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Salhi A.,Faculte des science de Tunis | Cambon C.,Ecole Centrale Lyon
Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

We study the stability problem of unbounded shear flow, with velocity Ui =S x3 δi1, subjected to a uniform vertical density stratification, with Brunt-Väisälä frequency N, and system rotation of rate Ω about an axis aligned with the spanwise (x2) direction. The evolution of plane-wave disturbances in this shear flow is governed by a nonhomogeneous second-order differential equation with time-dependent coefficients. An analytical solution is found to be described by Legendre functions in terms of the nondimensional parameter σφ2 =R (R+1) sin2 φ+ Ri, where R= (2Ω/S) is the rotation number, φ is the angle between the horizontal wave vector and the streamwise axis, and Ri = N2 / S2 is the Richardson number. The long-time behavior of the solution is analyzed using the asymptotic representations of the Legendre functions. On the one hand, linear stability is analyzed in terms of exponential growth, as in a normal-mode analysis: the rotating stratified shear flow is stable if Ri >1/4, or if 0< Ri <1/4 and R (R+1) >0, or if R (R+1) <0 Source

Essafi W.,Institute National Of Recherche Et Danalyse Physico Chimique | Abdelli A.,Institute National Of Recherche Et Danalyse Physico Chimique | Abdelli A.,University of Carthage | Bouajila G.,Faculte des science de Tunis | Boue F.,CEA Saclay Nuclear Research Center
Journal of Physical Chemistry B

We investigate in this paper the influence of the improvement of the solvent quality on the structure and the viscous properties of solutions of an hydrophobic polyelectrolyte, poly(styrene-co-sodium styrenesulfonate): PSS. The solvent used is a mixture of water and an organic solvent, THF, which is also slightly polar. We use small angle neutron scattering in the semidilute regime and viscosimetry as a function of concentration in dilute and semidilute unentangled regime. The structure, namely the scattering from all chains, is characterized by a maximum ("polyelectrolyte peak"). Its position and amplitude depends, at a given sulfonation rate of PSS, on the solvent quality through the added amount of organic solvent (THF). These evolutions with the THF amount are more pronounced when the sulfonation rate f is low (more hydrophobic polyelectrolyte) and the amount of added THF is high. Adding THF to hydrophobic PSS (f = 0.50 or f = 0.38), diminishes also the "shoulder" visible in the log I - log q plot and associated with the pearl size. It is therefore proposed that when THF is added to aqueous polyelectrolyte solutions, the pearls are dissolved and the chain conformation evolves from the pearl-necklace shape already reported in pure water toward the string-like conformation in pure water for fully sulfonated PSS. An addition of THF also reduces the important low q upturn found with hydrophobic polyelectrolyte solutions: the large aggregates are dissolved by THF. The upturn can become for PSSNa f = 0.38, after adding enough THF (50%), even smaller than that for the charged hydrophilic case PSSNa f = 0.82, in water. This can mean that in the quasi-fully charged PSS at f = 0.82 there are still hydrophobic effects in water, which is disagreeing with our recent reports, or that the electrostatics contribution to the upturn is reduced due to a lower dielectric permittivity. Concerning the hydrophilic polyelectrolyte, poly(sodium-2-acrylamido-2-methylpropanesulfonate)-co- (acrylamide): AMAMPS, no evolution in structure occurs until 25% THF. The viscosimetry variation with THF fraction is in good agreement with the scattering one up to 25%: though little dependent on THF for AMAMPS, and for hydrophilic PSSNa, it increases for hydrophobic PSSNa in agreement with the chain expansion signaled by scattering. At 50% THF concentration, the hydrophilic polyelectrolyte shows new surprising behaviors: the scattering of PSSNa is no longer characteristic of polyelectrolytes, and AMAMPS solutions display an unexpected viscosity decrease. © 2012 American Chemical Society. Source

Jouini A.,Faculte des science de Tunis | Lemarie-Rieusset P.G.,DeVry University
Physica D: Nonlinear Phenomena

