On the validity of Stokes equations in ciliary flows*

Flagellar and ciliary flows at low Reynolds number are usually modeled by solving Stokes equations under given boundary conditions. These equations are time-invariant, which are valid when the viscous forces dominate over the inertia forces. The validity of this assumption for ciliary flows of high frequency (10-100 Hz) requires careful experimental examination. To investigate ciliary flows, we developed a time-resolved optical-tweezers based velocimetry (OTV). Our flow velocity measurements differ dramatically from that predicted by a stokeslet field. In particular, the periodic flow velocity is gradually phase delayed at increasing distances from the cilium; and the amplitude of the oscillatory flow component decays at a much faster rate (1/r^3) than the stokes prediction (1/r).  This indicates that the quasi-steady approximation and the use of Stokes equations for unsteady ciliary flows are often not justified, and the finite timescale for vorticity diffusion cannot be neglected. Our results
have significant implications in studies of ciliary synchronization.