Ref HAL: hal-00749940_v1
DOI: 10.1016/j.jnnfm.2012.01.010
WoS: 000303943800004
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23 Citations
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An analytical and numerical study of the linear Saffman-Taylor instability for a Maxwell viscoelastic fluid is presented. Results obtained in a rectangular Hele-Shaw cell are complemented by experiments in a circular cell corroborating the universality of our main result: The base flow becomes unstable and the propagating disturbances develop into crack-like features. The full hydrodynamics equations in a regime where viscoelasticity dominates show that perturbations to the pressure remain Laplacian. Darcy's law is expressed as an infinite series in the cell thickness. An unique dimensionless parameter Delta-bar, equivalent to a relaxation time, controls the growth rate of the perturbation. Delat-bar depends on the applied gradient of pressure, the surface tension, the cell thickness, and the elastic modulus of the fluid. For small values of Delta-bar, Newtonian behavior dominates whereas for higher values of Delta-bar viscoelastic effects appear. For the critical value Dalta-bar ~= 10 a blowup is predicted and fracture-like patterns are observed.

Ref HAL: hal-00749901_v1
Ref Arxiv: 1206.1491
DOI: 10.1016/j.coastaleng.2013.02.008
WoS: WOS:000318325400005
Ref. & Cit.: NASA ADS
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11 Citations
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In order to study the growth of wind waves in finite depth we extend Miles' theory to the finite depth domain. A depth-dependent wave growth rate is derived from the dispersion relation of the wind/water interface. A suitable dimensionless finite depth wave age parameter allows us to plot a family of wave growth curves, each family member characterized by the water depth. Two major results are that for small wave age, the wave growth rates are comparable to those of deep water and for large wave age, a finite-depth wave-age-limited growth is reached, with wave growth rates going to zero. The corresponding limiting wave length and limiting phase speed are explicitely calculated in the shallow and in the deep water cases. A qualitative agreement with well-known empirical results is established and shows the robust consistency of the linear theoretical approach.

Ref HAL: hal-00716088_v1
Ref Arxiv: 1207.2246
DOI: 10.1063/1.4768530
WoS: 000312833500042
Ref. & Cit.: NASA ADS
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64 Citations
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A nonlinear Schrödinger equation for the envelope of two dimensional surface water waves on finite depth with non zero constant vorticity is derived, and the influence of this constant vorticity on the well known stability properties of weakly nonlinear wave packets is studied. It is demonstrated that vorticity modifies significantly the modulational instability properties of weakly nonlinear plane waves, namely the growth rate and bandwidth. At third order we have shown the importance of the coupling between the mean flow induced by the modulation and the vorticity. Furthermore, it is shown that these plane wave solutions may be linearly stable to modulational instability for an opposite shear current independently of the dimensionless parameter kh, where k and h are the carrier wavenumber and depth respectively.

Ref HAL: hal-00749957_v1
Ref Arxiv: 1101.5773
DOI: 10.1088/1751-8113/47/2/025208
WoS: WOS:000329041500012
Ref. & Cit.: NASA ADS
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11 Citations
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In order to describe the dynamics of monochromatic surface waves in deep water, we derive a nonlinear and dispersive system of equations for the free surface elevation and the free surface velocity from the Euler equations in infinite depth. From it, and using a multiscale perturbative methods, an asymptotic model for small-aspect-ratio waves is derived. The model is shown to be completely integrable. The Lax pair, the first conserved quantities as well as the symmetries are exhibited. Theoretical and numerical studies reveal that it supports periodic progressive Stokes waves which peak and break in finite time. Comparison between the limiting wave solution of the asymptotic model and classical irrotational results is performed.

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Date du colloque : 08/2010

Ref HAL: hal-00542956_v1
PMID 20365649
DOI: 10.1103/PhysRevE.81.026305
WoS: 000275053800033
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31 Citations
Résumé:

We study the linear stability of an air front pushing on a viscoelastic upper convected Mawxell fluid inside a Hele-Shaw cell. Both theory and experiments involving several viscoelastic fluids prove that a unique dimensionless time parameter lambda approximate to controls all elastic effects. For small values of lambda approximate to, Newtonian behavior dominates, while for higher values of lambda approximate to viscoelastic effects appear. We show that the linear growth rate of a small initial perturbation diverges for a critical value lambda approximate to=lambda(c)approximate to similar or equal to 10. Experiments prove that this divergence is associated to a fracturelike pattern instability of the interface. We conclude that the observed fractures come from the Saffman-Taylor instability and that they directly emerge from the linear regime of it.

DOI: 10.1017/S0022112010004349
WoS: 000285472300007
52 Citations
Résumé:

The modulational instability of gravity wave trains on the surface of water acted upon by wind and under influence of viscosity is considered. The wind regime is that of validity of Miles' theory and the viscosity is small. By using a perturbed nonlinear Schrodinger equation describing the evolution of a narrow-banded wavepacket under the action of wind and dissipation, the modulational instability of the wave group is shown to depend on both the frequency (or wavenumber) of the carrier wave and the strength of the friction velocity (or the wind speed). For fixed values of the water-surface roughness, the marginal curves separating stable states from unstable states are given. It is found in the low-frequency regime that stronger wind velocities are needed to sustain the modulational instability than for high-frequency water waves. In other words, the critical frequency decreases as the carrier wave age increases. Furthermore, it is shown for a given carrier frequency that a larger friction velocity is needed to sustain modulational instability when the roughness length is increased.