Evolution and breaking of parametrically forced capillary waves in a circular cylinder

Abstract

We present results on parametrically forced capillary waves in a circular cylinder, obtained in the limit of large fluid depth, using two low-viscosity liquids whose surface tensions differ by an order of magnitude. The evolution of the wave patterns from the instability to the wave-breaking threshold is investigated in a forcing frequency range (f= ω/2π = 25–100 Hz) that is around the crossover frequency (ωot) from gravity to capillary waves (ωot/2≤ω/2≤4ωot). As expected, near the instability threshold the wave pattern depends on the container geometry, but as the forcing amplitude is increased the wave pattern becomes random, and the wall effects are insignificant. Near breaking, the distribution of random wavelengths can be fitted by a Gaussian. A new gravity–capillary scaling is introduced that is more appropriate, than the usual viscous scaling, for low-viscosity fluids and forcing frequencies <103Hz. In terms of these scales, a criterion is derived to predict the crossover from capillary- to gravity-dominated breaking. A wave-breaking model is developed that gives the relation between the container and the wave accelerations in agreement with experiments. The measured drop size distribution of the ejected drops above the breaking threshold is well approximated by a gamma distribution. The mean drop diameter is proportional to the wavelength determined from the dispersion relation; this wavelength is also close to the most likely wavelength of the random waves at drop ejection. The dimensionless drop ejection rate is shown to have a cubic power law dependence on the dimensionless excess acceleration ε′dan inertial–gravitational ligament formation model is consistent with such a power law.