Advances in Mathematical Fluid Mechanics pp 329-338 | Cite as

# Streaming Flow Effects in the Nearly Inviscid Faraday Instability

## Abstract

We study the weakly nonlinear evolution of Faraday waves in a two dimensional container that is vertically vibrated. It is seen that the surface wave evolves to a drifting standing wave, namely a wave that is standing in a moving reference frame. In the small viscosity limit, the evolution of the surface waves is coupled to a non-oscillatory mean flow that develops in the bulk of the container. A system of equations is derived for the coupled slow evolution of the spatial phase of the surface wave and the streaming flow. These equations are numerically integrated to show that the simplest reflection symmetric steady state (the usual array of counter rotating eddies below the surface wave) becomes unstable for realistic values of the parameters. The new states include limit cycles, drifted standing waves and some more complex attractors.We also consider the effect of surface contamination, modelled by Marangoni elasticity with insoluble surfactant, in promoting drift instabilities in spatially uniform standing Faraday waves. It is seen that contamination enhances drift instabilities that lead to various steadily propagating and (both standing and propagating) oscillatory patterns. In particular, steadily propagating waves appear to be quite robust, as in the experiment by Douady et al., Europhysics Letters, pp. 309-315, 1989

## Keywords

Faraday waves Weakly nonlinear evolution Hydrodynamic instabilities Surface contamination Marangoni elasticity Navier-Stokes numerical simulation## Preview

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## Notes

### Acknowledgments

This work was supported by the National Aeronautics and Space Administration (Grant NNC04GA47G) and the Spanish Ministry of Education (Grant MTM2004-03808)

## References

- 1.Cross, M. and Hohenberg, P.C.: Pattern formation outside of equilibrium. Rev. Mod. Phys.,
**65**, pp. 851–1112 (1993)CrossRefGoogle Scholar - 2.Douady, S. Fauve, S. and Thual, O.: Oscillatory phase modulation of parametrically forced surface waves. Europhys. Lett.,
**10**, pp. 309–315 (1989)CrossRefGoogle Scholar - 3.Faraday, M.: On the forms and states assumed by fluids in contact with vibrating elastic surfaces. Phil. Trans. R. Soc. Lond.,
**121**, pp. 319–340 (1831)Google Scholar - 4.Lapuerta, V., Martel, C. and Vega, J.M.: Interaction of nearly-inviscid Faraday waves and mean flows in 2-D containers of quite large aspect ratio. Physica D,
**173**, pp. 178–203 (2002)MATHCrossRefMathSciNetGoogle Scholar - 5.Martel C. and Knobloch, E.: Damping of nearly-inviscid water waves. Phys. Rev. E,
**56**, pp. 5544–5548 (1997)CrossRefGoogle Scholar - 6.Martín, E., Martel, C. and Vega, J.M.: Drift instability of standing Faraday waves. J. Fluid Mech.,
**467**, pp. 57–79 (2002)MATHCrossRefMathSciNetGoogle Scholar - 7.Martín, E. and Vega, J.M.: The effect of surface contamination on the drift instability of standing Faraday waves. J. Fluid Mech.,
**546**, pp. 203–225 (2006)MATHCrossRefMathSciNetGoogle Scholar - 8.Miles, J. and Henderson D.: On the forms and states assumed by fluids in contact with vibrating elastic surfaces. Annu. Rev. Fluid Mech.,
**22**, pp. 143–165 (1990)CrossRefMathSciNetGoogle Scholar - 9.Thual, O., Douady, S. and Fauve, S.: Intabilities and Nonequilibrium Structures II. Ed. Tirapegui, E. and Villaroel, D., Kluwer, Dordrecht p. 227 (1989)Google Scholar
- 10.Vega, J.M., Knobloch, E. and Martel, C.: Nearly inviscid Faraday waves in annular containers of moderately large aspect ratio. Physica D,
**154**, pp. 147–171 (2001)CrossRefGoogle Scholar