Settling disks in a linearly stratified fluid

2010

A complete solution is obtained for the wave field generated by the time-harmonic edgewise oscillations of a horizontal circular disk in an incompressible stratified viscous fluid. The linearized equations of viscous internal waves and the no-slip condition on the rigid disk are used to derive sets of dual integral equations for the fluid velocity and vorticity. The dual integral equations are solved by analytic reduction to sets of linear algebraic equations. Asymptotic results confirm that this edgewise motion no longer excites waves in the small-viscosity limit. Broadside oscillations and the effect of density diffusion are also considered.

Journal of Fluid Mechanics, 2017

We analyse the linear stability of stably stratified fluids whose density depends on two scalar fields where one of the scalar fields is unstably stratified and involves a settling velocity. Such conditions may be found, for example, in flows involving the transport of sediment in addition to heat or salt. A linear stability analysis for constant-gradient base states demonstrates that the settling velocity generates a phase shift between the perturbation fields of the two scalars, which gives rise to a novel, settling-driven instability mode. This instability mechanism favours the growth of waves that are inclined with respect to the horizontal. It is active for all density and diffusivity ratios, including for cases in which the two scalars diffuse at identical rates. If the scalars have unequal diffusivities, it competes with the elevator mode waves of the classical double-diffusive instability. We present detailed linear stability results as a function of the governing dimensionl...

Journal of Fluid Mechanics, 1986

The time-dependent settling of a dilute monodisperse suspension in a centrifugal force field is considered. The settling takes place between two axisymmetric narrowly spaced conical disks that are rotating rapidly. Clear fluid and suspension are assumed to behave as Newtonian fluids of different densities. The viscosities are, for simplicity, assumed to be the same. All effects of the sediment are neglected. The fluid motion is assumed to be almost parallel with the disks and is computed by using lubrication theory. This leads to a nonlinear hyperbolic equation of first order for the location of the interface between clear fluid and suspension. Local multivaluedness of the solution is removed by inserting shocks. Such a shock is a model for a small region where the slope of the interface, as scaled in the lubrication approximation, is large. Two problems are solved: batch settling and a case where the suspension is pumped into a conical channel that is initially filled with clear fluid. In the batch-settling case, the solution is quite similar to that computed by Herbolzheimer t Acrivos for settling due to a constant gravity field in a narrow tilted channel. For large values of the Taylor number, it is found that the blocking of radial flow outside the Ekman layers leads to a somewhat slower separation process than expected. In the Glling problem, the character of the solution is distinctly different for Taylor numbers of order unity and large values of this parameter.

Physical review fluids, 2019

The low-Reynolds-number gravitational settling of a sphere through a fluid interface is investigated experimentally. By varying the viscosity ratio between the two fluids and the Bond number, two different modes of interfacial deformation are observed: a tailing mode and a film drainage mode. In the tailing mode, the interface deforms significantly as the sphere approaches, and the sphere becomes enveloped by a layer of the upper fluid. A tail forms, connecting the sphere to the bulk of the upper phase. In the film drainage mode, the interface deforms much less and the sphere impacts onto the interface, which either ruptures to form a contact line on the sphere or leaves a very thin wetting film. Additionally, two types of sinking profiles are observed: steady sinking, where the sphere velocity changes monotonically as it sinks, and stalled sinking, where the sphere's progress is inhibited by the interface, before it accelerates into the lower fluid. We present a regime diagram showing the different behaviors. Finally, the dependence of the sinking time on the Bond number and viscosity ratio is investigated. For the film drainage regime a simple scaling law is deduced; the tailing regime exhibits more complicated dynamics, possibly explained by a multistage sinking process.

