On the brink of jamming: granular convection in densely filled containers

Granulates are ubiquitous in nature and technology, but, despite their great importance, their dynamics are by far less well understood than those of liquids. We demonstrate in an almost compactly filled flat (Hele-Shaw) cell, where slow horizontal rotation simulates a variable gravitational force, that unexpected dynamic structures may arise under geometrical restrictions. The cell motion drives regular flow in the compact interior, and convection rolls combine with segregation. The container fill level is crucial for the dynamic regime. A transition from chute flow at lower fill levels to convection in densely packed containers is found. These observations suggest the existence of comparable phenomena in situations where so far no systematic search for dynamic patterns has been performed.     

Rare fluctuations and large-scale circulation cessations in turbulent convection

In turbulent Rayleigh-Bénard convection, a large-scale circulation (LSC) develops in a nearly vertical plane and is maintained by rising and falling plumes detaching from the unstable thermal boundary layers. Rare but large fluctuations in the LSC amplitude can lead to extinction of the LSC (a cessation event), followed by the reemergence of another LSC with a different (random) azimuthal orientation. We extend previous models of the LSC dynamics to include momentum and thermal diffusion in the azimuthal plane and calculate the tails of the probability distributions of both the amplitude and azimuthal angle. Our analytical results are in very good agreement with the experimental data.     

Size segregation and convection of granular mixtures almost completely packed in a thin rotating Box

We simulate size segregation in granular mixtures which are almost completely packed in a rotating drum. Instead of a 3D drum, we simulate a 2D thin rotating box which is almost completely packed with granular mixtures. The phase inversion of a radially segregated pattern which was found in a 3D experiment is qualitatively reproduced with this simulation. A global convection appears after a radial segregation pattern is formed, and this convection induces an axially segregated pattern.     

Sloping convection in a rotating fluid

Laboratory experiments on thermal convection in a fluid which rotates about a vertical axis and is subject to a horizontal temperature gradient show that when the rotation rate Ω exceeds a certain critical value Ω_R (which depends on the acceleration of gravity, the shape and dimensions of the apparatus, the physical properties of the fluid and the distribution and intensity of the applied differential heating) Coriolis forces inhibit overturning motion in meridian planes and promote a completely different type of flow which has been termed ‘sloping convection’ or ‘baroclinic waves’. The motion is then non-axisymmetric and largely confined to meandering ‘jet streams’, with trajectories of individual fluid elements inclined at only very small (though essentially non-zero) angles to the horizontal. The kinetic energy of the waves derives from the interaction of slight vertical motions with the potential energy field maintained by differential heating, and it is dissipated by friction arising largely in boundary layers on the walls of the apparatus.

Provided that Ω, though greater than Ω_R, does not exceed a second critical value Ω_I, the waves are characterized by great regularity; they are either steady or undergo periodic ‘vacillation’ in their amplitude, shape and other properties. The azimuthal wavelength decreases with increasing Ω until at Ω=Ω_I it reaches a sufficiently low value, ～1.5 times the radial dimension of the wave, for nonlinear processes to overcome various constraints associated with the anisotropy of the flow, thereby rendering the main baroclinic wave barotropically unstable by transferring kinetic energy to larger as well as smaller scales of motions.

Theoretical investigations of sloping convection have their origin in ideas concerning the large-scale mid-latitude circulation of the Earth's atmosphere, modern work on which includes important studies based on numerical models. Conditions favouring sloping convection should be fairly common in natural systems and the process is expected to underlie various phenomena of interest to oceanographers, geophysicists, planetary scientists and astronomers.     

Realistic rotating convection in a DNA suspension

In this work we report theoretical and numerical results on convection in a viscoelastic binary mixture under rotation for realistic rigid-rigid boundary conditions. We focus our analysis in the DNA aqueous suspensions. Instability thresholds for oscillatory convection are calculated. Finally, we analyze the stabilizing effect for the onset of convection.     

