3.2.1 Transport and mixing zone

Figure 7.

Normalized average displacement in the $x$ -direction as a function of $z/L$ for (a)  $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=\pm \unicode[STIX]{x03C0}/9$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=\pm \unicode[STIX]{x03C0}/4$ with $a/L=0.8$ , and (b)  $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=\pm \unicode[STIX]{x03C0}/4$ and $a/L=1.67$ . As in Ding et al. (Reference Ding, Nawroth, McFall-Ngai and Kanso2014), the mixing zone extends from 0 to $1L$ and the transport zone from $1L$ to $1.4L$ . In both cases (a) and (b), the mucus–PCL interface is located at $z/L=0.9$ and is indicated by a dashed horizontal line.

The displacement field is calculated as $\boldsymbol{d}(\boldsymbol{x})=\int _{0}^{\text{T}}\boldsymbol{u}(\boldsymbol{x}(t),t)\,\text{d}t$ , where $\boldsymbol{x}$ is the position vector and $\boldsymbol{u}$ is the fluid velocity. Its component in the $x$ -direction as a function of the axial position is averaged over several beating cycles and displayed in figure 7. The transport and mixing zones defined in Ding et al. (Reference Ding, Nawroth, McFall-Ngai and Kanso2014) are clearly visible. This shows that the antipleptic MCWs perform better than the other kind of coordination in transporting the particles. It should be noted that, in figure 7(b), the two curves with $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=\pm \unicode[STIX]{x03C0}/4$ have a similar shape, whereas in figure 7(a), symplectic MCWs induce a negative transport (i.e. a counter-flow) in the mixing zone for the same phase lag. For a given value of the phase lag, the transport depends on the density of cilia. One can also notice the importance of the phase lag by looking at figure 7(a): for a phase lag $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=-\unicode[STIX]{x03C0}/9$ , the symplectic MCWs induce no counter-flow in the mixing zone, while the opposite happens for $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=-\unicode[STIX]{x03C0}/4$ . Finally, it is worth noticing that, as the cilium spacing increases, the influence of the kind of metachrony decreases. Eventually, for very large cilium spacings, one would expect that the three curves displayed in figure 7(b) would merge into a single one corresponding to the displacement generated by an isolated cilium. Additionally, even with a larger cilium spacing, the particles are better transported in case (b). As in Ding et al. (Reference Ding, Nawroth, McFall-Ngai and Kanso2014), the displacement reaches a plateau for an axial position of $1.4L$ in both cases (a) and (b). The beating pattern used in Ding et al. (Reference Ding, Nawroth, McFall-Ngai and Kanso2014) is different, but similar trends are observed. Nevertheless, a major difference must be pointed out: for $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=-\unicode[STIX]{x03C0}/4$ and $a/L=1.67$ (figure 7 b), the present results show that the displacement induced by the symplectic wave has the same shape as the synchronized and antipleptic cases, i.e. it induces transport in the mucus phase even if it is smaller. On the contrary, Ding et al. (Reference Ding, Nawroth, McFall-Ngai and Kanso2014) obtained no transport at all for the symplectic wave in the mucus phase, but instead a peak of transport under the cilium tips in the PCL at $z=0.7L$ due to the presence of a vortex-like structure both below and above the cilium tips for this particular value of the phase lag $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}$ . Since the main difference between the two studies is the beating pattern of the cilia (same phase lag and cilium spacing), this highlights the sensitivity of the system to the beating pattern.

The displacement in the $z$ - and $y$ -directions has also been analysed. The results show that the $y$ -component of the displacement can be neglected and that the $z$ -component of the displacement is almost null above the cilium tips, but not zero under them. As will be discussed later, the $z$ -component of the displacement reaches its maximal value in regions where the shear rate is maximal too. This agrees with the existence of a mixing zone under the cilium tips.

