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Air-induced inverse Chladni patterns

Published online by Cambridge University Press:  05 December 2011

Henk Jan van Gerner ,
Ko van der Weele ,
Martin A. van der Hoef  and
Devaraj van der Meer
Henk Jan van Gerner
Affiliation:
Faculty of Science and Technology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands National Aerospace Laboratory, P.O. Box 153, 8300 AD Emmeloord, The Netherlands
Ko van der Weele
Affiliation:
Department of Mathematics, University of Patras, 26500 Patras, Greece
Martin A. van der Hoef
Affiliation:
Faculty of Science and Technology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands Department of Chemical Engineering and Chemistry, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
Devaraj van der Meer*
Affiliation:
Faculty of Science and Technology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
*
†Email address for correspondence: d.vandermeer@utwente.nl
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Abstract

When very light particles are sprinkled on a resonating horizontal plate, inverse Chladni patterns are formed. Instead of going to the nodal lines of the plate, where they would form a standard Chladni pattern, the particles are dragged to the antinodes by the air currents induced by the vibration of the plate. Here we present a detailed picture of the mechanism using numerical simulations involving both the particles and the air. Surprisingly, the time-averaged Eulerian velocity, commonly used in these type of problems, does not explain the motion of the particles: it even has the opposite direction, towards the nodal lines. The key to the inverse Chladni patterning is found in the averaged velocity of a tracer particle moving along with the air: this Lagrangian velocity, averaged over a vibration cycle, is directed toward the antinodes. The Chladni plate thus provides a unique example of a system in which the Eulerian and Lagrangian velocities point in opposite directions.

JFM classification

Granular media: Granular media Multiphase and Particle-laden Flows: Particle/fluid flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 689 , 25 December 2011 , pp. 203 - 220
Copyright
Copyright © Cambridge University Press 2011

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van Gerner et al. supplementary movies

Classic Chladni pattern: Top view of a flexible plate of dimensions 40 mm × 40 mm, resonating in its 2 × 2 mode and sprinkled with 80,000 heavy particles (density ρ = 20,000 kg/m3, diameter d = 0.075 mm). After a few seconds most particles have collected at the nodal lines, forming a classic Chladni pattern.

Download van Gerner et al. supplementary movies(Video)
Video 7.9 MB
Supplementary material: PDF

van Gerner et al. supplementary material

Supplementary material

Download van Gerner et al. supplementary material(PDF)
PDF 154.6 KB

van Gerner et al. supplementary movies

Inverse Chladni pattern: Top view of a flexible plate of dimensions 40 mm × 40 mm, resonating in its 2 × 2 mode and sprinkled with 80,000 very light particles (density ρ = 20 kg/m3, diameter d = 0.075 mm). Due to the presence of air, the particles now migrate to the anti-nodes and after about 4 seconds an inverse Chladni pattern has formed.

Download van Gerner et al. supplementary movies(Video)
Video 7.4 MB