Start-up Flow of Nanoscale Particles and their Periodic Arrays: Insights from Fundamental Solutions of the Unsteady Stokes Equations

At nanoscopic lengths and time scales, the interplay between inertial and viscous forces in the fluid results in time dependent (unsteady) flows important in the study of nanoporous materials and microswimmer locomotion. Here, these unsteady flows are studied theoretically by developing fundamental solutions of the unsteady Stokes equations for isolated particles and particles in periodic arrays. It is found that the approach to steady state is characterized by a time dependent viscous penetration depth. For an isolated particle, fluid inertia leads to vortex flows with a rotation axis whose distance from the particle is roughly equal to the penetration depth. As time increases, the vortex distance to the particle increases diffusively, with the limit where the vortex is infinitely far from the particle corresponding to steady state. In a periodic array, inertia also leads to vortex flows. Furthermore, the presence of the other array particles results in an unsteady back flow that develops simultaneously to the local flow around a test particle. The development of the back flow occurs with a characteristic time scale that is proportional to the time it takes the vortex to diffusive a distance on the order of the unit cell size.