Monthly Notices of the Royal Astronomical Society, 1997

In this paper we present an analytical and numerical study of the response of a circumbinary disc subject to the tidal-forcing of a binary with a fixed circular orbit. We consider isentropic fluid discs with a range of thicknesses and binaries with a range of mass ratios, orbital separations and inclination angles. Our numerical simulations are implemented using an SPH code such that we can consider the hydrodynamics fully in three-dimensions. For our unperturbed disc models, we find the numerical shear viscosity to be equivalent to a constant kinematic viscosity and we calibrate its magnitude. Writing a scaling relation for the shear viscosity manifest in our models, we deduce that the disc thickness cannot be varied without affecting the viscosity in these kinds of SPH disc models, as is supported by our numerical results. It is found that maintainance of an inner cavity owing to the tidal truncation of the disc is effective for non-zero orbital inclinations. Also we show that our model discs may precess approximately like rigid bodies, provided that the disc is able to communicate on a length scale comparable to the inner boundary radius by either sonic or viscous effects, in a sufficiently small fraction of the local precession period. It is found also that the surface density in the disc should not decrease too rapidly with increasing radius, otherwise the disc may separate into disconnected annuli. Furthermore, the disc precession period may tend to infinity if the disc outer edge is allowed to become arbitrarily large, the disc suffering only a modest quasi-steady warp near the inner boundary. When the disc response is linear, or weakly non-linear, the precession periods and the forms of warping that we measure in our numerical results yield reasonable quantitative agreement with the analytical expressions that we derive from a linear response calculation. For a stronger disc response the results can agree poorly with our linear analysis, although some qualitative features of the response remain intact. We demonstrate that the response of a disc of noninteracting particles is qualitatively different, showing a much larger kinematic

Astronomy and Astrophysics, 2005

We discuss the possibility that astrophysical accretion disks are dynamically unstable to nonaxisymmetric disturbances with characteristic scales much smaller than the vertical scale height. The instability is studied using three methods: one based on the energy integral, which allows the determination of a sufficient condition of stability, one using a WKB approach, which allows the determination of the necessary and sufficient condition for instability and a last one by numerical solution. This linear instability occurs in any inviscid stably stratified differential rotating fluid for rigid, stress-free or periodic boundary conditions, provided the angular velocity Ω decreases outwards with radius r. At not too small stratification, its growth rate is a fraction of Ω. The influence of viscous dissipation and thermal diffusivity on the instability is studied numerically, with emphasis on the case when d ln Ω/d ln r = −3/2 (Keplerian case). Strong stratification and large diffusivity are found to have a stabilizing effect. The corresponding critical stratification and Reynolds number for the onset of the instability in a typical disk are derived. We propose that the spontaneous generation of these linear modes is the source of turbulence in disks, especially in weakly ionized disks.

Physics of Fluids, 2009

Clusters made of a small number of close solid spherical particles at a random configuration, sedimenting through a viscous fluid at small Reynolds number, were experimentally investigated at a short-time scale. The cluster settling velocities were measured and shown to be well approximated by the ensemble-averaged formula derived earlier for the uniform distribution of the point particles inside a spherical volume. It was emphasized that the "effective radius" of this volume in general should be smaller than the actual radius of a cluster made of the spheres, and the relation between both radii was determined. The formula was also shown to account well for the gravitational settling of rigid conglomerates, measured and computed elsewhere.

Physical Review E, 2014

Flow past a disk was investigated experimentally in a low-velocity water channel in the range of intermediate Reynolds numbers. Systematic experiments with flow visualization and particle image velocimetry measurements are presented. Different disks with the aspect ratio χ = d/ h varying from 1 to 24 were investigated. The measurements were made in the range of Reynolds numbers from 0 to 500, where stationary and oscillatory instabilities appear. The influence of the aspect ratio on the value of onset instability, the evolution of perturbation, and the obtained vorticity bifurcation branches of the instability were identified.

Oceanography, 2006

Journal of Non-Newtonian Fluid Mechanics, 2014

The stability of the flow due to a rotating disk is considered for non-Newtonian fluids, specifically shearthinning fluids that satisfy the power-law (Ostwald-de Waele) relationship. In this case the basic flow is not an exact solution of the Navier-Stokes equations, however, in the limit of large Reynolds number the flow inside the three-dimensional boundary layer can be determined via a similarity solution. An asymptotic analysis is presented in the limit of large Reynolds number. It is shown that the stationary spiral instabilities observed experimentally in the Newtonian case can be described for shear-thinning fluids by a linear stability analysis. Predictions for the wavenumber and wave angle of the disturbances suggest that shear-thinning fluids may have a stabilising effect on the flow.