By making an analogy with the strain-rate tensor classically used in solid mechanics, the gradient of the displacement field $\unicode[STIX]{x1D735}d$ can be computed by considering each Eulerian node of the Cartesian grid as a passive tracer. Since the displacement in the transverse direction is very small, the $y$ -component of the displacement field can be neglected. Henceforth, the gradient is computed over every ( $x,z$ ) plane, and an average is then performed over the $y$ -direction. Following the methodology described in Ding et al. (Reference Ding, Nawroth, McFall-Ngai and Kanso2014), $\unicode[STIX]{x1D735}d$ is decomposed into an antisymmetric component $\unicode[STIX]{x1D64D}=(\unicode[STIX]{x1D735}d-(\unicode[STIX]{x1D735}d)^{\text{T}})/2$ corresponding to rotation and a symmetric component $\unicode[STIX]{x1D64E}=(\unicode[STIX]{x1D735}d+(\unicode[STIX]{x1D735}d)^{\text{T}})/2$ corresponding to shear deformation. The two eigenvalues of $\unicode[STIX]{x1D64E}$ are of the form $\pm \unicode[STIX]{x1D6FE}$ and indicate the rates of stretching ( $+\unicode[STIX]{x1D6FE}$ ) and compression ( $-\unicode[STIX]{x1D6FE}$ ). The unit eigenvector $\boldsymbol{e}_{\unicode[STIX]{x1D6FE}}$ corresponding to the positive eigenvalue indicates the direction of stretching and the other eigenvalue the direction of compression. Because of the incompressibility condition, both eigenvectors are orthogonal, and so plotting only one of them is sufficient to have the complete set of information. In figure 8, the stretching rate and its direction are presented in an ( $x,z$ ) plane for the three cases studied. The stretching rate is maximal near the upper part, and at small distance above the cilium tips in all three cases. For the synchronized motion (case (a)), there is almost no stretching away from the cilia during the recovery phase, whereas for cases (b) and (c), corresponding respectively to symplectic and antipleptic motion, a weak stretching can always be observed (see the right-hand side of cases (b) and (c) in figure 8). A complex shape of the stretching rate is observed at the cilium tips during the stroke phase in antipleptic motion (see cilium 3 for instance), during the stroke phase in symplectic motion (see cilia 6 and 7) and during the stroke phase in synchronized motion (results not shown). From their orientations, one can expect an enhancement in the mixing in this region. As in Ding et al. (Reference Ding, Nawroth, McFall-Ngai and Kanso2014), the stretching direction is a nonlinear function of space. Except for zones where the shear rate is maximal, the $\unicode[STIX]{x1D6FE}\boldsymbol{e}_{\unicode[STIX]{x1D6FE}}$ field is then oriented at $45^{\circ }$ compared with the $x$ -direction, and almost uniform. This is reminiscent of a linear shear profile $d_{x}=cz$ , where $c$ is a constant. The nonuniform aspect of the stretching orientation indicates the presence of ‘folding’ in the displacement field $\boldsymbol{d}$ , which also plays a role in the mixing, as explained by Kelley & Ouellette (Reference Kelley and Ouellette2011). Ding et al. (Reference Ding, Nawroth, McFall-Ngai and Kanso2014) obtained that the stretching rate was maximal for the antipleptic case for $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=\unicode[STIX]{x03C0}/4$ , which is exactly what is obtained here, even if the beating patterns used are different. One can then expect that this value of the phase lag $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}$ is the most efficient in mixing fluids by enhancing the stretching near the upper part of the cilia.

Figure 8.

Stretching rate and direction for (a) synchronized motion ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=0$ ), (b) symplectic MCW ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=-\unicode[STIX]{x03C0}/4$ ) and (c) antipleptic MCW ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=\unicode[STIX]{x03C0}/4$ ). In each case, the cilium spacing is $a/L=1.67$ and the size of the computational domain is ( $N_{x}=201$ , $N_{y}=26$ , $N_{z}=50$ ).

In order to assess the reliability and robustness of the solver, a comparison with experimental data is also performed. The average clearance velocity was computed on the plane ( $x$ , $y$ , $3.2L$ ) for all simulations. The highest clearance velocity is reached for the case $a/L=1.67$ with $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=\unicode[STIX]{x03C0}/4$ and equals $33.47~\unicode[STIX]{x03BC}\text{m}~\text{s}^{-1}$ . This is in good agreement with the experimental results of Matsui et al. (Reference Matsui, Randell, Peretti, Davis and Boucher1998), who observed a clearance velocity of $39.8\pm 4.2~\unicode[STIX]{x03BC}\text{m}~\text{s}^{-1}$ for mucus. Matsui et al. (Reference Matsui, Randell, Peretti, Davis and Boucher1998) also observed that the PCL flows in the same direction as the mucus, which is the case in the present study. Finally, the PCL–mucus interface remains approximately flat in most of the simulations presented, as is the case in the micrographs performed by Sanderson & Sleigh (Reference Sanderson and Sleigh1981) on rabbit tracheal epithelium.

