Extreme singular events associated with inertial-viscous cusp formation in fluids

Abstract

Cusp singularities in fluids have been experimentally demonstrated in the past only at a low Reynolds number, Re ≪ 1, and large capillary number, Ca ≫ 1, in Newtonian or non-Newtonian fluids. Here, we show that the collapse of a free surface wave depression cavity can lead to inertial-viscous cusp formation at local Re > 1 and Ca > 1, which gives rise to extreme events, i.e., very high-velocity surface jets. The cavities are generated in a cylindrical container (2R = 10 cm), partially filled with glycerine–water solution, by parametrically forcing the axi-symmetric wave mode beyond the breaking limit. By varying the forcing amplitude and the fluid viscosity, parabolic or cusp singularities manifest, depending on the last stable wave amplitude b that determines the cavity shape. Cusp formation in collapse without bubble pinch-off, leading to very high-velocity surface jets, is obtained when b is close to the singular wave amplitude bs and Ca > 1. The free surface shape is self-similar, changing from an inertial to a viscous regime when the singularity is approached. At cusp singularity, the cavity shape takes the form of (z − Z0)/R ∼ −(r/R)2/3, where Z0 is the final cavity depth. Cavity collapse with bubble pinch-off, which occurs when b > bs, also exhibits a cusp singularity when bs < b ≤ 1.14 bs and Ca > 1, but surface jet velocities are much less because about half of the wave energy is lost.