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The Faraday instability in rectangular and annular geometries: comparison of experiments with theory

Abstract

A longstanding question on the mechanically forced Faraday instability in rectangular geometries arises from a disparity between theory and experiments (Craik and Armitage in Fluid Dyn Res 15:129–143, 1995). It can be stated as: do corners in a rectangular geometry Faraday experiment cause the disparity between prediction and experiment? This study is an attempt to settle this question by comparing the Faraday instability for two fluids in rectangular geometries where corners must be present with equivalent annular geometries where such corners are necessarily absent. Non-idealities, i.e., damping, arising from a slight meniscus wave motion and induced sidewall damping are observed for thin gaps, causing discrepancies between the predicted instability thresholds and those determined experimentally. However, even for thin-gap geometries, experimental agreement between the tested rectangular geometry and its corresponding annular geometry remains excellent. This suggests that corner effects on the system stability are negligible for rectangular geometries and thus any disparity between theory and experiment is due principally to the damping caused by proximity of the lateral walls.

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References

  1. Abramson HN, Dodge FT, Kana D (1965) Liquid surface oscillations in longitudinally excited rigid cylindrical containers. AIAA J 3(4):685–695

  2. Batson W, Zoueshtiagh F, Narayanan R (2013) The Faraday threshold in small cylinders and the sidewall non-ideality. J Fluid Mech 729:496–523

  3. Bechhoefer J, Manneville S, Ego V, Johnson B (1995) An experimental study of the onset of parametrically pumped surface waves in viscous fluids. J Fluid Mech 288:325–350

  4. Benjamin TB, Ursell F (1954) The stability of the plane free surface of a liquid in vertical periodic motion. Proc R Soc Lond A 225(1163):505–515

  5. Courant R, Hilbert D (1937) Methods of mathematical physics, vol 1. Wiley Interscience, New York

  6. Craik ADD, Armitage JGM (1995) Faraday excitation, hysteresis and wave instability in a narrow rectangular wave tank. Fluid Dyn Res 15:129–143

  7. Das SP, Hopfinger EJ (2008) Parametrically forced gravity waves in a circular cylinder and finite-time singularity. J Fluid Mech 599:205–228

  8. Douady S (1990) Experimental study of the Faraday instability. J Fluid Mech 221:383–409

  9. Edwards WS, Fauve S (1994) Patterns and quasi-patterns in the Faraday experiment. J Fluid Mech 278:123–148

  10. Ezerskii AB, Korotin PI, Rabinovich MI (1985) Random self-modulation of two-dimensional structures on a liquid during parametric excitation. Sov Phys JETP 41:157–160

  11. Faraday M (1831) On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces. Philos Trans 121:299–340

  12. Henderson DM, Miles JW (1990) Single-mode Faraday waves in small cylinders. J Fluid Mech 213:95–109

  13. Kudrolli A, Gollub JP (1996) Patterns and spatiotemporal chaos in parametrically forced surface waves: a systematic survey at large aspect ratio. Phys D 97(1):133–154

  14. Kumar K, Tuckerman LS (1994) Parametric instability of the interface between two fluids. J Fluid Mech 279:49–68

  15. Lam KD Nguyem Thu, Caps H (2011) Effect of a capillary meniscus on the Faraday instability threshold. Eur Phys J E Soft Matter 34(10):112

  16. Miles JW (1967) Surface-wave damping in closed basins. Proc R Soc Lond A 297:459–475

  17. Müller HW (1993) Periodic triangular patterns in the Faraday experiment. Phys Rev Lett 71(20):3287–3290

  18. Rayleigh JWS (1883) On the crispations of fluid resting upon a vibrating support. Philos Mag 16(97):50–58

  19. Tipton CR, Mullin T (2004) An experimental study of Faraday waves formed on the interface between two immiscible liquids. Phys Fluids 16(7):2336–2341

  20. Weinberger HF (1995) A first course in partial differential equations with complex variables and transform methods. Dover, New York

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Acknowledgements

Support is acknowledged from the National Science Foundation via a Graduate Research Fellowship under Grant no. DGE-1315138, CASIS NNH11CD70A, NASA NNX17AL27G, and a Chateaubriand Fellowship. RN acknowledges support from the Institute of Advanced Study, Durham University for a Fellowship. FZ acknowledges support from CNES. The authors also acknowledge Dipin Pillai for helpful conversations about the manuscript.

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Correspondence to Kevin Ward.

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Ward, K., Zoueshtiagh, F. & Narayanan, R. The Faraday instability in rectangular and annular geometries: comparison of experiments with theory. Exp Fluids 60, 53 (2019). https://doi.org/10.1007/s00348-019-2695-4

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