On the Hamiltonian structure and three-dimensional instabilities of rotating liquid bridges

We consider a rotating inviscid liquid drop trapped between two parallel plates. The liquid–air interface is a free surface and the boundaries of the wetted regions in the plates are also free. We assume that the two contact angles at the plates are equal. We present drop shapes that generalize the catenoids, nodoids and unduloids in the presence of rotation. We describe profile curves of these drops and investigate their stability to three-dimensional perturbations. The instabilities are associated with degeneracies of eigenvalues of the corresponding Hamiltonian linear stability problem. We observe that these instabilities are present even in the case when the analogue of the Rayleigh criterion for two-dimensional stability is satisfie

Analytical and phenomenological studies of rotating turbulence

A framework, which combines mathematical analysis, closure theory, and phenomenological treatment, is developed to study the spectral transfer process and reduction of dimensionality in turbulent flows that are subject to rotation. First, we outline a mathematical procedure that is particularly appropriate for problems with two disparate time scales. The approach which is based on the Green's method leads to the Poincare velocity variables and the Poincare transformation when applied to rotating turbulence. The effects of the rotation are now reflected in the modifications to the convolution of a nonlinear term. The Poincare transformed equations are used to obtain a time-dependent analog of the Taylor-Proudman theorem valid in the asymptotic limit when the non-dimensional parameter mu is identical to Omega(t) approaches infinity (Omega is the rotation rate and t is the time). The 'split' of the energy transfer in both direct and inverse directions is established. Secondly, we apply the Eddy-Damped-Quasinormal-Markovian (EDQNM) closure to the Poincare transformed Euler/Navier-Stokes equations. This closure leads to expressions for the spectral energy transfer. In particular, an unique triple velocity decorrelation time is derived with an explicit dependence on the rotation rate. This provides an important input for applying the phenomenological treatment of Zhou. In order to characterize the relative strength of rotation, another non-dimensional number, a spectral Rossby number, which is defined as the ratio of rotation and turbulence time scales, is introduced. Finally, the energy spectrum and the spectral eddy viscosity are deduced

Bursting Dynamics of the 3D Euler Equations in Cylindrical Domains

A class of three-dimensional initial data characterized by uniformly large vorticity is considered for the Euler equations of incompressible fluids. The fast singular oscillating limits of the Euler equations are studied for parametrically resonant cylinders. Resonances of fast swirling Beltrami waves deplete the Euler nonlinearity. The resonant Euler equations are systems of three-dimensional rigid body equations, coupled or not. Some cases of these resonant systems have homoclinic cycles, and orbits in the vicinity of these homoclinic cycles lead to bursts of the Euler solution measured in Sobolev norms of order higher than that corresponding to the enstrophy

The Cauchy problem for the Navier-Stokes equations with spatially almost periodic initial data

A unique classical solution of the Cauchy problem for the Navier-Stokes equations is considered when the initial velocity is spatially almost periodic. It is shown that the solution is always spatially almost periodic at any time provided that the solution exists. No restriction on the space dimension is imposed. This fact follows from continuous dependence of the solution with respect to initial data in uniform topology. Similar result is also established for Cauchy problem of the three-dimensional Navier-Stokes equations in a rotating frame

Uniform local solvability for the Navier-Stokes equations with the Coriolis force

The unique local existence is established for the Cauchy problem of the incompressible Navier-Stokes equations with the Coriolis force. The Coriolis operator restricted to divergence free vector fields is a zero order pseudodifferential operator with the skew-symmetric matrix symbol related to the Riesz operator. It leads to the additional term in the Navier-Stokes equations which has real parameter being proportional to the speed of rotation. For initial data as Fourier preimage of the space of all finite Radon measures with no point mass at the origin we prove uniform estimate for the existence time in the speed of rotation

On time analyticity of the Navier-Stokes equations in a rotating frame with spatially almost periodic data

We consider the Navier-Stokes equations with the Coriolis force when intial data may not decrease at spatial infinity so that almost periodic data is allowed. We prove that the local-in-time solution is analytic in time when initial data is in $FM_0$, Fourier preimage of the space of all finite Radon measures with no point mass at the origin. When the initial data is almost periodic, this implies that the complex amplitude is analytic in time. In particular, a new mode cannot be created at any positive time