View source for Ordered clusters and dynamical states of particles in a vibrated fluid

Original entry: Hsin-I Lu, APPHY 225, Fall 2009

"Ordered clusters and dynamical states of particles in a vibrated fluid"

Greg A. Voth, B. Bigger, M. R. Buckley, W. Losert, M. P. Brenner, H. A. Stone, and J. P. Gollub, PRL '''88''', 234301 (2002)

==Summary==
This paper studied the clustering and dynamical states of small stainless steel spheres when they were vibrated in fluid. The long-range attractive interaction and short-range repulsive interaction between particles are fluid mediated. The resulting patterns include hexagonally ordered microcrystallites, time-periodic structures, and chaotic fluctuating patterns with complex dynamics. 

==Soft Matter Keywords==
Interstitial fluid, self-assembly, viscosity, ordered clusters, microcrystallites

==Soft Matter==
[[Image:apparatus.jpg|thumb|Fig. 1]]
[[Image:clustering.jpg|thumb|Fig. 2]]
* Experiment: 
Fig. 1 shows the experimental setup. A submonolayer of uniform stainless steel spheres together with a water/glycerol mixture are sealed inside an aluminum container vibrated vertically by an electromagnetic vibrator. Both the frequency (<math>v=2\pi f</math>) and amplitude (<math>S</math>) of
the vibration of the container are controlled externally. The entire system is imaged from above by a fast CCD camera.

The time evolution of an initially random distribution of particles is shown in Fig. 2. After the vibrator is started, the particles quickly collect into localized clusters, and the clusters then slowly coalesce in a manner similar to coarsening in phase transitions.

* Interactions between particles:
The authors proposed a model to explain the interactions by considering the flow produced by each individual particle’s motion. The vibrations produce an oscillatory flow around each particle, which is well approximated as a potential flow. The potential flow near each oscillating particle is <math>\mathbf u = \bigtriangledown \phi</math>, where <math>\phi = -1/2 A \omega</math>sin(<math>\omega t</math>)<math>a^3</math>cos(<math>\theta</math>)/<math>r^2</math>; here the origin is at the middle of the particle, and u is the angle
between a given location on the sphere and the forcing direction. At the particle surface, the component of the
velocity field parallel to its surface is <math>u_{||} (\theta)= 1/2 A \omega</math>sin(<math>\omega t</math>)<math>a^3</math>sin(<math>\theta</math>). 

Lord Rayleigh pointed out that when the magnitude of the oscillatory flow <math>u_{||}</math>=<math>u_{||} (\theta) </math> varies along a solid surface, such as that of the particle, a steady secondary
flow is generated [1, 2]. For flow near the surface, mass conservation then requires that there is also a flow perpendicular to the boundary with magnitude <math>u_{||}  \delta_{osc} \sim \partial u_{||}</math>, where <math>\delta_{osc}</math> is the thickness of the boundary layer. Since this flow is not entirely out of phase with <math>u_{||}</math> in the boundary layer, every oscillation cycle transports a finite amount of momentum into the boundary layer. Hence there is a time independent force density on the fluid in the boundary layer, which is parallel to the boundary. This forcing produces a steady flow, which when balanced against the viscous force gives the steady flow <math>u_{steady} \sim - u_{||} \partial u_{||} \sim -A^2 \omega /a </math>sin(<math>\theta</math>)cos(<math>\theta</math>), where <math>a</math> is the radius of sphere, at the edge of the oscillatory boundary layer.

This steady flow pushes fluid away from the poles of each particle; and so there is a perpendicular inflow
velocity towards the equator. This steady inflow is the origin of the attractive interactions between two particles.

The authors found that an analytic matching argument, connecting the flows in the boundary layers to a potential flow far away, yields an explicit formula for the inflow velocity <math>v(r) =-0.53 A \sqrt{\omega \nu} a^2 / r^3</math>, where <math>\nu</math> is the kinematic viscosity of the fluid.


==Reference==
[1]Lord Rayleigh, Philos. Trans. R. Soc. London A 175, 1 (1883).
[2]N. Riley, Annu. Rev. Fluid Mech. 33, 43 (2001).