Thermal convection in a rotating layer of a magnetic fluid

We consider Brownian particles with the ability to take up energy from the environment, to store it in an internal depot, and to convert internal energy into kinetic energy of motion. Provided a supercritical supply of energy, these particles are able to move in a “high velocity” or active mode, which allows them to move also against the gradient of an external potential. We investigate the critical energetic conditions of this self-driven motion for the case of a linear potential and a ratchet potential. In the latter case, we are able to find two different critical conversion rates for the internal energy, which describe the onset of a directed net current into the two different directions. The results of computer simulations are confirmed by analytical expressions for the critical parameters and the average velocity of the net current. Further, we investigate the influence of the asymmetry of the ratchet potential on the net current and estimate a critical value for the asymmetry in order to obtain a positive or negative net current. Received 20 September 1999     

Oscillations of a bose-einstein condensate rotating in a harmonic plus quartic trap

We study the normal modes of a two-dimensional rotating Bose-Einstein condensate confined in a quadratic plus quartic trap. Hydrodynamic theory and sum rules are used to derive analytical predictions for the collective frequencies in the limit of high angular velocities Omega where the vortex lattice produced by the rotation exhibits an annular structure. We predict a class of excitations with frequency sqrt[6]Omega in the rotating frame, irrespective of the mode multipolarity m, as well as a class of low energy modes with frequency proportional to |m|/Omega. The predictions are in good agreement with results of numerical simulations based on the 2D Gross-Pitaevskii equation. The same analysis is also carried out at even higher angular velocities, where the system enters the giant vortex regime.     

Spatial localization in rotating convection and magnetoconvection

Stationary spatially localized states are present in both rotating convection and magnetoconvection. In two-dimensional convection with stress-free boundary conditions, the formation of such states is due to the interaction between convection and a large scale mode: zonal velocity in rotating convection and magnetic potential in magnetoconvection. We develop a higher order theory, a nonlocal fifth order Ginzburg-Landau equation, to describe the effects of spatial modulation near a codimension-two point. Two different bifurcation scenarios are identified. Our results shed light on numerical studies of two-dimensional convective systems with stress-free boundary conditions. This paper is dedicated to Professor Helmut Brand on the occasion of his 60th birthday.     

Oscillatory convection and chaos in a Lorenz-type model of a rotating fluid

A four-mode model of convection in a rotating fluid layer is studied. The model is an extension of the Lorenz model of Rayleigh-Bénard convection, the extra mode accounting for the regeneration of vorticity by rotation. Perturbation theory is applied to show that the Hopf bifurcations from conductive and steady convective solutions can be either supercritical or subcritical. Perturbation theory is also used at large Rayleigh numbersr to predict novel behavior. Supercritical oscillatory convection of finite amplitude is found by numerical integration of the governing equations. The general picture is of a series of oscillatory solutions stable over larger intervals, interspersed by short bursts of chaos.     

Thermal convection in a rotating binary viscoelastic liquid mixture

In this work we report theoretical and numerical results on convection in a viscoelastic binary mixture under rotation. In particular, we focus in the Maxwelian case of viscoelastic fluid. We obtain explicit expressions for the convective thresholds in terms of the mixture parameters of the system in the case of idealized boundary conditions. We also calculate numerically the convective thresholds for the case of realistic rigid–rigid boundary conditions.     

Thermal and inertial modes of convection in a rapidly rotating annulus

The nature of the primary instabilities that arise in a fluid contained in a fast rotating cylindrical annulus with slightly inclined plane top and bottom boundaries, radial gravity, and internal heating is numerically analyzed. It is shown that for moderate and high Prandtl numbers, the onset of convection is described by a competition of azimuthal thermal modes with different radial structure, which dominate in different regions of the parameter space. By the combined effect of the inclined ends and rotation, there are modes that are attached to the heated wall and slanted to the prograde direction of rotation, and others which are straight and fill the convective layer. Nevertheless, for very small Prandtl numbers the velocity field of the dominant modes corresponds essentially to the inertial solution of the Poincare equation, and the temperature perturbation is forced by this velocity field. In addition, a detailed exploration of the critical Rayleigh numbers and precession frequencies of the convective modes versus the radius ratio and the Coriolis parameter, for different Prandtl numbers, is presented.     

