  A new mathematical analysis of exhaled respiratory droplet dynamics and
settling distances in the vicinity of vortical environments is presented.
Recent experimental and theoretical studies suggest that vortical flow
structures may enhance the settling distances of exhaled respiratory droplets
beyond the two-meter distancing rule recommended by health authorities lately.
We propose a mathematical framework to study the underlying physical mechanism
responsible for the entrapment and subsequently delayed settling times of
evaporating droplets and solid particles. A dipolar vortex is considered
self-propelling through a cloud of micron-sized evaporating droplets. This
configuration might be utilized to approximate an indoor environment in which
similar unsteady vortical flow structures interact with exhaled respiratory
droplets. We demonstrate the vortex dipole effect on droplet and solid
particles settling distances, depending on the evaporation rate, the vorticity
of the dipole, and the droplet's initial diameter and location relative to the
vortex core. Our theoretical analysis reveals non-intuitive interactions
between the vortex dipole, droplet relaxation time, gravity, and mass transfer.
The existence of optimal conditions for maximum displacement is suggested,
where the droplet entrainment reaches up to an order of magnitude larger than
the vortex core length scale. We present a basic model that may be applied for
evaluating the spread of exhaled respiratory droplets in vortical environments.
Our theoretical study suggests that exhaled respiratory droplets initially at
rest can translate to significant distances, hence implying that vortical flow
might enhance the transmission of airborne pathogens.
