Drops in the wind: their dispersion and COVID-19 implications

Most of the works on the dispersion of droplets and their COVID-19 (Coronavirus disease) implications address droplets' dynamics in quiescent environments. As most droplets in a common situation are immersed in external flows (such as ambient flows), we consider the effect of canonical flow profiles namely, shear flow, Poiseuille flow, and unsteady shear flow on the transport of spherical droplets of radius ranging from 5$\mu$m to 100 $\mu$m, which are characteristic lengths in human talking, coughing or sneezing processes. The dynamics we employ satisfies the Maxey-Riley (M-R) equation. An order-of-magnitude estimate allows us to solve the M-R equation to leading order analytically, and to higher order (accounting for the Boussinesq-Basset memory term) numerically. Discarding evaporation, our results to leading order indicate that the maximum travelled distance for small droplets ($5\mu m$ radius) under a shear/Poiseuille external flow with a maximum flow speed of $1m/s$ may easily reach more than 250 meters, since those droplets remain in the air for around 600 seconds. The maximum travelled distance was also calculated to leading and higher orders, and it is observed that there is a small difference between the leading and higher order results, and that it depends on the strength of the flow. For example, this difference for droplets of radius $5\mu m$ in a shear flow, and with a maximum wind speed of $5m/s$, is seen to be around $2m$. In general, higher order terms are observed to slightly enhance droplets' dispersion and their flying time.