Scaling and excitation of combined convection in a rapidly rotating plane layer

The optimum (to my mind) scaling of the combined thermal and compositional convection in a rapidly rotating plane layer is proposed.This scaling follows from self-consistent estimates of typical physical quantities. Similarity coefficients are introduced for the ratio convection dissipation/convection generation (s) and the ratio thermal convection/compositional convection (r). The third new and most important coefficient δ is the ratio of the characteristic size normal to the axis of rotation to the layer thickness. The faster the rotation, the lower δ. In the case of the liquid Earth core, δ ~ 10^–3 substitutes for the generally accepted Ekman number (E ~ 10^–15) and s ~ 10^–6 substitutes for the inverse Rayleigh number 1/Ra ~ 10^–30. It is found that, at turbulent transport coefficients, number s and the Prandtl number are on the order of unity for any objects and δ is independent of transport coefficients. As a result of expansion in powers of δ, an initially 3D system of six variables is simplified to an almost 2D system of four variables without δ. The problem of convection excitation in the main volume is algebraically solved and this problem for critical values is analytically solved. Dispersion relations and general expressions for critical wavenumbers, numbers s (which determine Rayleigh numbers), other critical parameters, and asymptotic solutions are derived. Numerical estimates are made for the liquid cores in the planets that resemble the Earth. Further possible applications of the results obtained are proposed for the interior of planets, moons, their oceans, stars, and experimental objects.     

Penta-hepta defect chaos in a model for rotating hexagonal convection

In a model for rotating non-Boussinesq convection with mean flow, we identify a regime of spatiotemporal chaos that is based on a hexagonal planform and is sustained by the induced nucleation of dislocations by penta-hepta defects. The probability distribution function for the number of defects deviates substantially from the usually observed Poisson-type distribution. It implies strong correlations between the defects in the form of density-dependent creation and annihilation rates of defects. We extract these rates from the distribution function and also directly from the defect dynamics.     

Analysis of radial segregation of granular mixtures in a rotating drum

This paper considers the segregation of a granular mixture in a rotating drum. Extending a recent kinematic model for grain transport on sandpile surfaces to the case of rotating drums, an analysis is presented for radial segregation in the rolling regime, where a thin layer is avalanching down while the rest of the material follows rigid body rotation. We argue that segregation is driven not just by differences in the angle of repose of the species, as has been assumed in earlier investigations, but also by differences in the size and surface properties of the grains. The cases of grains differing only in size (slightly or widely) and only in surface properties are considered, and the predictions are in qualitative agreement with observations. The model yields results inconsistent with the assumptions for more general cases, and we speculate on how this may be corrected. Received 4 June 1999 and Received in final form 28 September 1999     

Time of avalanche mixing of granular materials in a half filled drum

The avalanche mixing of granular solids in a slowly rotated 2D upright drum is studied. We demonstrate that the account of the difference δ between the angle of marginal stability and the angle of repose of the granular material leads to a restricted value of the mixing time τ for a half filled drum. The process of mixing is described by a linear discrete difference equation. We show that the mixing looks like linear diffusion of fractions with the diffusion coefficient vanishing when δ is an integer part of π. Introduction of fluctuations of δ suppresses the singularities of τ(δ) and smoothes the dependence τ(δ). Received 27 October 2000 and Received in final form 13 March 2001     

Radial convection of plasma structures in a turbulent rotating magnetized-plasma column

The turbulent regime of a rotating magnetized plasma column has been studied. The detection and the spatiotemporal analysis of structures by means of conditional sampling techniques is performed. Because of the overall rotation and centrifugal effects, the structures inside the turbulence move on average along a spiral trajectory leading to a net radial convection of the charged particles to the walls. The development of a poloidal electric field inside the structures has been measured. It leads to the observed outwards radial E x B drift in agreement with the expectations of recent theoretical works.