First-passage-time problem for tracers in turbulent flows applied to virus spreading

We study the spreading of viruses, such as SARS-CoV-2, by airborne aerosols, via a first-passage-time problem for Lagrangian tracers that are advected by a turbulent flow: By direct numerical simulations of the three-dimensional (3D) incompressible Navier-Stokes equation, we obtain the time t(R) at which a tracer, initially at the origin of a sphere of radius R, crosses the surface of the sphere for the first time. We obtain the probability distribution function P(R, t(R)) and show that it displays two qualitatively different behaviors: (a) for R << L-I, P(R, t(R)) has a power-law tail similar to t(R)(-alpha), with the exponent alpha = 4 and L-I the integral scale of the turbulent flow; (b) for L-I less than or similar to R, the tail of P(R, t(R)) decays exponentially. We develop models that allow us to obtain these asymptotic behaviors analytically. We show how to use P(R, t(R)) to develop social-distancing guidelines for the mitigation of the spreading of airborne aerosols with viruses such as SARS-CoV-2.