Figure 3.4: Measured velocity of a tracer particle (circles) and drift velocity of the surface waves (crosses) over the normalized acceleration Γ. Throughout the measurement the driving amplitude was kept constant at 1.47 mm.
3.4 Conclusion
In the framework of investigations into the phenomenon of granular surface waves upon circular oscillation of the container, the wavelength dependence on the forcing acceleration Γ was measured in the regime of low horizontal grain mobility. A drift of the standing wave pattern was found to occur, and the drift velocity matches the transport velocity of the granular material. Both velocities strongly depend on the forcing acceleration and even reverse sign at Γ ≈3.3. The detected waves are similar to those seen in containers oscillating purely in the vertical direction. Obviously the horizontal oscillation component has little influence on the wave phenomenon, except for the drift.
When the experiments which are presented in this chapter were carried out, the trans-parent container was not yet available. A channel with Plexiglas walls later allowed to uncover the shape of the lower layer boundary during the flight phase (see Fig. 3.5). It turned out to be flat in the range 2.4<Γ<4.8 and waves only showed up at the surface.
In contrast, for a bending wave one would have expect arches underneath peaks of the surface undulations. Thus, if horizontal material displacement is ruled out the surface wave must be due to periodic lateral density variations. It would further be interesting to know the depth down to which the particle motion is affected by the surface wave and to have further information about the fluidization state of the granular bed. One could speculate that the wave motion is restricted to an upper fluidized part of the granular bed, while the lower part is in a solid-like state unaffected by the wave.
luminated from the front they exhibit a bright reflection spot. The particle diameter is 1 mm.
For surface waves in purely vertically vibrated granular beds there exists a continuum model devised by Eggers and Riecke (2006) that reproduces experimenal results fairly well. In order to reproduce and explain the experimental data on circularly vibrated granular beds, presented here, Grevenstette and Linz (2006) generalized and modified this model. They used the following phenomenological equations to describe the spacio-temporal evolution of the local bed height H(x, t) and the vertically averaged local flow velocity in the horizontal direction v(x, t):
∂H
In these equations D1 and D2 are constants. The terms fh and fv contain the coupling of the granular bed to the horizontal and vertical driving components:
fh =gΓ cos (2πf t) , fv =−gΓ sin (2πf t)
The term fd accounts for the friction between granular material and container:
fd=fs−Bv , fs=
( −µsgn(v)aN if aN >0
0 otherwise , aN =g−gΓ sin(2πf t) , whereµandB are constants. Simulations of this model yield drifting subharmonic surface waves as found in the experiments. Yet in contrast to the experiments, in the simulations the wavelength increases with increasing forcing strength. Another shortcoming of the model is that it shows no reversal of the drift velocity.
Of some practical interest for industrial applications are the mixing properties in the surface wave regime. One might expect enhanced self diffusion in the presence of an un-dulated surface, compared to the state with a flat surface at lower and higher acceleration.
Chapter 4
Localized subharmonic waves
Localized period doubling waves arise in circularly shaken granular beds contained in an annular channel. These solitary wave packets are accompanied by a locally increased particle density. The width and velocity of the granular wave pulse were measured as a function of the bed height. A continuum model for the material distribution, based on the measured granular transport velocity as a function of the bed thickness, captures the essence of the experimental findings.
The simulations of the continuum model were conducted by Daniel Svenˇsek from the Department of Physics at the University of Ljubljana, who spent a year in Bayreuth as Humbold fellow working at the chair Theoretische Physik III.
The obtained results have been submitted for publication to Physical Review Letters (G¨otzendorfer et al. 2006).
4.1 Introduction
La Ola, the Mexican wave, investigated by Farkas et al. (2002), serves as a good illus-tration for solitary waves in annular systems. On a smaller scale, the confined state in binary mixture convection is a prime example of solitary waves in dissipative systems.
These pulses of traveling waves, which appear in the neighborhood of a bistable regime, were experimentally investigated by Kolodner et al. (1988, 1991), Niemela et al. (1990), and Anderson and Behringer (1990). The physical mechanism behind the phenomenon was uncovered by Barten et al. (1991, 1995) and Jung and L¨ucke (2002), who solved the full hydrodynamic field equations. Riecke (1992) derived a set of amplitude equa-tions to describe the self-trapping of traveling-wave pulses in these systems. The so-called
“worms” in electroconvection, which were discovered by Dennin et al. (1996) and ex-plained by Riecke and Granzow (1998), represent a different type of phenomenon in the sense that the associated bifurcation from the flat surface to extended traveling waves is
35
Figure 4.1: Solitary waves. (a) Mexican wave circulating in a sports stadium.
(b) Confined state in a simulation of binary mixture convection (Image from http://www.uni-saarland.de/fak7/luecke/main de.html). (c) “Worms” observed in electroconvection by Dennin et. al (1996). (d) Oscillons (Image from http://chaos.ph .utexas.edu/research/granular.html). (e) Propagating solitary state discovered by Lioubashevski et al. (1996).
4.2. EXPERIMENT 37 supercritical. Moreover, periodically driven systems are known to exhibit highly localized stationary structures, coined oscillons (Umbanhowar et al. 1996), or propagating solitary waves with the periodicity of the driving (Lioubashevski et al. 1996, 1999). Fig. 4.1 shows examples of the mentioned solitary waves. Here we present drifting localized pulses that envelop parametrically driven, subharmonic, standing waves. They arise in a circularly vibrated granular bed. The results of our experiments are described by hydrodynamic equations.
4.2 Experiment
A detailed description of the experimental setup and the driving mechanism is given in Chapter 2. The vertical component of the container oscillation had an amplitude Av of 2.06 mm at a frequency f of 26.0 Hz. From these values the normalized peak container acceleration in the vertical direction Γ = Av(2πf)2/g is calculated to be 5.60. The horizontal oscillation amplitude in the center of the channelAh was 1.87 mm. Accordingly every point of the channel floor followed approximately a circular trajectory in a plane tangent to the ring.
The amount of material in the container is given by the number of particle layers H0 that form when the material is evenly distributed around the ring and the particles are close-packed. A densely packed monolayer consists of 21,500 particles. In the experiments H0 was varied from 3 up to 19 in steps of 0.5.