3.2.2 Directional pushing efficiency

Inspired by the works of Kim & Netz (Reference Kim and Netz2006), Gauger et al. (Reference Gauger, Downton and Stark2009) and Khaderi, Den-Toonder & Onck (Reference Khaderi, Den-Toonder and Onck2011), a positive flux $Q_{p}$ and a negative flux $Q_{n}$ are defined as follows. The $x$ -component of the velocity field is considered over the ( $N_{x},y,z$ ) plane and, at each time step of a full beating cycle, the negative and positive velocity values are separated. The instantaneous negative and positive fluxes are then computed over a sufficiently large number of cycles. The difference $(Q_{p}-Q_{n})$ gives the net instantaneous flux, and the directional pushing efficiency $\unicode[STIX]{x1D716}_{PN}$ is defined as $\unicode[STIX]{x1D716}_{PN}=(Q_{p}-Q_{n})/(Q_{p}+Q_{n})$ . It should be noted that here, in contrast to the previous works of the aforementioned authors, the domain is not restricted to the transport area but covers the whole flow region instead. Such a choice is justified by the fact that it has been experimentally observed, using confocal microscopy and fluorescent markers, that the PCL and mucus are in reality transported at approximately the same rate (Matsui et al. Reference Matsui, Randell, Peretti, Davis and Boucher1998). As a matter of fact, PCL transport seems to depend on the presence of a mucus layer above it, and ciliary mixing is thought to be responsible for the diffusion of momentum from mucus to PCL. Taking into account the transport zone and the mixing zone in the computation of $\unicode[STIX]{x1D716}_{PN}$ then allows one to collect information about how transport and mixing work together during a beating cycle.

The variation of $\unicode[STIX]{x1D716}_{PN}$ over one beating cycle for five different phase lags $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}$ , all for an antipleptic MCW with a cilium spacing of $a/L=2$ , is illustrated in figure 9. Let us recall that, although the quantities considered here are the results of the motion of all cilia, the calculations have been made through the beating cycle of a single cilium of reference. Therefore, the fact that the corresponding curves have a huge decrease around $t=\unicode[STIX]{x03C0}$ is related to the choice of this particular cilium. Indeed, the plane used to compute the positive and negative fluxes is located close to a particular cilium. This cilium is in the recovery phase at $t=\unicode[STIX]{x03C0}$ , inducing a small value of $\unicode[STIX]{x1D716}_{PN}$ at this particular time. The choice of another cilium or another plane would horizontally shift the drop observed around $t=\unicode[STIX]{x03C0}$ .

To interpret the evolution of $\unicode[STIX]{x1D716}_{PN}$ in figure 9, it is necessary to examine the topology of the flow during this beating cycle, shown in figure 10. As cilium 1 begins its recovery phase (case (a) of figure 10), there is no reversal flow at the periodic boundary frontier. Hence, $\unicode[STIX]{x1D716}_{PN}=1$ , as is visible in figure 9 at the point (a). Then, when cilium 1 carries on its recovery phase (case (b) of figure 10), no reversal flow can be observed since cilium 1 is still at the beginning of its recovery phase and cilium 8 is finishing its stroke phase. At $t=\unicode[STIX]{x03C0}/2$ (case (c) of figure 10), both cilia 1 and 8 are in their early recovery phase and the directional pushing efficiency begins to decrease (see points (b) and (c) on figure 9). At $t=3\unicode[STIX]{x03C0}/4$ (case (d) of figure 10), cilium 1 is reaching the maximal speed of its recovery phase. The negative flow generated is important, and the efficiency quickly drops: see point (d) on figure 9. Between $t=3\unicode[STIX]{x03C0}/4$ and $t=\unicode[STIX]{x03C0}$ , the directional pushing efficiency weakly increases. This is due to the positive velocity near the base of cilium 1 which is performing a ‘whip-like’ motion: while its free end is still going to the left, its base starts moving to the right. Case (e) of figure 10 corresponds to the end of the recovery phase of the first cilium, and at the same moment, cilium 8 is at its maximal negative velocity: henceforth the directional pushing efficiency is still decreasing. At $t=5\unicode[STIX]{x03C0}/4$ (case (f) of figure 10), cilium 8 has finished its recovery phase and cilium 1 is in the middle of its stroke phase. It counteracts the reverse flow created earlier, and $\unicode[STIX]{x1D716}_{PN}$ finally increases. At $t=3\unicode[STIX]{x03C0}/2$ and $t=7\unicode[STIX]{x03C0}/4$ (cases (g) and (h) of figure 10), both cilia 1 and 8 are in their stroke phases, and the flow generated is purely in the positive $x$ -direction: see points (g) and (h) on figure 9.

