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Faraday instability on a sphere: numerical simulation

  • A. Ebo-Adou (a1) (a2) (a3), L. S. Tuckerman (a1), S. Shin (a4), J. Chergui (a2) and D. Juric (a2)...

Abstract

We consider a spherical variant of the Faraday problem, in which a spherical drop is subjected to a time-periodic body force, as well as surface tension. We use a full three-dimensional parallel front-tracking code to calculate the interface motion of the parametrically forced oscillating viscous drop, as well as the velocity field inside and outside the drop. Forcing frequencies are chosen so as to excite spherical harmonic wavenumbers ranging from 1 to 6. We excite gravity waves for wavenumbers 1 and 2 and observe translational and oblate–prolate oscillation, respectively. For wavenumbers 3 to 6, we excite capillary waves and observe patterns analogous to the Platonic solids. For low viscosity, both subharmonic and harmonic responses are accessible. The patterns arising in each case are interpreted in the context of the theory of pattern formation with spherical symmetry.

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Corresponding author

Email address for correspondence: laurette@pmmh.espci.fr

References

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Arbell, H. & Fineberg, J. 2002 Pattern formation in two-frequency forced parametric waves. Phys. Rev. E 65, 036224.
Basaran, O. A. 1992 Nonlinear oscillations of viscous liquid drops. J. Fluid Mech. 241, 169198.
Busse, F. H. 1975 Patterns of convection in spherical shells. J. Fluid Mech. 72, 6785.
Busse, F. H. & Riahi, N. 1982 Patterns of convection in spherical shells. Part 2. J. Fluid Mech. 123, 283301.
Chorin, A. J. 1968 Numerical simulation of the Navier–Stokes equations. Math. Comput. 22, 745762.
Chossat, P., Lauterbach, R. & Melbourne, I. 1991 Steady-state bifurcation with O (3) symmetry. Arch. Rat. Mech. Anal. 113, 313376.
Douady, S. 1990 Experimental study of the Faraday instability. J. Fluid Mech. 221, 383409.
Ebo-Adou, A. & Tuckerman, L. S. 2016 Faraday instability on a sphere: Floquet analysis. J. Fluid Mech. 805, 591610.
Edwards, W. S. & Fauve, S. 1994 Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech. 278, 123148.
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. A 121, 299340.
Golubitsky, M., Stewart, I. & Schaeffer, D. G. 1988 Singularities and Groups in Bifurcation Theory: Vol. II. Springer.
Harlow, F. H. & Welch, J. E. 1965 Numerical calculation of time dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8, 2182.
Ihrig, E. & Golubitsky, M. 1984 Pattern selection with O (3) symmetry. Physica D 13, 133.
Kahouadji, L., Périnet, N., Tuckerman, L. S., Shin, S., Chergui, J. & Juric, D. 2015 Numerical simulation of supersquare patterns in Faraday waves. J. Fluid Mech. 772, R2.
Kudrolli, A., Pier, B. & Gollub, J. P. 1998 Superlattice patterns in surface waves. Physica D 123, 99111.
Kumar, K. 1996 Linear theory of Faraday instability in viscous fluids. Proc. R. Soc. Lond. 452, 11131126.
Kumar, K. & Tuckerman, L. S. 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 4968.
Kwak, D. Y. & Lee, J. S. 2004 Multigrid algorithm for the cell-centered finite-difference method II: discontinuous coefficient case. Numer. Methods Partial Differential Equations 20, 723741.
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Lundgren, T. S. & Mansour, N. N. 1988 Oscillations of drops in zero gravity with weak viscous effects. J. Fluid Mech. 194, 479510.
Matthews, P. C. 2003 Pattern formation on a sphere. Phys. Rev. E 67, 036206.
Meradji, S., Lyubimova, T. P., Lyubimov, D. V. & Roux, B. 2001 Numerical simulation of a liquid drop freely oscillating. Cryst. Res. Technol. 36, 729744.
Patzek, T., Benner, R., Basaran, O. & Scriven, L. 1991 Nonlinear oscillations of inviscid free drops. J. Comput. Phys. 97, 489515.
Périnet, N., Juric, D. & Tuckerman, L. S. 2012 Alternating hexagonal and striped patterns in Faraday surface waves. Phys. Rev. Lett. 109, 164501.
Peskin, C. S. 1977 Numerical analysis of blood flow in the heart. J. Comput. Phys. 25, 220252.
Politis, A.2013 Real/complex spherical harmonic transform, Gaunt coefficients and rotations. http://www.mathworks.com/matlabcentral/fileexchange/43856-real-complex-spherical-harmonic-transform-gaunt-coefficients-and-rotations/.
Politis, A.2016 Microphone array processing for parametric spatial audio techniques. PhD thesis, Aalto University, Finland.
Popinet, S. 1993 Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. J. Comput. Phys. 190, 572600.
Rajchenbach, J., Leroux, A. & Clamond, D. 2011 New standing solitary waves in water. Phys. Rev. Lett. 107, 024502.
Rayleigh, L. 1879 On the capillary phenomena of jets. Proc. R. Soc. Lond. 29, 7197.
Riahi, N. 1984 Nonlinear convection in a spherical shell. J. Phys. Soc. Japan 53, 25062512.
Rucklidge, A. M. & Skeldon, A. C. 2015 Can weakly nonlinear theory explain Faraday wave patterns near onset? J. Fluid Mech. 777, 604632.
Shin, S. 2007 Computation of the curvature field in numerical simulation of multiphase flow. J. Comput. Phys. 222, 872878.
Shin, S., Chergui, J. & Juric, D. 2017 A solver for massively parallel direct numerical simulation of three-dimensional multiphase flows. J. Mech. Sci. Technol. 31, 17391751.
Shin, S. & Juric, D. 2007 High order level contour reconstruction method. J. Mech. Sci. Technol. 21, 311326.
Shin, S. & Juric, D. 2009 A hybrid interface method for three-dimensional multiphase flows based on front-tracking and level set techniques. Intl J. Numer. Meth. Fluids 60, 753778.
Shu, C. W. & Osher, S. 1989 Efficient implementation of essentially non-oscillatory shock capturing schemes, II. J. Comput. Phys. 83, 3278.
Silber, M. & Proctor, M. R. E. 1998 Nonlinear competition between small and large hexagonal patterns. Phys. Rev. Lett. 81, 24502453.
Trinh, E. & Wang, T. G. 1982 Large-amplitude free and driven drop-shape oscillations: experimental observations. J. Fluid Mech. 122, 315338.
Tryggvason, G., Scardovelli, R. & Zaleski, S. 2011 Direct Numerical Simulation of Gas–Liquid Multiphase Flows. Cambridge University Press.
Tsamopoulos, J. A. & Brown, R. A. 1983 Nonlinear oscillations of inviscid drops and bubbles. J. Fluid Mech. 127, 514537.
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JFM classification

