On Shear-Free perturbations of FLRW Universes

A surprising exact result for the Einstein Field Equations is that if pressure-free matter is moving in a shear-free way, then it must be either expansion-free or rotation-free. It has been suggested this result is also true for any barotropic perfect fluid, but a proof has remained elusive. We consider the case of barotropic perfect fluid solutions linearized about a Robertson-Walker geometry, and prove that the result remains true except for the case of a specific highly non-linear equation of state. We argue that this equation of state is non-physical, and hence the result is true in the linearized case for all physically realistic barotropic perfect fluids. This result, which is not true in Newtonian cosmology, demonstrates that the linearized solutions, believed to result in standard local Newtonian theory, do not always give the usual behaviour of Newtonian solutions

Motion of a sphere in the presence of a plane interface. Part 2. An exact solution in bipolar co-ordinates

A general solution for Stokes’ equation in bipolar co-ordinates is derived, and then applied to the arbitrary motion of a sphere in the presence of a plane fluid/fluid interface. The drag force and hydrodynamic torque on the sphere are then calculated for four specific motions of the sphere; namely, translation perpendicular and parallel to the interface and rotation about an axis which is perpendicular and parallel, respectively, to the interface. The most significant result of the present work is the comparison between these numerically exact solutions and the approximate solutions from part 1. The latter can be generalized to a variety of particle shapes, and it is thus important to assess their accuracy for this case of spherical particles where an exact solution can be obtained. In addition to comparisons with the approximate solutions, we also examine the predicted changes in the velocity, pressure and vorticity fields due to the presence of the plane interface. One particularly interesting feature of the solutions is the fact that the direction of rotation of a freely suspended sphere moving parallel to the interface can either be the same as for a sphere rolling along the interface (as might be intuitively expected), or opposite depending upon the location of the sphere centre and the ratio of viscosities for the two fluids

A new family of exact and rotating solutions of fireball hydrodynamics

A new class of analytic, exact, rotating, self-similar and surprisingly simple solutions of non-relativistic hydrodynamics are presented for a three-dimensionally expanding, spheroidally symmetric fireball. These results generalize earlier, non-rotating solutions for ellipsoidally symmetric fireballs with directional, three-dimensional Hubble flows. The solutions are presented for a general class of equations of state that includes the lattice QCD equations of state and may feature inhomogeneous temperature and corresponding density profiles.Comment: Dedicated to T. Kodama on the occasion of his 70th birthday. 15 pages, no figures. Accepted for publication at Phys. Rev. C. Minor rewritings from previous versio

Large Scale Features of Rotating Forced Turbulence

Large scale features of a randomly isotropically forced incompressible and unbounded rotating fluid are examined in perturbation theory. At first order in both the random force amplitude and the angular velocity we find two types of modifications to the fluid equation of motion. The first correction transforms the molecular shear viscosity into a (rotation independent) effective viscosity. The second perturbative correction leads to a new large scale non-dissipative force proportional to the fluid angular velocity in the slow rotation regime. This effective force does no net work and alters the dispersion relation of inertial waves propagating in the fluid. Both dynamically generated corrections can be identified with certain components of the most general axisymmetric ``viscosity tensor'' for a Newtonian fluid.Comment: 12 pages, 2 figures, RevTeX, and accepted for publication in Phys. Rev.

Rotation and pseudo-rotation

Eigenvectors of stress-energy tensor (the source in Einstein's equations) form privileged bases in description of the corresponding space-times. When one or more of these vector fields are rotating (the property well determined in differential geometry), one says that the space-time executes this rotation. Though the rotation in its proper sense is understood as that of a timelike congruence (vector field), the rotation of a spacelike congruence is not a less objective property if it corresponds to a canonical proper basis built of the just mentioned eigenvectors. In this last case, we propose to speak on pseudo-rotation. Both properties of metric, its material sources, and space-time symmetries are considered in this paper.Comment: 13 pages, no figures, contains parts of the PhD Thesis of H. Vargas Rodr\'igue

Weak turbulence theory for rotating magnetohydrodynamics and planetary dynamos

A weak turbulence theory is derived for magnetohydrodynamics under rapid rotation and in the presence of a large-scale magnetic field. The angular velocity $\Omega_0$ is assumed to be uniform and parallel to the constant Alfv\'en speed ${\bf b_0}$. Such a system exhibits left and right circularly polarized waves which can be obtained by introducing the magneto-inertial length $d \equiv b_0/\Omega_0$. In the large-scale limit ($kd \to 0$; $k$ being the wave number), the left- and right-handed waves tend respectively to the inertial and magnetostrophic waves whereas in the small-scale limit ($kd \to + \infty$) pure Alfv\'en waves are recovered. By using a complex helicity decomposition, the asymptotic weak turbulence equations are derived which describe the long-time behavior of weakly dispersive interacting waves {\it via} three-wave interaction processes. It is shown that the nonlinear dynamics is mainly anisotropic with a stronger transfer perpendicular ($\perp$) than parallel ($\parallel$) to the rotating axis. The general theory may converge to pure weak inertial/magnetostrophic or Alfv\'en wave turbulence when the large or small-scales limits are taken respectively. Inertial wave turbulence is asymptotically dominated by the kinetic energy/helicity whereas the magnetostrophic wave turbulence is dominated by the magnetic energy/helicity. For both regimes a family of exact solutions are found for the spectra which do not correspond necessarily to a maximal helicity state. It is shown that the hybrid helicity exhibits a cascade whose direction may vary according to the scale $k_f$ at which the helicity flux is injected with an inverse cascade if $k_fd < 1$ and a direct cascade otherwise. The theory is relevant for the magnetostrophic dynamo whose main applications are the Earth and giant planets for which a small ($\sim 10^{-6}$) Rossby number is expected.Comment: 4 figures, 33 page