Extended models of non-linear waves in liquid with gas bubbles

Kudryashov, Nikolay A.; Sinelshchikov, Dmitry I.

doi:10.1016/j.ijnonlinmec.2014.03.011

Cited by 43 publications

(35 citation statements)

References 51 publications

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“…We seek for solutions of (20) assuming that η is a polynomial in u. In this way we find infinitesimals which are presented in Table 1, where c i , i = 6, .…”

Section: Nonclassical Methodsmentioning

confidence: 99%

“…In Refs. [19,20] it was shown that the Burgers equation with high order corrections can be used for the description of nonlinear waves in a liquid with gas bubbles. This equation has the form [19,20]: u t + αuu x − µu xx = ε (2µα 2 + µα + ν)uu xx + (2µα 1 + µα + ν)u…”

Section: Introductionmentioning

confidence: 99%

“…It is known that at the derivation of the Burgers equation with high order corrections appears an obstacle to the integrability (see Refs. [8,14,19,20,32]). Eq.…”

Section: Introductionmentioning

confidence: 99%

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Analytical and numerical studying of the perturbed Korteweg-de Vries equation

Kudryashov

Sinelshchikov

2014

Journal of Mathematical Physics

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The perturbed Korteweg-de Vries equation is considered. This equation is used for the description of one-dimensional viscous gas dynamics, nonlinear waves in a liquid with gas bubbles and nonlinear acoustic waves. The integrability of this equation is investigated using the Painlevé approach. The condition for parameters for the integrability of the perturbed Korteweg-de Vries equation equation is established. New classical and nonclassical symmetries admitted by this equation are found. All corresponding symmetry reductions are obtained. New exact solutions of these reductions are constructed. They are expressed via trigonometric and Airy functions. Stability of the exact solutions of the perturbed Korteweg-de Vries equation is investigated numerically.

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“…We seek for solutions of (20) assuming that η is a polynomial in u. In this way we find infinitesimals which are presented in Table 1, where c i , i = 6, .…”

Section: Nonclassical Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Analytical and numerical studying of the perturbed Korteweg-de Vries equation

Kudryashov

Sinelshchikov

2014

Journal of Mathematical Physics

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“…Exact solutions (49) and (51) are the solitary wave solutions in the form of kinks. We can look for the periodic solutions of equation (36) if we apply the method developed in works [50][51][52].…”

Section: Painlevé Analysis and Exact Solution Of Equation At M = 2 Anmentioning

confidence: 99%

Method of the Logistic Function for Finding Analytical Solutions of Nonlinear Differential Equations

Kudryashov¹

2015

Model. anal. inf. sist.

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“…Kudryashov and Sinelshchikov further extended their model in Ref. [29], where small effects of liquid compressibility and surface tension were considered, leading to third-and fourth-order partial differential equations. In parallel, Kanagawa et al [30] proposed a systematic derivation of nonlinear equations in bubbly liquids, based on parameter scaling.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear dynamics of a viscous bubbly fluid

Thiessen

Cheviakov

2019

Communications in Nonlinear Science and Numerical Simulation

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A physical model of a three-dimensional flow of a viscous bubbly fluid in an intermediate regime between bubble formation and breakage is presented. The model is based on mechanics and thermodynamics of a single bubble coupled to the dynamics of a viscous fluid as a whole, and takes into account multiple physical effects, including gravity, viscosity, and surface tension. Dimensionless versions of the resulting nonlinear model are obtained, and values of dimensionless parameters are estimated for typical magma flows in horizontal subaerial lava fields and vertical volcanic conduits.Exact solutions of the resulting system of nonlinear equations corresponding to equilibrium flows and traveling waves are analyzed in the one-dimensional setting. Generalized Su-Gardner-type perturbation analysis is employed to study approximate solutions of the model in the long-wave ansatz. Simplified nonlinear partial differential equations (PDE) satisfied by the leading terms of the perturbation solutions are systematically derived. It is shown that for specific classes of perturbations, approximate solutions of the bubbly fluid model arise from solutions of the classical diffusion, Burgers, variable-coefficient Burgers, and Korteweg-de Vries equations. A The Numerical MethodSince the dimensionless bubbly fluid model equations (2.34) is not given by evolution equations, in order to solve it numerically, a non-standard procedure must be used. (Throughout this section, tildes in the equations of (2.34) and related formulas are omitted.) We employ a modified method of lines, as follows.

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Extended models of non-linear waves in liquid with gas bubbles

Cited by 43 publications

References 51 publications

Analytical and numerical studying of the perturbed Korteweg-de Vries equation

Analytical and numerical studying of the perturbed Korteweg-de Vries equation

Method of the Logistic Function for Finding Analytical Solutions of Nonlinear Differential Equations

Nonlinear dynamics of a viscous bubbly fluid

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