Type Description Title
VIDEO
Movies

Ebo-Adou et al. supplementary movie 1
Visualisation of $\ell=1$ mode for spherical drop. The drop is displaced alternately to the left and the right. Length and colors of arrows indicate the velocity of the surrounding air.

 Video (6.2 MB)
6.2 MB
VIDEO
Movies

Ebo-Adou et al. supplementary movie 2
Visualisation of $\ell=2$ prolate-oblate pattern of gravitational harmonic waves. Drop interface and velocity field on and outside drop are shown.

 Video (49.5 MB)
49.5 MB
VIDEO
Movies

Ebo-Adou et al. supplementary movie 3
Visualisation of $\ell=3$ tetrahedral pattern of capillary subharmonic waves.

 Video (2.3 MB)
2.3 MB
VIDEO
Movies

Ebo-Adou et al. supplementary movie 4a
Visualisation of $\ell=4$ cubic-octahedral pattern of capillary subharmonic waves.

 Video (1.4 MB)
1.4 MB
VIDEO
Movies

Ebo-Adou et al. supplementary movie 4b
Visualisation of $\ell=4$ axisymmetric pattern for subharmonic capillary waves. In this stroboscopic film, only snapshots at a single temporal phase are included, emphasizing the overall drift of the pattern

 Video (4.0 MB)
4.0 MB
VIDEO
Movies

Ebo-Adou et al. supplementary movie 5
Visualisation of $\ell=5$ mode for subharmonic capillary waves, showing evolution from axisymmetric to $D_4$ pattern.

 Video (14.8 MB)
14.8 MB
VIDEO
Movie

Ebo-Adou et al. supplementary movie 6
Visualisation of $\ell=6$ icosahedral-dodecahedral pattern for subharmonic capillary waves.

 Video (2.6 MB)
2.6 MB

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