Figure 9.

Directional pushing efficiency $\unicode[STIX]{x1D716}_{PN}$ over one beating cycle for three different phase lags $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}$ . The results are obtained for an antipleptic MCW and $a/L=2$ .

Figure 10.

Snapshots of the fluid velocity taken at eight different instants during the beating cycle of the cilium located at the left of the images. The cilia are beating in an antipleptic motion with a phase lag $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=\unicode[STIX]{x03C0}/4$ and a spacing of $a/L=2$ . In (a),  $t=0$ , and the recovery phase of the first cilium begins; (b–e) recovery phase; (f–h) stroke phase. Once (h) is completed, a new cycle begins: (h)  $\rightarrow$  (a). The figure shows contours of the norm of the normalized fluid velocity $\Vert \boldsymbol{V}_{f}\Vert$ such that $\boldsymbol{V}_{f}=\boldsymbol{V}_{f}^{dim}/V_{f}^{ref}$ , where $V^{ref}=\unicode[STIX]{x1D706}/T$ is the reference speed of the present system.

It is now interesting to compare the average directional pushing efficiency $\langle \unicode[STIX]{x1D716}_{PN}\rangle$ for the different kinds of synchronization over a full beating cycle (see figure 11 a,b). Before going further into detail, let us recall the fact that the current efficiency $\langle \unicode[STIX]{x1D716}_{PN}\rangle$ is a criterion qualifying the non-isotropy of the transport. Thus, it only gives an insight into the capacity of the cilia to transport the flow in a unidirectional way and does not give any quantitative information on the volume of fluid flow that is effectively displaced. Figure 11(a,b) shows an interesting phenomenon: systems with larger cilium spacing have a better capacity to transport the flow in the same direction. Another intriguing fact is that, for the smallest cilium spacing ( $a/L=1.67$ , figure 11), antipleptic MCWs seem to have a better ability to create a unidirectional flow compared with synchronized or symplectic motion, with a peak for the average efficiency around $\unicode[STIX]{x03C0}/2$ . This agrees particularly well with the results of Gauger et al. (Reference Gauger, Downton and Stark2009), who reported that antipleptic MCWs are more efficient than the synchronized motion of cilia, which itself is more efficient that the symplectic case. This also partially agrees with the results of Ding et al. (Reference Ding, Nawroth, McFall-Ngai and Kanso2014), who obtained peaks of efficiency for phase lags around $\pm \unicode[STIX]{x03C0}/2$ , with a stronger one for the antipleptic case. In the present study, the minimal efficiency for metachronal motion is reached for a symplectic MCW with a phase lag of approximately $-\unicode[STIX]{x03C0}/2$ .

When the cilium spacing is increased to $a/L=2$ (figure 11), a peak of efficiency appears for the antipleptic MCW for $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=2\unicode[STIX]{x03C0}/3$ , and the lowest values are reached in the symplectic case for $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}\approx -\unicode[STIX]{x03C0}/2$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}\approx -\unicode[STIX]{x03C0}/3$ . These trends will be found again in the results shown in § 3.2.3.

Then, if the cilium spacing is increased above a limit value ( $a/L=3.33$ , figure 11), the directional pushing efficiency becomes equal to 1 for all cases (antipleptic, symplectic and synchronized). This agrees well with the results obtained by Khaderi et al. (Reference Khaderi, Den-Toonder and Onck2011), who concluded that ‘the amount of flow enhancement depends on the inter-cilia spacing [but] the efficiency is not significantly influenced’. Indeed, as the cilia are set away from each other, the influence of their neighbours becomes negligible. Nevertheless, the authors reported a very low efficiency for synchronized cilia, whereas the present results always show a positive flow in the mucus phase for synchronized motion. As mentioned in § 2.1, this is a direct consequence of the inertial effects ( $Re>1$ ) in the present simulations. It is recalled that they only affect the synchronized case. It has been carefully checked that all other conclusions drawn for antipleptic and symplectic metachrony remain the same for $Re<1$ and $Re=20$ . For a detailed study of the inertial effects at $Re=20$ , see § 3.2.7. The present results are different from the results of Gueron et al. (Reference Gueron, Levit-Gurevich, Liron and Blum1997), as it appears that even when the cilium spacing is higher than 2 cilium lengths, the influence of neighbouring cilia cannot be neglected (see the case $a/L=2.53$ , for example).

