volume 203, Issue 1, Pages 512–523
DOI: https://doi.org/10.1134/S0040577920040078
Bibliographic databases:
Document Type: Article
Language: Russian
Citation: J. O. Oloo, V. I. Shrira, “Boundary layer collapses described by the two-dimensional intermediate long-wave equation”, TMF, 203:1 (2020), 91–105; Theoret. and Math. Phys., 203:1 (2020), 512–523
Citation in format AMSBIB
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    Теоретическая и математическая физика Theoretical and Mathematical Physics
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