Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability
Zhi-Min Chen Philip A. Wilson
Discrete & Continuous Dynamical Systems - B 2012, 17(7): 2329-2341 doi: 10.3934/dcdsb.2012.17.2329
The Benjamin-Feir instability describes the instability of a uniform oscillatory wave train in an irrotational flow subject to small perturbation of wave number, amplitude and frequency. Their instability analysis is based on the perturbation around the second order Stokes wave which satisfies the dynamic and kinematic free-surface boundary conditions up to the second order. In the same irrotational flow and perturbation framework of the Benjamin-Feir analysis, the perturbation in the present paper is around a nonlinear oscillatory wave train which solves exactly the dynamic free-surface boundary condition and satisfies the kinematic free-surface boundary condition up to the third order. It is shown that the nonlinear oscillatory wave train is stable with respect to the perturbation when the irrotational flow involves small Rayleigh energy dissipation.
keywords: Benjamin-Feir instability gravity wave stability. oscillatory wave potential flow
Straightforward approximation of the translating and pulsating free surface Green function
Zhi-Min Chen
Discrete & Continuous Dynamical Systems - B 2014, 19(9): 2767-2783 doi: 10.3934/dcdsb.2014.19.2767
The translating and pulsating free surface Green function represents the velocity potential of a three-dimensional free surface source advancing in waves. This function involves singular wave integral, which is troublesome in numerical computation. In the present study, a regular wave integral approach is developed for the discretisation of the singular wave integral in a whole space harmonic function expansion, which permits the free surface wave produced by the fluid motion to be decomposed by plane regular propagation waves. This approximation gives rise to a simple and straightforward evaluation of the Green function. The algorithm is validated from comparisons between present numerical results and existing numerical data.
keywords: hydrodynamics. numerical approximation surface gravity wave Free surface Green function
Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows
Bo-Qing Dong Zhi-Min Chen
Discrete & Continuous Dynamical Systems - A 2009, 23(3): 765-784 doi: 10.3934/dcds.2009.23.765
This paper deals with asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows in the whole space $R^2$. Based on the spectral decomposition of linearized micropolar fluid flows, the sharp algebraic time decay estimates of the micropolar fluid flows in $L_2$ and $L_\infty$ norms are obtained.
keywords: micropolar fluid flows; spectral decomposition; asymptotic profiles

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