On the motion of an evaporating respiratory droplet

TL;DR

This paper models the motion and evaporation of respiratory droplets in stagnant air, analyzing how size, humidity, and initial velocity affect their trajectories and potential for infection spread.

Contribution

It introduces a detailed mathematical model accounting for nonlinear air resistance and evaporation effects, providing new insights into droplet behavior and transmission risk.

Findings

Droplet size and humidity significantly influence evaporation time.

Trajectories vary with initial velocity and angle, affecting ground reach.

Evaporation reduces droplet size before reaching the ground, impacting infection potential.

Abstract

An analysis of the projectile motion in stagnant air is presented for an evaporating respiratory micro-droplet which has been ejected from the mouth as an isolated droplet. It is assumed that the air resistance is a nonlinear function of the droplet's velocity and that the rate of decrease of the droplet's external surface area depends only on the relative humidity and the ambient temperature. The droplet's initial content is considered to be 98 wt% water, 1 wt% salt and 1 wt% protein. The change of the average density of the droplet due to water evaporation is determined, up to the instant when the droplet reduces to its nucleus, consisting of salt and dry protein only. The numerical solution of the governing differential equations of droplet's motion gives the trajectories of different-sized droplets ejected at different velocities and angles, and under different relative humidities and rates of evaporation. The evaporation times are compared with the times for droplets to reach the ground after being ejected from a given height. The maximum horizontal and vertical distances reached by the droplet are evaluated in the presence of wind and discussed in the context of possible infection spreading.