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>i.e.</i> 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 <i>l</i> = (4<i>νt</i>)^1/2, where <i>ν</i> is the kinematic viscosity and <i>t</i> is time. For an isolated particle, fluid inertia leads to vortex flows with a rotation axis located roughly a distance <i>l</i> from the particle. As time increases, the vortex distance to the particle increases diffusively as <i>l</i>² ∼ 4<i>νt</i>, with the limit <i>l</i> → <i>∞</i> 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 <i>L</i>²/<i>ν</i>, where <i>L</i> is the size of the unit cell.