Using Lagrangian coherent structures to analyze fluid mixing by cilia

Abstract

Motivated by the desire to understand the fluid flow within the airway surface liquid of the lung, we consider the flow generated by a computational model of a motile, internally actuated cilium. The cilium, along with a mucus layer modeled by linear elastic elements, is coupled to a viscous, incompressible fluid. The evolution of this coupled system is captured using an immersed boundary method. The Eulerian velocity field computed on a grid is used to compute finite-time Lyapunov exponent fields, whose maximal ridges identify Lagrangian coherent structures (LCSs). The computed LCS uncovers a barrier that separates a recirculation region of fluid that remains near the beating cilium from fluid that is advected downstream. Moreover, periodic stretching and folding of this region gives rise to complex mixing. Flow structures around a cilium propelling a mucus layer are compared to flow structures around a cilium with no mucus load.

Mixing and transport by ciliary carpets: a numerical study

“…The possibility of mixing by chaotic advection below the cilia tips is suggested in the experimental studies of Supatto et al (2008) on a single beating cilium in zebrafish embryo and Shields et al (2010) on arrays of artificial cilia beating in synchrony. More concrete evidence of mixing by chaotic advection was reported in the computational studies of Lukens et al (2010) on a single cilium in a two-dimensional fluid and Khatavkar et al (2007) on single and two cilia in a two-dimensional rectangular channel. In the case of two cilia (micro-actuators) beating at a phase difference ∆φ, Khatavkar et al (2007) reported exponential growth in the length of a fluid strip for ∆φ between π/3 and 2π/3 with the optimal value being π/2, which is consistent with our results.…”

“…At this micro scale, lack of turbulence makes mixing by chaotic advection a nontrivial task (Aref, 1990;Ottino, 1989). The computational results of Lukens et al (2010) showed some evidence of mixing in the flow around a single cilium. A transition from unidirectional flow above the cilium to vortical flow below the cilium was reported by Supatto et al (2008) based on in-vivo experiments that mapped the velocity field surrounding a single beating cilium in zebrafish embryo.…”

“…The flow fields generated by beating cilia were examined numerically using the Stokeslet method (see Smith et al, 2007;Ainley et al, 2008;Cortez, 2001, and references therein) as well as the immersed boundary method, see, e.g. Lukens et al (2010) for a description of the flow field generated In the synchronized case, i.e. , with a phase difference of zero, no metachronal wave is generated.…”

Fluid flows and forces in development: functions, features and biophysical principles

“…Interestingly, the beating frequency of cilia is rather similar between species and organs, and usually varies between 10 and 30 Hz (Supatto and Vermot, 2011). Mathematical models (Lukens et al, 2010) suggest that such a planar motion promotes mixing near the cilia.…”

The backward phase flow method for the Eulerian finite time Lyapunov exponent computations

“…For example, it has been recently applied to study flow data in oceans, 15,27 hurricane, 26 flight data, 4,32 gravity wave propagation, 33 some bio-inspired fluid flows, 10,19,21 etc. The idea is to partition the space-time domain into different regions according to a Lagrangian quantity advected along with passive tracers.…”