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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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References

1. Açikalin, T., Raman, A. & Garimella, S. V. 2003 Two-dimensional streaming flows induced by resonating, thin beams. J. Acoust. Soc. Am. 114, 1785.CrossRefGoogle ScholarPubMed
2. Banerjee, S. & Law, S. E. 1998 Characterization of chargeability of biological particulates by triboelectrification. IEEE Trans. Indust. Appl. 34, 1201.CrossRefGoogle Scholar
3. Beetstra, R., van der Hoef, M. A. & Kuipers, J. A. M. 2007 Drag force of intermediate Reynolds number flow past mono- and bidisperse arrays of spheres. AIChE J. 53, 489.CrossRefGoogle Scholar
4. Boluriaan, S. & Morris, P. J. 2003 Acoustic streaming: from Rayleigh to today. Intl J. Aeroacoust. 2, 255.CrossRefGoogle Scholar
5. Brenner, H. 1961 The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Engng Sci. 16, 242.CrossRefGoogle Scholar
6. Chladni, E. F. F. 1802 Die Akustik. Breitkopf & Härtel.Google Scholar
7. Chladni, E. F. F. 1809 Traité d’Acoustique. Courcier.Google Scholar
8. Deen, N. G., van Sint-Annaland, M. & Kuipers, J. A. M. 2004 Multi-scale modelling of dispersed gas–liquid two-phase flows. Chem. Engng Sci. 59, 1853.CrossRefGoogle Scholar
9. Dorrestijn, M., Bietsch, A., Açikalin, T., Raman, A., Hegnerand, M., Meyer, E. & Gerber, CH. 2007 Chladni figures revisited based on nanomechanics. Phys. Rev. Lett. 98, 026102.CrossRefGoogle ScholarPubMed
10. Elghobashi, S. 1994 On predicting particle-laden turbulent flows. Appl. Sci. Res. 52, 309.CrossRefGoogle Scholar
11. Ergun, S. 1954 Fluid flow through packed columns. Chem. Engng Prog. 48, 89.Google Scholar
12. Faraday, M. 1831 On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 121, 299.Google Scholar
13. van Gerner, H. J. 2009 Newton versus Stokes: competing forces in granular matter. PhD thesis, Enschede, http://purl.org/utwente/61070.Google Scholar
14. Goldman, A. J., Cox, R. G. & Brenner, H. 1967 Slow viscous motion of a sphere parallel to a plane wall – I Motion through a quiescent fluid. Chem. Engng Sci. 22, 637.CrossRefGoogle Scholar
15. Hamilton, M. F., Ilinskii, Y. A. & Zabolotskaya, E. A. 2003 Acoustic streaming generated by standing waves in two-dimensional channels of arbitrary width. J. Acoust. Soc. Am. 113, 153.CrossRefGoogle ScholarPubMed
16. van der Hoef, M. A., Ye, M., van Sint Annaland, M., Andrews IV, A. T., Sundaresan, S. & Kuipers, J. A. M. 2006 Multi-scale modelling of gas-fluidized beds. Adv. Chem. Engng 31, 65.CrossRefGoogle Scholar
17. van der Hoef, M. A., van Sint Annaland, M., Deen, N. G. & Kuipers, J. A. M. 2008 Numerical simulation of dense gas–solid fluidized beds: a multiscale modelling strategy. Ann. Rev. Fluid Mech. 40, 47.CrossRefGoogle Scholar
18. Lighthill, J. 1978 Acoustic streaming. J. Sound Vib. 61, 391.CrossRefGoogle Scholar
19. Loh, B.-G., Hyun, S., Ro, P. I. & Kleinstreuer, C. 2002 Acoustic streaming induced by ultrasonic flexural vibrations and associated enhancement of convective heat transfer. J. Acoust. Soc. Am. 111, 875.CrossRefGoogle ScholarPubMed
20. Maxey, M. & Riley, J. 1983 Equation of motion for a small rigid sphere in a non-uniform flow. Phys. Fluids 26, 883.CrossRefGoogle Scholar
21. Nyborg, W. L. 1953 Acoustic streaming due to attenuated plane waves. J. Acoust. Soc. Am. 25, 68.CrossRefGoogle Scholar
22. Peskin, C. S. 2002 The immersed boundary method. Acta Numerica 11, 480.CrossRefGoogle Scholar
23. Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
24. Rayleigh, L. 1884 On the circulation of air observed in Kundt’s tubes, and on some allied acoustical problems. Phil. Trans. R. Soc. Lond. 125, 1.Google Scholar
25. Rayleigh, L. 1894 The Theory of Sound, vol. I. Macmillan.Google Scholar
26. Riley, N. 2001 Steady streaming. Annu. Rev. Fluid Mech. 33, 43.CrossRefGoogle Scholar
27. Savage, S. B. 1988 Streaming motions in a bed of vibrationally fluidized dry granular material. J. Fluid Mech. 194, 457.CrossRefGoogle Scholar
28. Stöckmann, H.-J. 2006 Ein Nomade der Wissenschaft. Physik J. 5, 47.Google Scholar
29. Stöckmann, H.-J. 2007 Chladni meets Napoleon. Eur. Phys. J. Special Topics 145, 15.CrossRefGoogle Scholar
30. Stokes, G. G. 1847 On the theory of oscillatory waves. Camb. Phil. Soc. Trans. 8, 441.Google Scholar
31. Uhlmann, M. 2005 An immersed boundary method with direct forcing for the simulation of particulate flows. J. Comput. Phys. 209, 448.CrossRefGoogle Scholar
32. Waller, M. D. 1955 Air circulations about a vibrating plate. Br. J. Appl. Phys. 6, 347.CrossRefGoogle Scholar

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