Averaging method and long-wave asymptotics in vibrational convection in layers with an interface

Zen'kovskaya, S. M.; Novosiadliy, Vasiliy

doi:10.1007/s10665-010-9415-7

Cited by 6 publications

(2 citation statements)

References 31 publications

(44 reference statements)

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“…Let us mention also the works [150,151], where the longwave asymptotics of the neutral curve of the Marangoni instability was studied for a two-layer system. which describes a smooth transition between the deformational and non-deformational modes [54].…”

Section: High-frequency Parameter Modulationsmentioning

confidence: 99%

Longwave oscillatory patterns in liquids: outside the world of the complex Ginzburg–Landau equation

Nepomnyashchy¹,

Shklyaev²

2015

J. Phys. A: Math. Theor.

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A new class of pattern forming systems is identified and investigated: anisotropic systems that are spatially inhomogeneous along the direction perpendicular to the preferred one. By studying the generic amplitude equation of this new class and a model equation, we show that branched stripe patterns emerge, which for a given parameter set are stable within a band of different wave numbers and different numbers of branching points (defects). Moreover, the branched patterns and unbranched ones (defect-free stripes) coexist over a finite parameter range. We propose two systems where this generic scenario can be found experimentally, surface wrinkling on elastic substrates and electroconvection in nematic liquid crystals, and relate them to the findings from the amplitude equation.

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Section: High-frequency Parameter Modulationsmentioning

confidence: 99%

Longwave oscillatory patterns in liquids: outside the world of the complex Ginzburg–Landau equation

Nepomnyashchy¹,

Shklyaev²

2015

J. Phys. A: Math. Theor.

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“…The present issue is concerned with body motion in a two-layer fluid [6], interaction of water waves with a porous plate [7], displacement of one fluid by another in a porous medium [8], liquid-film flows over a step topography with external electric field [9], turbulent flows produced on rotating blades [10], a water-impact problem [11], the Maslov canonical operator for shallow-water equations with localized initial data [12], internal gravity waves over a bottom topography [13], a reaction-diffusion model for concrete corrosion in sewer pipes [14] and convection in a two-layer fluid with external high-frequency vibration [15]. Different methods of asymptotic analysis such as the homogenization method, the Maslov canonical operator, the method of matched asymptotic expansions, generalized methods from geometric optics, and their combinations with numerical methods are used to demonstrate their practical importance in the analysis of practical problems.…”

mentioning

confidence: 99%

Preface to the fifth special issue on practical asymptotics

Korobkin

2010

J Eng Math

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mathematics seems, at first sight anyway, to become more and more dominated by direct numerical simulations. Admittedly, this leads to new insights which, it would seem, could not have been attained by other means" and ". . . many believe that asymptotics deals with exceptional cases which are usually outside the practical domain. However, this is a misconception!" The Guest-Editors of the following, second, third and fourth issues on practical asymptotics, argued that asymptotic solutions and asymptotic methods remain and will be valuable because they "provide important complements to numerical simulation" [2], "can offer ways to systematically study problems that may otherwise be inaccessible" [3] and "may provide quantitative and/or qualitative information that computer simulations can not" [4]. It is seen that the issues on practical asymptotics are a forum for discussion "Does the asymptotic analysis have a future in the era of supercomputers?". The answer to the question is obvious: "Yes, it has."Researchers and engineers, who use asymptotic methods and/or approximations inspired by these methods to solve practical problems, well understand that asymptotic analysis is a powerful tool, the range of applicability of which is very wide but does not cover everything we need. However, the situation is not so simple. There is a difficulty which is not about the application of asymptotic methods to practical problems but about the "public" opinion that asymptotic and, in general, mathematical tools are "old-fashioned" compared with direct numerical simulations. These days even academics sometimes ask the question "Why do we need this complicated analysis, if the problem can be easily solved by computer simulations?" Before the 90s numerical calculations were complementary to rigorous or approximate analysis of a problem. At the present time the "center of mass" in research is still moving towards computations. However, along this way, it is getting more and more clear that computer simulations being applied to practical problems is not a universal panacea. Euphoria about computers is still high but a more practical point of view is becoming at least visible these days: computation is a research tool as are many other approximate or rigorous tools of mathematics which help us to gain new knowledge, understand important phenomena and solve challenging practical problems.It would be perfect if our problems could be solved in a rigorous way. But this is an ideal situation which happens quite rarely, especially in new formulations. In order to obtain a solution, we need rigorous mathematical methods, approximate methods, with asymptotic methods being the most powerful of them, and computations. It is wrong A. Korobkin (B)

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Control of Longwave Instabilities

Shklyaev

Nepomnyashchy

2017

Longwave Instabilities and Patterns in Fluids

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Averaging method and long-wave asymptotics in vibrational convection in layers with an interface

Cited by 6 publications

References 31 publications

Longwave oscillatory patterns in liquids: outside the world of the complex Ginzburg–Landau equation

Longwave oscillatory patterns in liquids: outside the world of the complex Ginzburg–Landau equation

Preface to the fifth special issue on practical asymptotics

Control of Longwave Instabilities

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