1. Introduction
Mucociliary clearance driven by ciliary beating in the human airways has received much attention due to its critical role in the capture and clearance of foreign pollutants and pathogens (Wanner, Salathé & O'Riordan Reference Wanner, Salathé and O'Riordan1996; Grotberg Reference Grotberg2021; Sedaghat, Behnia & Abouali Reference Sedaghat, Behnia and Abouali2023). Human airways are protected by two fluid layers, a periciliary layer (PCL) covering the epithelial surface and a mucus layer on top of the PCL (Chilvers & O'Callaghan Reference Chilvers and O'Callaghan2000; Knowles & Boucher Reference Knowles and Boucher2002; Choudhury et al. Reference Choudhury, Filoche, Ribe, Grenier and Dietze2023). The mucus is often described as a yield stress and shear thinning fluid (Banerjee, Bellare & Puniyani Reference Banerjee, Bellare and Puniyani2001; Nordgard & Draget Reference Nordgard and Draget2011; Chatelin et al. Reference Chatelin, Anne-Archard, Murris-Espin, Thiriet and Poncet2017). Cilia are almost immersed in the PCL, and interact with the mucus through their tips. The force generated by the cilia propels the mucus flow, and the mucus in turn affects the orientation of the ciliary beating (Gsell et al. Reference Gsell, Loiseau, D'Ortona, Viallat and Favier2020; Loiseau et al. Reference Loiseau, Gsell, Nommick, Jomard, Gras, Chanez, D'Ortona, Kodjabachian, Favier and Viallat2020; Pellicciotta et al. Reference Pellicciotta, Hamilton, Kotar, Faucourt, Delgehyr, Spassky and Cicuta2020). This hydrodynamic coupling between mucus and millions of microscopic cilia has a major impact on ciliary coordination (self-organization) and mucus transport. Understanding the hydrodynamic mechanism of cilia–mucus interaction is desirable for the study of various respiratory diseases caused by the impairment of mucus transport.
Considerable research effort has been devoted to the ciliary coordination and cilia-induced flow. For the former, the proposal of a cilia model can be traced back to an analytical study by Barton & Raynor (Reference Barton and Raynor1967), where the cilium was simplified as an oscillating cylinder mounted on a plate. The relationship between the flow rate and the geometric parameters of the cilium was determined. Since then, many other studies involving modelling flexible cilia have been reported. Two asymmetric beating phases of a cilium were identified, i.e. an effective stroke characterized by almost straight cilium to better drive the mucus, and a recovery stroke characterized by large deformed cilium to reduce the retarding effect on the mucus (Blake Reference Blake1972; Xu & Jiang Reference Xu and Jiang2019). Cilia in an array can coordinate with each other to generate metachronal waves instead of repeating two beating phases synchronously (Hussong, Breugem & Westerweel Reference Hussong, Breugem and Westerweel2011; Elgeti & Gompper Reference Elgeti and Gompper2013; Meng et al. Reference Meng, Bennett, Uchida and Golestanian2021; Mesdjian et al. Reference Mesdjian, Wang, Gsell, D'Ortona, Favier, Viallat and Loiseau2022; Wang, Tang & Zhang Reference Wang, Tang and Zhang2022). This usually depends on the phase difference between adjacent cilia (Chateau et al. Reference Chateau, Favier, D'Ortona and Poncet2017; Hall & Clarke Reference Hall and Clarke2020), and is related to ciliary flexibility (Kim & Netz Reference Kim and Netz2006) and ciliary density (Chateau et al. Reference Chateau, D'Ortona, Poncet and Favier2018). Various metachronal waves have been observed, such as the antiplectic and symplectic waves. These two waves move in the opposite and same directions as the flow, respectively. Cilia beating with an antiplectic wave were found to be more efficient in transporting and mixing fluid than cilia beating synchronously or with a symplectic wave. On the other hand, various experimental and numerical studies have investigated the cilia-induced flow, usually focusing on local flow characteristics and flow rate (Brumley et al. Reference Brumley, Wan, Polin and Goldstein2014; Wei et al. Reference Wei, Dehnavi, Aubin-Tam and Tam2019, Reference Wei, Dehnavi, Aubin-Tam and Tam2021; Boselli et al. Reference Boselli, Jullien, Lauga and Goldstein2021; Hu & Meng Reference Hu and Meng2023). Among them, Brumley et al. (Reference Brumley, Wan, Polin and Goldstein2014) measured the flow around a single cilium and a pair of cilia, and calculated the instantaneous forces generated by the cilia using a Stokeslet model. They highlighted the importance of hydrodynamic coupling; a synchronized beating of two cilia can be realized even when only hydrodynamic interactions exist. Furthermore, Ding et al. (Reference Ding, Nawroth, McFall-Ngai and Kanso2014) observed a transport region and a mixing (shear) region above and below the ciliary tips, respectively. The asymmetric stroke of the cilia and the no-slip epithelial surface resulted in a shear-like flow field. The enhancement of fluid transport and mixing was mainly attributed to the increase in the shear rate. Fluid transport was also found to be continuous even though the epithelial surface was not completely covered by the cilia (Juan et al. Reference Juan, Mathijssen, He, Jan, Marshall and Prakash2020).
