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1

On nonexistence of global in time solution for a mixed problem for a nonlinear evolution equation with memory generalizing the Voigt-Kelvin rheological model

Pukach P., Il'kiv V., Nytrebych Z., Vovk M.

Opuscula Mathematica

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2017

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Vol. 37, no. 5

735--753

EN

The paper deals with investigating of the first mixed problem for a fifth-order nonlinear evolutional equation which generalizes well known equation of the vibrations theory. We obtain sufficient conditions of nonexistence of a global solution in time variable.

2

Fourth order nonlinear evolution equation for interfacial gravity waves in the presence of air flowing over water and a basic current shear

Majumder D. P., Dhar A. K.

International Journal of Applied Mechanics and Engineering

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2015

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Vol. 20, no. 3

517--530

EN

A fourth order nonlinear evolution equation, which is a good starting point for the study of nonlinear water waves as first pointed out by Dysthe (1979) is derived for gravity waves propagating at the interface of two superposed fluids of infinite depth in the presence of air flowing over water and a basic current shear. A stability analysis is then made for a uniform Stokes gravity wave train. Graphs are plotted for the maximum growth rate of instability and for wave number at marginal stability against wave steepness for different values of air flow velocity and basic current shears. Significant deviations are noticed from the results obtained from the third order evolution equation, which is the nonlinear Schrödinger equation.

3

Effect of capillarity on fourth order nonlinear evolution equation for two Stokes wave trains in deep water in the presence of air flowing over water

Dhar A. K., Mondal J.

International Journal of Applied Mechanics and Engineering

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2015

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Vol. 20, no. 2

267--282

EN

Fourth order nonlinear evolution equations, which are a good starting point for the study of nonlinear water waves, are derived for deep water surface capillary gravity waves in the presence of second waves in which air is blowing over water. Here it is assumed that the space variation of the amplitude takes place only in a direction along which the group velocity projection of the two waves overlap. A stability analysis is made for a uniform wave train in the presence of a second wave train. Graphs are plotted for the maximum growth rate of instability wave number at marginal stability and wave number separation of fastest growing sideband component against wave steepness. Significant improvements are noticed from the results obtained from the two coupled third order nonlinear Schrödinger equations.

4

Evolution of a random field of surface gravity waves in a two fluid domain

Senapati S., Debsarma S., Das K. P.

International Journal of Applied Mechanics and Engineering

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2012

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Vol. 17, no. 2

481-493

EN

A spectral transport equation is derived here that governs the evolution of a random field of surface gravity waves in a two layer fluid model. This equation is used to study the stability of an initially homogeneous Lorentz spectrum under long wave length perturbations. It is observed that the effect of randomness is to reduce the growth rate of instability. An increase in the thickness of the upper fluid results in an increase in the extent of instability. It is also found that the extent of instability becomes less for a smaller density difference of the two fluids.

5

Stability of small but finite amplitude interfacial capillary gravity waves for perturbations in two horizontal directions

Majumder D. P., Dhar A. K.

International Journal of Applied Mechanics and Engineering

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2011

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Vol. 16, no 2

425-434

EN

By exact numerical computation Yuen (1984) obtained regions of type-I instability for waves propagating at the interface of two superposed fluids of infinite thickness in which the upper fluid has a constant streaming velocity. In the present paper it is shown that the long wavelength part of these instability regions can be obtained analytically from a fourth order nonlinear evolution equation for small but finite amplitude interfacial capillary gravity waves in the presence of air flowing over water.

6

Stability analysis from fourth order nonlinear evolution equation for two stokes wave trains in deep water in the presence of air flowing over water

Majumder D. P., Dhar A. K.

International Journal of Applied Mechanics and Engineering

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2009

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Vol. 14, no 4

989-1008

EN

Fourth order nonlinear evolution equations are derived for two Stokes wave trains in deep water in the presence of air flowing over water. The importance of the fourth order term in the evolution equation was pointed out by Dysthe (1979). Stability analysis is then made for uniform two Stokes wave trains in the presence of air flowing over water. From these evolution equations the expressions for the maximum growth rate of instability, the wave number at marginal stability and the wave number separation of fastest growing side band are derived and graphs are plotted for the above three expressions against the wave steepness. Significant improvements can be achieved from the results obtained from the two coupled third order nonlinear Schrödinger equations.

7

Stability analysis from fourth order evolution equation for deep water capillary-gravity waves in the presence of air flowing over water

Majumder D. P., Dhar A. K.

International Journal of Applied Mechanics and Engineering

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2009

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Vol. 14, no 2

433-442

EN

Fourth order nonlinear evolution equations, which are a good starting point for the study of nonlinear water waves as first pointed out by Dysthe (1979) and later elaborated by Janssen (1983), are derived for deep water capillary-gravity waves in the presence of air flowing over water. Stability analysis is then made for a uniform Stokes capillary gravity wave train. Graphs are plotted for the maximum growth rate of instability, the frequency at marginal stability and the frequency separation for fastest growing side-band component as a function of wave steepness. Significant deviations are noticed from the results obtained from the third-order evolution equation, which is the nonlinear Schrödinger equation.