On impulsively generated inviscid axisymmetric surface jets, waves and drops

Abstract

The evolution of an unbounded inviscid free surface subjected to a velocity potential of Gaussian form and also to the influence of inertial, interfacial and gravitational forces is considered. This construct was motivated by the occurrence of lung haemorrhage resulting from ultrasonic imaging and pursues the notion that bursts of ultrasound act to expel droplets that puncture the soft air-filled sacs in the lung plural surface, allowing them to fill with blood. The tissue adjacent to the sacs is modelled as a liquid and the air–tissue interface in the sacs as a free surface. The evolution of the free surface is described by a boundary-integral formulation and, since the free surface evolves slowly relative to the bursts of ultrasound, they are realized as an impulse at the free surface, represented by the velocity potential. As the free surface evolves, it is seen to form axisymmetric surface jets, waves or droplets, depending upon the levels of gravity and surface tension. Moreover the droplets may be spherical and ejected away from the surface or an inverted tear shape and fall back to the surface. These conclusions are expressed in a phase diagram of inverse Froude number Fr−1 versus inverse Weber number We−1. Specifically, while axisymmetric surface jets form in the absence of surface tension and gravity, gravity acts to bound their height, rendering them waves, although instability overrides the calculation prior to its reaching that bound. Surface tension acts to suppress the instability (provided that We−1 > 0.045) and to form drops; if sufficiently strong it can also damp the evolving wave, causing it to collapse. The pinchoff which effects spherical drops is of power-law type with exponent 2/3, and the universal constant that relates the necking radius to the time from pinchoff, thereby realizing a finite-time singularity, has the value ${\mathfrak{K}} \,{=}\, 0.45 \pm 0.025$. Finally, drops can occur once the mechanical index, a dimensional measure used in ultrasonography, exceeds 0.5.