A long-range ciliary coordination distinct from metachronal waves has been discovered (Matsui et al. Reference Matsui, Grubb, Tarran, Randell, Gatzy, Davis and Boucher1998; Tarran et al. Reference Tarran2005; Shapiro et al. Reference Shapiro, Fernandez, Garren, Guasto, Debaillon-Vesque, Kramarsky-Winter, Vardi and Stocker2014; Khelloufi et al. Reference Khelloufi, Loiseau, Jaeger, Molinari, Chanez, Gras and Viallat2018), characterized by large-scale mucus swirls accompanied by cilia beating in a circular pattern. Several experimental studies have shown the existence of hydrodynamic coupling between the ciliary coordination and the circular mucus flow (Mitchell et al. Reference Mitchell, Jacobs, Li, Chien and Kintner2007; Guirao et al. Reference Guirao2010; Faubel et al. Reference Faubel, Westendorf, Bodenschatz and Eichele2016). Recently, Loiseau et al. (Reference Loiseau, Gsell, Nommick, Jomard, Gras, Chanez, D'Ortona, Kodjabachian, Favier and Viallat2020) and Gsell et al. (Reference Gsell, Loiseau, D'Ortona, Viallat and Favier2020) experimentally investigated the hydrodynamic coupling of the cilia–mucus system in detail, and proposed a two-dimensional model to predict the ciliary coordination and the Newtonian mucus flow. They demonstrated that the hydrodynamic coupling of cilia and mucus dominates the long-range coordination. The formation of mucus swirls was closely related to the density and the interaction length of the cilia. As mentioned above, mucus is a non-Newtonian fluid with yield stress and shear thinning properties. Some researchers have investigated the effect of non-Newtonian properties on mucus transport (Chatelin & Poncet Reference Chatelin and Poncet2016; Sedaghat, George & Abouali Reference Sedaghat, George and Abouali2021; Sedaghat et al. Reference Sedaghat, Farnoud, Schmid and Abouali2022; Modaresi Reference Modaresi2023; Wang et al. Reference Wang, Gsell, D'Ortona and Favier2023). The present study aims to focus first on the yield stress and shear thinning properties, although the mucus rheology also exhibits other properties, e.g. viscoelasticity (Vasquez et al. Reference Vasquez, Jin, Palmer, Hill and Forest2016; Guo & Kanso Reference Guo and Kanso2017; Choudhury et al. Reference Choudhury, Filoche, Ribe, Grenier and Dietze2023). Most of the studies employed numerical methods because the non-Newtonian properties can be well controlled. However, few studies have been reported on the effect of non-Newtonian properties on long-range ciliary coordination, which warrants a more detailed investigation.
