Symplectic and antiplectic waves in an array of beating cilia attached to a closed body

By taking into account the hydrodynamic interactions in a one dimensional array of model cilia attached to a no-slip cylinderical surface, we investigate their synchronized motion. We show how the emergence of metachronal waves depends on the initial state of the system and investigate the conditions under which the formation of symplectic and antiplectic waves are possible.

Figure 1

Left: geometry of a single cilium and right: an array of cilia attached to the surface of a cylinder. The direction of the power stroke and the average flow are shown in the figure.

Figure 2

Phase diagram for the state of synchronization for two or many cilia. Synchronization of two cilia depends on initial phase differences $\mathrm{\Delta }\varphi \left(0\right)$ and $\alpha$. Different values of $\mathrm{\Delta }\varphi \left(0\right)$ are shown by points on a unit circle. The synchronized states of two cilia are represented by the long time value of their phase differences $\mathrm{\Delta }\varphi \left(\infty \right)$ and are shown by bold dots on the circles. The arrows show the direction of the phase evolution from the initial values toward their final synchronized state, and different colors are used to show the evolution to different final states. For many cilia, both symplectic and antiplectic waves can emerge. For any $\beta \ne 0$, antiplectic (symplectic) waves emerge for $0<\alpha <\pi /2$ and $\pi <\alpha <3\pi /2(\pi /2<\alpha <\pi \mathrm{and}3\pi /2<\alpha <2\pi )$. Numerical parameters are as follows: $a/h=0.2,h/\ell =0.19,B/\ell =0.19,\beta =\pi /4$, and $e=0.87$.

Figure 3

The state of synchronization in a collection of cilia can be seen in a time-phase portrait. The vertical axes show the cilium number $n$, the horizontal axes show the time and the value of $sin\left[{\varphi }_n\left(t\right)\right]$ are encoded via colors. A symplectic wave appears for (a) $\alpha =2\pi -\frac{\pi }{4}$, and an antiplectic wave appears for (b) $\alpha =\frac{\pi }{4}$. The numerical parameters are $a/h=0.2,h/\ell =0.35,B/\ell =0.22,e=0.87$, and $\beta =\pi /4$.