Will we ever wash our hands of lubrication theory?

Lubrication theory is used to investigate how weakly bound particles can be transported away from the vicinity of the wall when two spatially periodic rough surfaces are sheared relative to one another at constant velocity U while immersed in fluid. The aim is to model what could be an important process during decontamination of hands by washing and is motivated by Mittal et al. ["The flow physics of COVID-19," J. Fluid Mech. 894, F2 (2020)] who remark "Amazingly, despite the 170+ year history of hand washing in medical hygiene, we were unable to find a single published research article on the flow physics of hand washing." Under the assumption that the roughness wavelength $$ is large compared with the spacing of the surfaces, a, the lubrication approximation permits closed-form expressions to be found for the time-varying velocity components. These are used to track the motion of a particle that is initially trapped in a potential well close to one of the surfaces, and experiences a drag force proportional to the difference between its velocity and that of the surrounding fluid. Complications such as particle-wall hydrodynamic interactions, finite size effects, and Brownian motion are ignored for now. Unsurprisingly, particles remain trapped unless the flow driven by the wall motion is strong compared to the depth of the trapping potential well. Perhaps less obvious is that for many starting positions the process of escape to large distances from the wall takes place over a large number of periods $$ , essentially because the no-slip boundary condition means that fluid velocities relative to the wall are small close to the wall, and thus the velocities of particles along or away from the wall are also small. With reasonable estimates for the various dimensional parameters, the escape times in these cases are found to be comparable in magnitude to the washing times recommended in hand washing guidelines.