We present in this paper two elementary constructions of multiresolution analyses on the L-shaped domain D. In the first one, we shall describe a direct method to define an orthonormal multiresolution analysis. In the second one, we use the decomposition method for constructing a biorthogonal multiresolution analysis. These analyses are adapted for the study of the Sobolev spaces Hs(D)(s N). © 2013 Elsevier B.V. All rights reserved. Source

Salhi A.,Faculte des science de Tunis | Salhi A.,Ecole Centrale Lyon | Lehner T.,University of Paris Descartes | Godeferd F.,Ecole Centrale Lyon | Cambon C.,Ecole Centrale Lyon
Astrophysical Journal

We examine accretion disk flow under combined radial and vertical stratification utilizing a local Cartesian (or "shearing box") approximation. We investigate both axisymmetric and nonaxisymmetric disturbances with the Boussinesq approximation. Under axisymmetric disturbances, a new dispersion relation is derived. It reduces to the Solberg-Hoïland criterion in the case without vertical stratification. It shows that, asymptotically, stable radial and vertical stratification cannot induce any linear instability; Keplerian flow is accordingly stable. Previous investigations strongly suggest that the so-called bypass concept of turbulence (i.e., that fine-tuned disturbances of any inviscid smooth shear flow can reach arbitrarily large transient growth) can also be applied to Keplerian disks. We present an analysis of this process for three-dimensional plane-wave disturbances comoving with the shear flow of a general rotating shear flow under combined stable radial and vertical rotation. We demonstrate that large transient growth occurs for K 2/k1 ≫ 1 and k3 = 0 or k1 ∼ k3, where k1, K2, and k3 are the azimuthal, radial, and vertical components of the initial wave vector, respectively. By using a generalized "wave-vortex" decomposition of the disturbance, we show that the large transient energy growth in a Keplerian disk is mainly generated by the transient dynamics of the vortex mode. The analysis of the power spectrum of total (kinetic+potential) energy in the azimuthal or vertical directions shows that the contribution coming from the vortex mode is dominant at large scales, while the contribution coming from the wave mode is important at small scales. These findings may be confirmed by appropriate numerical simulations in the high Reynolds number regime. © 2013. The American Astronomical Society. All rights reserved. Source

Salhi A.,Faculte des science de Tunis | Salhi A.,Aix - Marseille University | Pieri A.B.,CNR Institute of atmospheric Sciences and Climate
Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Rotating stratified flows in thermal wind balance are at the center of geophysical fluid dynamics. Recently, endeavors were put on studying the linear response of such flows to potential vorticity perturbations. It has been shown that the initial potential vorticity (PV) distribution is fundamental and is responsible for important transient growth of the perturbation and gravity-wave generation. Using Pfeiffer's theorem [J. Differ. Equat. 11, 145 (1972)JDEQAK0022-039610.1016/0022-0396(72)90085-X], we give the mathematical demonstration of the stability of asymmetric perturbations k1≠0 of a uniform, unbounded flow in thermal wind balance. Incidentally, we prove that both the wave mode (that corresponds to a vanishing PV) and the vortex mode (corresponding to a nonzero PV) are stable. The emphasis is put on the nontrivial behavior of inertia-gravity waves (IGWs) when deformed by a background shear. In particular, we show that in the linear limit, sheared inertia-gravity waves asymptotically oscillate at the inertial waves frequency, but their amplitude is sensitive to shear, stratification, and rotation. Last, we study the development of the IGWs dynamics considering isotropic initial conditions. Computations indicate that both the vortex mode and the wave mode generate IGWs, but the energy of the IGWs generated by the vortex mode is more important than the energy of the IGWs generated by the wave mode. It is also found that, at large times, the energy of the IGWs generated by the vortex mode increases as the ratio kv/kh (initial vertical wavenumber over horizontal wavenumber) increases (like kv2/kh2), while the energy of the IGWs generated by the wave mode oscillates in function of kv/kh. © 2014 American Physical Society. Source

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