It is important to remember that this efficiency does not take into account the actual net flow volume transported.

Figure 11.

Mean directional pushing efficiency $\langle \unicode[STIX]{x1D716}_{PN}\rangle$ over a beating cycle as a function of the phase lag $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}$ for different values of the cilium spacing $a/L$ .

3.2.3 Comparison of hydrodynamic efficiency

After having investigated the capacity to transport the flow in the desired direction, it is now interesting to look at the actual flow volume displaced to see whether a better efficiency to transport the flow directionally corresponds to a better flow transport. To do so, the global volumetric flow rate $Q_{v}$ over a unit volume of size ( $1\times 1\times N_{z}$ ) is introduced,

where $\text{d}x=1$ using the classical LBM normalization, and $U^{\ast }=U^{av}/U^{ref}$ , where $U^{ref}=(\unicode[STIX]{x1D706}/N_{cil})/T$ is the reference velocity and $U^{av}=(1/n_{i}n_{j}n_{k})\sum _{i,j,k}U_{ijk}$ is the average fluid velocity inside the whole domain.

In figure 12, the dimensionless volumetric flow rate $Q_{v}$ is plotted over one beating cycle for arrays of cilia with different phase lags ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=0$ , $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=-\unicode[STIX]{x03C0}/4$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=\unicode[STIX]{x03C0}/4$ ) and different cilium spacings. For cilium spacings equal to $a/L=1.67$ and $a/L=2$ , the antipleptic MCW is the most efficient for transporting fluids. However, it should be noted that as the cilium spacing is increased, the ability of the antipleptic wave to transport fluids exhibits a huge decrease compared with the symplectic wave. For a larger cilium spacing ( $a/L=3.33$ ), the fluid transport is not impacted any more by metachrony. Finally, no flow reversal occurs for the synchronized cases; this is the consequence of working at $Re=20$ .

Figure 12.

Normalized volumetric flow rate $Q_{v}$ as a function of time over a beating cycle for different phase lags $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}$ and different cilium spacings $a/L$ . The recovery phase occurs for $t\in [0,\unicode[STIX]{x03C0}]$ and the stroke phase for $t\in [\unicode[STIX]{x03C0},2\unicode[STIX]{x03C0}]$ .

The total volume of fluid displaced during a beating cycle for the different phase lags is compared in figure 13(a,b). For a small cilium spacing ( $a/L=1.67$ ), the efficiency of the antipleptic metachrony is obvious. This agrees well with the results of Khaderi et al. (Reference Khaderi, Den-Toonder and Onck2011), who observed a larger net flow produced by antipleptic metachrony for this value of cilium spacing. Symplectic waves appear to be less efficient than or at best equally efficient to antipleptic motion, except for $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=-7\unicode[STIX]{x03C0}/8$ for $a/L=1.67$ , where there is a peak in the total displaced volume of flow. On the contrary, negative peaks are found for $a/L=1.67$ and $a/L=2$ for $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=-\unicode[STIX]{x03C0}/3$ . There are two neighbouring maxima at $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=\unicode[STIX]{x03C0}/4$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=\unicode[STIX]{x03C0}/2$ for $a/L=1.67$ and $a/L=2$ respectively, indicating that specific phase lags are more able to generate a strong flow. At this point, it is worth remembering that in the present model, the recovery and stroke phases occur in the same plane, which is not the case in real configurations. According to Downton & Stark (Reference Downton and Stark2009), who studied both 2D and 3D beating patterns, it is expected that in the presence of 3D beating patterns, both the directional pushing efficiency $\unicode[STIX]{x1D716}_{PN}$ and the total displaced volume of flow would increase. It is also expected that a small flow might occur in the $y$ -direction.

Figure 13.

Total dimensionless displaced flow volume generated by an array of cilia over a beating cycle for different phase lags and cilia spacings: $+$ , $a/L=1.67$ ; ○,  $a/L=2$ ; *,  $a/L=2.53$ ; ▫,  $a/L=3.33$ .