The objective of the present study is to explore numerically the hydrodynamic coupling of cilia and mucus with yield stress and shear thinning properties. The mucus flow is solved by the lattice Boltzmann method. The cilia–mucus interaction is handled by an alignment rule, and the non-Newtonian fluid is modelled by the Herschel–Bulkley model. The effects of ciliary density (
$\phi$), interaction length (
$\lambda$), Bingham number (
$Bn$, quantifying yield stress effect) and flow index (
$n$, quantifying shear thinning effect) on the formation of the mucus flow (or ciliary beating orientation) regime are examined. Three different mucus flow regimes are observed: a poorly organized (PO) regime, a swirly (S) regime, and a fully unidirectional (FU) regime (corresponding to the poorly aligned, swirly and fully aligned regimes in Gsell et al. Reference Gsell, Loiseau, D'Ortona, Viallat and Favier2020). Two physical parameters, i.e. polarization (
$P$) and integral length (
$\varLambda$), are used to identify the three regimes. The mechanism of regime formation caused by the yield stress and shear thinning effects is characterized by the evolution of the flow velocity, viscosity and shear-rate fields. In addition, a rescaling of 
$\lambda$ is proposed for different 
$Bn$ and 
$n$.
2. Computational model
Mucus flow is almost parallel to the epithelium and uniform along the direction perpendicular to the epithelium (Gsell et al. Reference Gsell, Loiseau, D'Ortona, Viallat and Favier2020). In addition, we concentrate on long-range fluid flow parallel to the epithelium, whose length scale is much larger than the typical fluid layer thickness. Therefore, the mucus flow is considered to be two-dimensional, and a two-dimensional hydrodynamic model is sufficient to describe the mucus motion. The flow is predicted based on a lattice Boltzmann solver, and the interaction between the ciliary beating and the mucus motion is handled by an alignment rule. A visualization of the computational domain for 
$\phi =0.3$ is shown in figure 1(a). The square domain is approximately 160
$D$ in both length and height. The cells are discretized using hexagonal elements, where the ciliated elements (black) are randomly placed during initialization. A hexagonal element represents a patch containing several ciliated cells with a common direction of ciliary beating. Approximately 
$10^4$ elements are contained in the domain. Here, 
$D$ represents the side length of the hexagonal elements, and 
$\phi =A_c/A$ represents the ciliary density, where 
$A_c$ is the ciliated area, and 
$A$ is the total area. A closer visualization of the hexagonal elements and the underlying lattice nodes is shown in figure 1(b). The computational domain is discretized on a uniform Cartesian grid. The black and grey dots represent ciliated and non-ciliated nodes, respectively. Note that cilia are simplified as ciliated nodes. Ciliary beating is modelled by a force that is constant in magnitude, whose orientation can change over time (Gsell et al. Reference Gsell, Loiseau, D'Ortona, Viallat and Favier2020). The initial condition is 
$\boldsymbol {u}=0$, with random orientation of ciliary forces and random placement of ciliated elements. Periodic boundary conditions are specified at the domain boundaries.
(a) Visualization of the computational domain (ciliary density 
$\phi =0.3$). Cells are discretized using hexagonal elements, with ciliated elements (black) randomly placed during initialization. (b) Closer visualization of the hexagonal elements and the underlying lattice nodes. The black and grey dots represent ciliated and non-ciliated nodes, respectively.
