Start-up flow of nanoscale particles and their periodic arrays: insights from fundamental solutions of the unsteady Stokes equations

Start-up flows induced by nanoscale particles and their periodic arrays are studied theoretically. At nanoscopic lengths and time scales, the interplay between inertial and viscous forces in a fluid results in time-dependent (unsteady) flows, which are important in the study of nanoporous materials and microswimmer locomotion. Here, these flows are studied by developing fundamental solutions, i.e. Green's functions considering point particles, of the unsteady Stokes equations. It is found that the approach to the steady state is characterized by a viscous penetration depth l = (4νt)^1/2, where ν is the kinematic viscosity and t is time. For an isolated particle, fluid inertia leads to vortex flows with a rotation axis located roughly a distance l from the particle. As time increases, the vortex distance to the particle increases diffusively as l² ∼ 4νt, with the limit l → ∞ corresponding to the steady-state limit. In a periodic array, inertia also leads to vortex flows. Furthermore, the presence of other array particles results in an unsteady back flow that develops simultaneously to the local flow around a test particle. The back flow develops with a characteristic time scale proportional to L²/ν, where L is the size of the unit cell.