3.2.4 Energetic cost of the MCWs

The previous sections have been dedicated to the study of the directional transport and the volume displacements of fluids. An energetic perspective is now introduced, to characterize the potential benefits of metachrony completely.

The average power spent by the cilia during a beating cycle is given by

using the forces illustrated in figure 2. The power spent is averaged over several beating cycles. To have a dimensionless power $P^{\ast }$ , the power $P^{\infty }$ spent by an isolated cilium ( $a/L=10$ ) is computed such that $P^{\ast }=P_{cil}/P^{\infty }$ .

Before going any further, it is important to remember here that the only parameter that differs between each value of the phase lag $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}$ for a given cilium spacing $a/L$ is the size of the domain $N_{x}$ over the $x$ -direction, and hence the number of cilia acting in one wavelength. Moreover, between different cilium spacings $a/L$ , the space $a$ between two cilia is modified in both the $x$ - and $y$ -directions, and the sizes $N_{x}$ and $N_{y}$ of the domain must be changed accordingly. As a consequence, if the cilium spacing is increased, the density of cilia in both the $x$ - and $y$ -directions is decreased. In all of the simulations presented in § 3.2, all of the other parameters (ratio of viscosity $r_{\unicode[STIX]{x1D708}}$ , beating frequency $f$ , length of the cilia $L$ , etc.) are fixed.

Figure 14(a,b) shows the average dimensionless power as a function of $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}$ . For small values of the phase lag ( $|\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}|\leqslant \unicode[STIX]{x03C0}/2$ ), the average power spent decreases to a minimal value for $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}\approx \unicode[STIX]{x03C0}/4$ . For this particular value, the system spends less power than the synchronized case: the antipleptic MCW allows the system composed of all cilia to encounter a smaller resistance from the surrounding fluids.

Nevertheless, it is obvious, from a fluid mechanics point of view, that when cilia beat together in a synchronized way, the viscous resistance felt by each cilium is reduced.

In the case of metachrony, the behaviour is dual. Indeed, cilia in the stroke phase of the antipleptic motion encounter a stronger viscous resistance from the flow due to their added respective remoteness. Therefore, compared with the synchronized motion, they will transfer more energy to the flow, and the energy transferred will be entirely used to propel the fluids in the desired direction. The opposite is true during the recovery stroke of the antipleptic motion: the clustered behaviour of the cilia allows them to experience a lower viscous resistance from the flow, and therefore to limit the amount of power transferred during this phase. However, globally, a system with an antipleptic MCW and $|\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}|<\unicode[STIX]{x03C0}/2$ encounters less viscous resistance than the synchronized motion of cilia, as can be seen from figure 14(a,b). This results in a better efficiency of the cilia with antipleptic metachronal motion in transferring their momentum to the flow, while in the meantime requiring less power. For the symplectic motion, a similar phenomenon occurss: cilia in their stroke phase are clustered and therefore unable to fully exert their pushing action, while cilia in their recovery phase are away from each other and generate a stronger reversal flow compared with the antipleptic motion. This results in a lower capacity of the symplectic MCW to transport mucus, while still allowing the system to spend less power than synchronously beating cilia for $|\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}|<\unicode[STIX]{x03C0}/2$ . An energetic ratio will be introduced in § 3.2.5 to quantify, for this value of $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}$ , the capacity of cilia with antipleptic or symplectic metachrony to transfer their momentum to the flow.

Figure 14.

Power spent by the system for different phase lags $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}$ and different cilium spacings: $+$ ,  $a/L=1.67$ ; ○,  $a/L=2$ ; ▫,  $a/L=3.33$ .

