Perturbation Theory for Translating Oblate-Spheroidal Droplets with Internal Circulation

Abstract

Liquid droplets deform from spherical shape due to aerodynamic variation of pressure along the surface as the droplet moves through a gas. The deformation is predicted for axisymmetric droplets translating through a gas with low Weber numbers, We < 1, and Reynolds number Re = O(10). That deformation analysis is based on the relations between local pressure jump and the two radii of curvature. A thin boundary layer on both sides of the gas-liquid interface is considered with a surface-velocity jump due to pressure-gradient-driven flow with a large density jump and a pressure jump due to surface tension. A near-ellipsoidal shape is predicted using $We$ as a perturbation parameter. Then, the quasi-steady internal liquid-phase stream function and velocity field are predicted, describing internal circulation and a vortex ring structure with vorticity distributed through an inviscid liquid. The gas-phase flow over the oblate droplet is described using a ring doublet as an image within the droplet. The ring-doublet radius is related to We. Gas potential flow results are presented and compared using both the exact analytical solution and a perturbation analysis based on the square root of We. The perturbation analysis provides a lower computational cost. Three analyses for local curvature, liquid circulation, and gas potential flow are matched to yield the velocity and pressure fields. The appropriate radius for the image ring doublet is matched to the square root of We. Liquid-phase stream function, two velocity components in each fluid, and gas potential field are predicted. S Some comments on droplet drag are presented.