An asymptotic averaged model of non-linear long waves propagation in media with a regular structure

Vakhnenko, Vyacheslav O.; Danylenko, V.A.; Michtchenko, A.

doi:10.1016/s0020-7462(98)00014-6

Cited by 9 publications

(11 citation statements)

References 9 publications

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“…Following a common practice [58], we shall calculate the Evans function for a system of ODEs with constant coefficients approximating the linearized system (34). Let us employ a trigonometric representation λ = |λ| e i ϕ for λ, assuming that |λ| >> 1, and − π 2 < ϕ < π 2 .…”

Section: Appendixmentioning

confidence: 99%

“…Thus, calculating N for sufficiently large B (usually B is a semicircle lying in C + ) and analyzing the behavior of E(λ) for large |λ| can give a hint regarding the location of σ discr . So, initializing (34) with k vectors v j + from U + and with n − k independent vectors v j − from U − and solving the system towards z = 0, we obtain two sets of vectors:…”

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confidence: 99%

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Stability and dynamical features of solitary wave solutions for a hydrodynamic-type system taking into account nonlocal effects

Vladimirov

̧czka

Sergyeyev

et al. 2014

Communications in Nonlinear Science and Numerical Simulation

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We consider a hydrodynamic-type system of balance equations for mass and momentum closed by the dynamical equation of state taking into account the effects of spatial nonlocality. We study higher symmetries and local conservation laws for this system and establish its nonintegrability for the generic values of parameters. A system of ODEs obtained from the system under study through the group theory reduction is investigated. The reduced system is shown to possess a family of the homoclinic solutions describing solitary waves of compression and rarefaction. The waves of compression are shown to be unstable. On the contrary, the waves of rarefaction are likely to be stable. Numerical simulations reveal some peculiarities of solitary waves of rarefaction, and, in particular, the recovery of their shape after the collisions Keywords nonlocal hydrodynamic-type model; integrability tests; spectral stability of soliton-like solutions; interaction of solitary waves

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Section: Appendixmentioning

confidence: 99%

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confidence: 99%

Stability and dynamical features of solitary wave solutions for a hydrodynamic-type system taking into account nonlocal effects

Vladimirov

̧czka

Sergyeyev

et al. 2014

Communications in Nonlinear Science and Numerical Simulation

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“…We describe the wave processes in nonequilibrium heterogeneous media in terms of an asymptotic averaged model [18][19][20][21][22]. The obtained integral differential system of equations cannot be reduced to the average terms (pressure, mass velocity, specific volume) and contains the terms with characteristic sizes of individual components.…”

Section: Asymptotic Averaged Model For Structured Mediummentioning

confidence: 99%

“…The immediate employment of a method of the asymptotic averaging in Eulerian variables is impossible because of the variability of the microstructure sizes. However, from the zero approximation in the equations of motion (11), which are presented by the averaged values p, u, and V hi , the equations can be rewritten in the Eulerian system of coordinates r, t E ðÞ utilizing a transformation from the Lagrangian system s, t ðÞ [18][19][20][21][22]:…”

Section: System Of Equations In Eulerian Coordinatesmentioning

confidence: 99%

Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves for the Structured Media

Vakhnenko¹,

Vengrovich²,

Michtchenko³

2020

Advances in Complex Analysis and Applications

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We have proven that the long wave with finite amplitude responds to the structure of the medium. The heterogeneity in a medium structure always introduces additional nonlinearity in comparison with the homogeneous medium. At the same time, a question appears on the inverse problem, namely, is there sufficient information in the wave field to reconstruct the structure of the medium? It turns out that the knowledge on the evolution of nonlinear waves enables us to form the theoretical fundamentals of the diagnostic method to define the characteristics of a heterogeneous medium using the long waves of finite amplitudes (inverse problem). The mass contents of the particular components can be denoted with specified accuracy by this diagnostic method.

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“…We consider a structured medium (Figure 1) in which separated components are considered as a homogeneous medium (the differentially small volume dv is much smaller than the characteristic size of a particular component ε ). We describe the wave processes in non-equilibrium heterogeneous media in terms of an asymptotic averaged model [19]- [23]. In this case the obtained integral differential system of equations cannot be reduced to the average terms (pressure, mass velocity, specific volume) and contains the terms with characteristic sizes of individual components.…”

Section: Introductionmentioning

confidence: 99%

Blast Waves in Multi-Component Medium with Thermal Relaxation

Vakhnenko¹

2014

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The mathematical models of relaxing media with a structure for describing nonlinear long-wave processes are explored. The wave processes in non-equilibrium heterogeneous media are studied in terms of the suggested asymptotic averaged model. On the microstructure level of the medium, the dynamical behavior is governed only by the laws of thermodynamics, while, on the macrolevel, the motion of the medium can be described by the wave-dynamical laws. It is proved rigorously that on the acoustic level, the propagation of long waves can be properly described only in terms of dispersive dissipative properties of the medium, and in this case, the dynamical behavior of the medium can be modeled by a homogeneous relaxing medium. At the same time, the dynamical behavior of the medium cannot be modeled by a homogeneous medium even for long waves, if they are nonlinear. For a finite-amplitude wave, the structure of medium produces nonlinear effects even if the individual components of the medium are described by a linear law. The heterogeneity of the structure of medium always introduces additional nonlinearity. It is shown that the solution of many problems for multi-component media with incompressible phases can be obtained through the known solution of a similar problem for a homogeneous compressible medium by means of the suggested transformation. It is not necessary to solve directly the problem for the medium with incompressible component, and it is sufficient just to transform the known solution of the similar problem for a homogeneous medium. The scope for the suggested transformation is demonstrated by the reference to the strong explosion state in a two-phase medium. The special attention is focused on the research of blast waves in multi-component media with thermal relaxation. The dependence of the shock damping parameters on the thermal relaxation time is analyzed in order to provide a deeper understanding of the damping of shock waves in such media and to determine their effectiveness as localizing media. This problem attracts the interest also in view of the practical possibility to estimate the efficiency of medium for damping the shock wave action. To find the nature of the relaxation interaction between the components of medium and to estimate the attenuation of shock waves generated by solid explosives, we have studied experimentally both the velocity field of shock waves and the pressure at front in an air foam. The comparison of experimental and theoretical investigations of the relaxation phenomena which accompany the propagation of shock waves in foam indicates that within the scope of relaxation hy-V. O. Vakhnenko 1056 drodynamics it is possible to explain the observed phenomena and estimate the efficiency of medium as localizer of the shock wave action.

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An asymptotic averaged model of non-linear long waves propagation in media with a regular structure

Cited by 9 publications

References 9 publications

Stability and dynamical features of solitary wave solutions for a hydrodynamic-type system taking into account nonlocal effects

Stability and dynamical features of solitary wave solutions for a hydrodynamic-type system taking into account nonlocal effects

Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves for the Structured Media

Blast Waves in Multi-Component Medium with Thermal Relaxation

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