This is not true for large phase lags ( $|\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}|\geqslant \unicode[STIX]{x03C0}/2$ ): a system with antipleptic or symplectic metachrony spends more power than the synchronous case. One can suppose that this is a direct consequence of the reduced number of cilia in a wavelength. Then, to explain the better efficiency of the antipleptic MCW over the symplectic one for large phase lags, an investigation of the flow topology is necessary. Figure 15 shows an antipleptic MCW (on the left) and a symplectic MCW (on the right) at the same time for a phase lag of $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=\pm 2\unicode[STIX]{x03C0}/3$ . For this value of $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}$ , the antipleptic wave is more efficient in transporting the mucus (see figure 13 a,b), although the average amounts of power spent by both systems are relatively similar (see figure 14 a,b). One can easily see the main difference between the two cases. In the symplectic case, the cilium in stroke phase encounters the reversal flow generated by the other cilium in recovery phase. When these two secondary flows meet, vortices are generated and the global transport of fluid is less efficient, as a fraction of the energy transferred to the flow is used to cancel this reversal flow. On the contrary, in the antipleptic case, the cilium in stroke phase does not immediately feel the influence of the reversal flow created by the other cilium in recovery phase, which is behind it. Hence, this cilium is able to fully exert its pushing effect on the mucus phase. Moreover, for the antipleptic case, there is a suction effect due to the combined motion of the cilia in the stroke and recovery phases, maximizing the propulsion of mucus. One can then expect a weak blowing effect between the cilia in the stroke and recovery phases of symplectic motion.

Figure 15.

Comparison of the flow generated by an antipleptic and a symplectic wave for the cilium spacing $a/L=1.67$ . (a) Antipleptic MCW with $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=2\unicode[STIX]{x03C0}/3$ . (b) Symplectic MCW with $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=-2\unicode[STIX]{x03C0}/3$ . The plane is coloured with the magnitude of the dimensionless fluid velocity.

When the cilium spacing is large (see the case $a/L=3.33$ of figure 14 a,b, for example), the power spent by the system is the same for all phase lags: the cilia are too far from each other to be impacted by the phase difference of their immediate neighbours. One can observe that for $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=0$ (i.e. for synchronized beating), the average power spent by a cilium, $P^{\ast }$ , is equal to 1 in figure 14(a) for the cilium spacings $a/L=1.67$ and $a/L=3.33$ , meaning that synchronous systems spend the same amount of power as an isolated cilium. This shows the beneficial cost of antipleptic metachrony for small phase lags.

3.2.5 Displacement ratio

Inspired by the work of Kim & Netz (Reference Kim and Netz2006), a displacement ratio, which can be seen as the transport efficiency of the waves, is introduced to quantify the capacity of a given system to transport particles, with respect to a given amount of power. In this context, $\unicode[STIX]{x1D702}_{1}$ is defined by the mean displacement in the $x$ -direction during one beating cycle, divided by the mean power that a cilium had to spend during this beating cycle. Since the main purpose of mucociliary clearance is to transport mucus, and since experimental data (Winters & Yeates Reference Winters and Yeates1997) report that the total thickness in the vertical direction of the mucus layer is in the range $[1.4L;10L]$ , values for the displacement were taken on an arbitrary plane $z/L=3.2$ near the extremity of the domain. To obtain a value for the displacement, the instantaneous average fluid velocity over the $x$ -direction is computed, and the resulting value is then multiplied by the period of a full beating cycle, giving the mean displacement $\langle d_{x}\rangle$ over one period on the ( $x,y,3.2L$ ) plane.

By dividing this mean displacement by appropriate quantities, a dimensionless expression for the displacement ratio is obtained,

In the synchronized case, i.e. $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=0$ , $\unicode[STIX]{x1D706}$ is infinite and thus the size of the domain over the $x$ -direction was used and divided by the number of cilia.

In figure 16(a,b), one can see, from an energetic point of view, the superiority of the antipleptic wave in transporting mucus for small cilium spacings, which confirms the previous findings. Moreover, for the smallest cilium spacing ( $a/L=1.67$ ), a clear peak of efficiency can be seen for an antipleptic MCW with $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=\unicode[STIX]{x03C0}/4$ . If the cilium spacing is increased ( $a/L=2$ ), similar results are found. Nevertheless, for $a/L=2.53$ , a different behaviour occurs: the displacement ratio is found to be worst for $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=\pm \unicode[STIX]{x03C0}/2$ . Then, for cilia far away from each other, the displacement ratio $\unicode[STIX]{x1D702}_{1}$ remains constant for all phase lags (the cilia do not feel the influence of the others any more).

Figure 16.

Displacement ratio $\unicode[STIX]{x1D702}_{1}$ as a function of the phase lag $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}$ for different cilium spacings $a/L$ .

Similar results regarding the efficiency of the antipleptic MCW are found in Osterman & Vilfan (Reference Osterman and Vilfan2011), where the optimal beating pattern is investigated. In their study, they observed that antipleptic MCWs were often the most efficient. They also found, as is the case in the present study, that an increase in the density of cilia results in an increase of the efficiency up to a critical point where clustering becomes counterproductive.

