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Mixtures of granular media often exhibit size segregation along the axis of a partially-filled, horizontal, rotating cylinder. Previous experiments have observed axial bands of segregation that grow from concentration fluctuations and merge in a manner analogous to spinodal decomposition. We have observed that a new dynamical state precedes this effect in certain mixtures: bi-directional traveling waves. By preparing initial conditions, we found that the wave speed decreased with wavelength. Such waves appear to be inconsistent with simple PDE models which are first order in time.Comment: 11 page

We have studied the early time evolution of granular segregation patterns in a horizontal rotating cylinder partially filled with a sand/salt mixture. The growth of concentration fluctuations starting from premixed initial conditions was analyzed using Fourier techniques. In one mixture, we observed generally merging dynamics in the segregated bands up to the onset of saturation. At the threshold of saturation, we found a spectrum of wavelengths with a broad peak. The peak position was nearly independent of the tube rotation rate.The overall growth rate of Fourier modes had a maximum at a particular value of the angular rotation frequency. In a slightly different mixture, we observed transient traveling waves when the larger grains were in the majority. We measured the wave speed as a function of several parameters using presegregated initial conditions to launch waves of various wavelengths.

We describe experiments on Bénard-Marangoni convection in horizontal layers of two immiscible liquids. Unlike previous experiments, which used gases as the upper fluid, we find a square planform close to onset which undergoes a secondary bifurcation to rolls at higher temperature differences. The scale of the convection pattern is that of the thinner lower fluid layer for which buoyancy and surface tension forces are comparable. The wavenumber of the pattern near onset agrees with the prediction of the linear stability analysis for the full two-layer problem. The square planform is in qualitative agreement with recent one-and two-layer nonlinear theories, which fail however to predict the transition to rolls.Submitted to Physical Review Letters, June 29, 1998. See also http://mobydick.physics.utoronto.ca.Convection in fluids has been a fruitful system for the study of nonlinear, nonequilibrium patterns for almost 100 years. [1,2,3] The original experiments of Bénard [1] used shallow layers of whale oil, heated from below and open to the air above. Many years passed before it was conclusively shown that surface tension gradients, or "Marangoni", forces were crucial. [4,5] In general, both Marangoni and buoyancy forces are present. [6,7] Beginning with Pearson [5], theories of this instability have traditionally neglected the dynamics of the upper fluid (air in Bénard's experiments), replacing it by a constant heat flux boundary condition on the lower fluid. This condition is experimentally unrealizable, however. In wellcontrolled experiments [8,9,10,11,12,13], the upper fluid is bounded above by a plate, on which a constant temperature boundary condition is maintained. The heat flux across the interface is then the result of conduction and convection in both fluids, and is not constant above the onset of convection. The two-fluid interface is deformable, and its distance from the upper plate also changes the local heat flux. [13] In addition, buoyancy forces in the upper fluid may actively assist or impede convection in the lower fluid via surface stresses. [7,14,15] Recent experiments have found it necessary to treat the dynamics of both fluids in order the quantitatively explain the data. [13] In order to gain insight into the more complex, but experimentally better posed, two-layer problem, we have undertaken an experimental study of convection in a system of two immiscible fluids sharing a deformable interface. To our knowledge, only one previous experiment has been done on this system, and it was restricted to locating the onset of convection, without visualization [14]. On the theoretical side, the two-layer problem has been the subject of a complete linear stability analysis [7,14,15] and some weakly nonlinear analyses [7,24] in certain limiting cases. The general problem, including surface deformations, should however be well within the range of modern weakly nonlinear theory and of numerical simulations.In our experiment, both fluids were liquids, and the conditions were such that the Marangoni forces w...

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