Epicyclic orbits in a viscous fluid about a precessing rod: Theory and experiments at the micro- and macro-scales

Abstract

We present experimental observations and quantified theoretical predictions of the nanoscale hydrodynamics induced by nanorod precession emulating primary cilia motion in developing embryos. We observe phenomena including micron size particles which exhibit epicyclic orbits with coherent fluctuations distinguishable from comparable amplitude thermal noise. Quantifying the mixing and transport physics of such motions on small scales is critical to understanding fundamental biological processes such as extracellular redistribution of nutrients. We present experiments designed to quantify the trajectories of these particles, which are seen to consist of slow orbits about the rod, with secondary epicycles quasicommensurate with the precession rate. A first-principles theory is developed to predict trajectories in such time-varying flows. The theory is further tested using a dynamically similar macroscale experiment to remove thermal noise effects. The excellent agreement between our theory and experiments confirms that the continuum hypothesis applies all the way to the scales of such submicron biological motions.

Biomimetic cilia arrays generate simultaneous pumping and mixing regimes

“…At short time scales, the cilia beat produces tracer epicycles that orbit at the frequency of the cilia beat, due to the strong coupling of the cilia beat with the low Reynolds number fluid (34). Because of the no-slip boundary at the cilium surface, tracers in close proximity to cilia should move with speeds comparable to the cilia tip speeds of approximately 700-800 μm∕s.…”

On the correspondence between creeping flows of viscous and viscoelastic fluids

“…Indeed, combining both correspondences leads to a transfer between elastic and viscous classical exact solutions [13]. Looking ahead to more sophisticated applications, by summing appropriate singularities, and exploiting linearity of the equations of motion, solutions of complex boundary value problems for viscous fluids have been developed; examples include slender body theory [14], the viscous flow around a rotating rod [3,15], and numerical methods for fluid-structure interactions (cf. the immersed boundary method [23], the blob projection method [7], manyparticle codes [4,29,27,22,25], and flows of slender filaments [26,31]).…”

A multirate time integrator for regularized Stokeslets

“…Specifically, a slender rigid rod precessing about its center sweeping out a double cone in an incompressible fluid containing a rigid sphere and flexible fibers in shear Stokes flow are used as examples in this discussion. These examples are motivated by current experimental fluid dynamics research being conducted at the University of North Carolina [23][24][25]. In the rod and sphere case introduced in Section 4.2, the calculation of the Stokeslet strengths along the rod requires the solution of a large dense linear system while those on the sphere can be easily computed with a linear spring law.…”

“…The numerical examples in this paper are motivated by current experimental fluid dynamics research being conducted at the University of North Carolina [23][24][25]. First, consider the case of a rigid rod precessing about its center in free space with a prescribed angular velocity and a rigid sphere moving and interacting with the fluid, as shown in Fig.…”

“…In Section 4, two application examples are discussed, both of that are motivated by current experimental fluid dynamics research being conducted at the University of North Carolina [23][24][25]. The first example models a slender rigid spheroidal rod precessing in an incompressible fluid that also contains a rigid sphere, while the second models multiple flexible fibers inspired by an experiment to investigate the flow through the endothelial glycocalyx.…”