3.2.6 Mixing

To investigate how the mixing can be enhanced by metachrony, the average stretching rate over the transport and mixing areas during a beating period is computed. Figure 17 shows the results obtained for all cilium spacings. Clearly, the antipleptic MCW is the most efficient for stretching fluids, and hence for mixing fluids. Clear peaks are visible for antipleptic waves with $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=\unicode[STIX]{x03C0}/4$ and $a/L=1.67$ , with $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}\approx \unicode[STIX]{x03C0}/3$ and $a/L=2$ , and with $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}\approx \unicode[STIX]{x03C0}/3$ and $a/L=2.53$ . On the contrary, symplectic MCWs are almost always less efficient for mixing than antipleptic MCWs, except for $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=-7\unicode[STIX]{x03C0}/8$ and $a/L=1.67$ , where there is an enhancement in the mixing. This peak is also present for the opposite phase lag $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=7\unicode[STIX]{x03C0}/8$ , where it reaches approximately the same value. This is perfectly coherent with all previous results, and partially with the ones of Ding et al. (Reference Ding, Nawroth, McFall-Ngai and Kanso2014), who observed, for a unique layer of fluid with a uniform viscosity and a cilium spacing of $a/L=1.67$ , the existence of two peaks in the shear rate: a weak one for symplectic waves for $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=-\unicode[STIX]{x03C0}/2$ and a stronger one for antipleptic waves for $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=\unicode[STIX]{x03C0}/2$ . The main difference here is that, in the present study, the symplectic MCWs with $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}\approx -\unicode[STIX]{x03C0}/2$ seem to have the worst capacity for mixing, while the antipleptic MCWs reach their full mixing capacity for $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=\unicode[STIX]{x03C0}/4$ . The combined study of figures 14 and 17 gives a good insight into the mixing efficiency of the system.

Figure 17.

(a) Average stretching rate in the transport and mixing areas as a function of the phase lag $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}$ for different cilium spacings: $+$ , $a/L=1.67$ ; ○, $a/L=2$ ; *, $a/L=2.53$ ; ▫, $a/L=3.33$ . (b) Three-dimensional view of the corresponding plot.

3.2.7 Quantification of inertial effects

In this section, the influence of the Reynolds number is considered. The present objective is to compare the qualitative behaviour of the MCWs between $Re<1$ and $Re>1$ . To reach Reynolds numbers of the order of $10^{-2}$ , the size of the cilia has been reduced to half the size of real cilia. The geometry is then divided by a factor of 2 in the three spatial directions. The main consequence of this choice is that the computed quantities (fluxes, displaced volume of fluid, etc.) displayed here are approximately eight times smaller compared with the results previously shown (as in figure 13). However, the qualitative behaviour of the MCWs remains the same, as well as the drag exerted by the cilia. Thus, a comparison of the MCW behaviour between $Re<1$ and $Re>1$ is possible. In Figure 18, one can see the total displaced volume of fluid for $Re=0.02$ , $Re=1$ and $Re=20$ , and a cilium spacing of $a/L=1.67$ . A transition between $Re<1$ and $Re>1$ for the synchronous cases is expected, and can be seen in figure 18. As mentioned in § 3.2.3, for $Re\leqslant 1$ , the synchronized cases are less efficient than the symplectic cases. Except for $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=0$ , no other notable quantitative differences are present between the cases $Re<1$ and $Re=20$ . The general behaviour of both the antipleptic and symplectic MCWs remains similar for all the other phase lags $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}$ . In particular, a peak around $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=\unicode[STIX]{x03C0}/4$ is present for all of the Reynolds numbers tested ( $Re=0.047$ , 2, 5, 10, not shown). The similarity of the metachronal cases (i.e. $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}\neq 0$ ) for this range of Reynolds numbers is certainly due to the fact that there are always cilia in the stroke phase. Hence, transport is observed during the whole beat cycle, and the inertial effects thus remain minor. This is no longer the case for fully synchronously beating cilia: despite being weak, inertial effects cancel the reversal of the flow.

Figure 18.

Total displaced volume of fluid for $Re=0.02$ , $Re=1$ and $Re=20$ as a function of the phase lag $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}$ . The results of these simulations are obtained with $L=7$  lu and $a/L=1.67$ .