Plane shearing waves of arbitrary form: exact solutions of the Navier-Stokes equations

We present exact solutions of the incompressible Navier-Stokes equations in a background linear shear flow. The method of construction is based on Kelvin's investigations into linearized disturbances in an unbounded Couette flow. We obtain explicit formulae for all three components of a Kelvin mode in terms of elementary functions. We then prove that Kelvin modes with parallel (though time-dependent) wave vectors can be superposed to construct the most general plane transverse shearing wave. An explicit solution is given, with any specified initial orientation, profile and polarization structure, with either unbounded or shear-periodic boundary conditions.Comment: 6 pages, 2 figures; version published in the European Physical Journal Plu

Kelvin Modes of a fast rotating Bose-Einstein Condensate

Using the concept of diffused vorticity and the formalism of rotational hydrodynamics we calculate the eigenmodes of a harmonically trapped Bose-Einstein condensate containing an array of quantized vortices. We predict the occurrence of a new branch of anomalous excitations, analogous to the Kelvin modes of the single vortex dynamics. Special attention is devoted to the excitation of the anomalous scissors mode.Comment: 7 pages, 3 figures, submitted to Phys. Rev.

Understanding the effect of sheared flow on microinstabilities

The competition between the drive and stabilization of plasma microinstabilities by sheared flow is investigated, focusing on the ion temperature gradient mode. Using a twisting mode representation in sheared slab geometry, the characteristic equations have been formulated for a dissipative fluid model, developed rigorously from the gyrokinetic equation. They clearly show that perpendicular flow shear convects perturbations along the field at a speed we denote by $Mc_s$ (where $c_s$ is the sound speed), whilst parallel flow shear enters as an instability driving term analogous to the usual temperature and density gradient effects. For sufficiently strong perpendicular flow shear, $M >1$, the propagation of the system characteristics is unidirectional and no unstable eigenmodes may form. Perturbations are swept along the field, to be ultimately dissipated as they are sheared ever more strongly. Numerical studies of the equations also reveal the existence of stable regions when $M < 1$, where the driving terms conflict. However, in both cases transitory perturbations exist, which could attain substantial amplitudes before decaying. Indeed, for $M \gg 1$, they are shown to exponentiate $\sqrt{M}$ times. This may provide a subcritical route to turbulence in tokamaks.Comment: minor revisions; accepted to PPC

From one cell to the whole froth: a dynamical map

We investigate two and three-dimensional shell-structured-inflatable froths, which can be constructed by a recursion procedure adding successive layers of cells around a germ cell. We prove that any froth can be reduced into a system of concentric shells. There is only a restricted set of local configurations for which the recursive inflation transformation is not applicable. These configurations are inclusions between successive layers and can be treated as vertices and edges decorations of a shell-structure-inflatable skeleton. The recursion procedure is described by a logistic map, which provides a natural classification into Euclidean, hyperbolic and elliptic froths. Froths tiling manifolds with different curvature can be classified simply by distinguishing between those with a bounded or unbounded number of elements per shell, without any a-priori knowledge on their curvature. A new result, associated with maximal orientational entropy, is obtained on topological properties of natural cellular systems. The topological characteristics of all experimentally known tetrahedrally close-packed structures are retrieved.Comment: 20 Pages Tex, 11 Postscript figures, 1 Postscript tabl

Deformation of Small Compressed Droplets

We investigate the elastic properties of small droplets under compression. The compression of a bubble by two parallel plates is solved exactly and it is shown that a lowest-order expansion of the solution reduces to a form similar to that obtained by Morse and Witten. Other systems are studied numerically and results for configurations involving between 2 and 20 compressing planes are presented. It is found that the response to compression depends on the number of planes. The shear modulus is also calculated for common lattices and the stability crossover between f.c.c.\ and b.c.c.\ is discussed.Comment: RevTeX with psfig-included figures and a galley macr

Weak turbulence of Kelvin waves in superfluid He

The physics of small-scale quantum turbulence in superfluids is essentially based on knowledge of the energy spectrum of Kelvin waves, E-k. Here we derive a new type of kinetic equation for Kelvin waves on quantized vortex filaments with random large-scale curvature which describes a step-by-step energy cascade over scales resulting from five-wave interactions. This approach replaces the earlier six-wave theory, which has recently been shown to be inconsistent owing to nonlocalization Solving the four-wave kinetic equation, we found a new local spectrum with a universal (curvature-independent) exponent, E-k proportional to k(-5/3), which must replace the nonlocal spectrum of the six-wave theory, E-k proportional to k(-7/5) in any future theory, e.g., when determining the quantum turbulence decay rate, found by Kosik and Svistunov under an incorrect assumption of locality of energy transfer in six-wave interactions. (c) 2010 American Institute of Physics. [doi:10.1063/1.3499242