In the lattice Boltzmann method (Krúger et al. Reference Krúger, Kusumaatmaja, Kuzmin, Shardt, Silva and Viggen2017; Ma et al. Reference Ma, Wang, Young, Lai, Sui and Tian2020; Lu et al. Reference Lu, Lei, Dai, Yang and Shu2022), the particle distribution function 
$f(\boldsymbol {x},\boldsymbol {\xi },t)$ is used to describe the mucus motion, representing the density of fluid particles moving with velocity 
$\boldsymbol {\xi }$ at location 
$\boldsymbol {x}$ and time 
$t$. The dynamics of 
$f(\boldsymbol {x},\boldsymbol {\xi },t)$ is governed by the Boltzmann equation:
\begin{equation} \frac{\partial f}{\partial t} + \boldsymbol{\xi} \boldsymbol{\cdot} \boldsymbol{\nabla} f= \varGamma (f), \end{equation}
where 
$\varGamma$ is the collision operator. Equation (2.1) is equivalent to the Navier–Stokes equations at the macroscopic level (Krúger et al. Reference Krúger, Kusumaatmaja, Kuzmin, Shardt, Silva and Viggen2017). The lattice Boltzmann equation is obtained by discretizing (2.1) in velocity space, physical space and time. A set of velocity vectors 
$\{\boldsymbol e_l, l=0,\ldots, Q-1\}$ is used to discretize the velocity space, where 
$Q$ is the number of discrete velocities. The 
$D2Q9$ scheme is employed as the discretization model to discretize the velocity space by nine velocities:
\begin{equation}
\boldsymbol{e}_l=\left\{\begin{array}{@{}ll@{}} (0,0), & l = 0, \\
c\left(
\cos{\left(\dfrac{{\rm \pi}(l-1)}{2}\right)},\sin{\left(\dfrac{{\rm \pi}(l-1)}{2}\right)}\right),
& l\in[1,4], \\
\sqrt{2}c\left(
\cos{\left(\dfrac{{\rm \pi}(2l-9)}{4}\right)},\sin{\left(\dfrac{{\rm \pi}(2l-9)}{4}\right)}\right),
& l\in[5,8], \end{array}\right.
\end{equation}
where 
$c=\Delta x/\Delta t=\Delta y / \Delta t$ is the lattice velocity. As mentioned before, the computational domain is discretized on a uniform Cartesian grid, i.e. 
$\Delta h=\Delta x=\Delta y$ and 
$\Delta h=\Delta t=D/5=1$. The lattice Boltzmann equation is written as follows, normalizing all the quantities by 
$c$ and 
$\Delta t$, and introducing an external body force:
\begin{equation} f_l(\boldsymbol{x}+\boldsymbol{e}_l,t + 1) -f_l(\boldsymbol{x},t)= \varGamma_{l}(\boldsymbol{x},t) + S^*_l(\boldsymbol{x},t), \end{equation}
where 
$S^*_l(\boldsymbol {x},t)$ is the external body force term. The left- and right-hand sides of (2.3) are the streaming and collision steps, respectively. These two steps can be treated separately due to the explicit (2.3). A two-relaxation-time collision operator is used in the present study:
\begin{equation} \varGamma_l ={-}\frac{1}{\tau^+}\,(f_l^+{-}f_l^{eq+})-\frac{1}{\tau^-}\,(f_l^-{-}f_l^{eq-}), \end{equation}
where 
$\tau ^+$ and 
$\tau ^-$ are the symmetric and antisymmetric relaxation times, and 
$f_l^+$ and 
$f_l^-$ are the symmetric and antisymmetric parts of 
$f_l$. The kinematic fluid viscosity 
$\nu =c_s^2(\tau ^+-\frac {1}{2})$ is determined by 
$\tau ^+$, where 
$c_s=1/\sqrt {3}$ is the lattice sound speed, and 
$\tau ^-$ is determined by the parameter 
$\varLambda _\tau =(\tau ^+-0.5)(\tau ^- -0.5)$, where 
$\varLambda _\tau$ is kept constant to ensure the viscosity independence (Gsell, D'Ortona & Favier Reference Gsell, D'Ortona and Favier2021), set to 
$1/4$ according to the previous study (Ginzburg, d'Humières & Kuzmin Reference Ginzburg, d'Humières and Kuzmin2010). Also, 
$f_l^{eq}$ is the equilibrium particle distribution function, expressed as
\begin{equation} f_l^{eq}(\boldsymbol{x},t) = w_l\rho \left[1+\frac{\boldsymbol{e}_l \boldsymbol{\cdot} \boldsymbol{u}}{c_s^2}+\frac{(\boldsymbol{e}_l \boldsymbol{\cdot} \boldsymbol{u})^2}{2c_s^4}-\frac{\boldsymbol{u}^2}{2c_s^2}\right], \end{equation}
where 
$w_l$ are the lattice weights, and 
$\rho$ is the fluid density. In the present 
$D2Q9$ scheme, 
$w_0=4/9$, 
$w_l=1/9$ for 
$l=1,\ldots,4$, and 
$w_l=1/36$ for 
$l=5,\ldots,8$ (Qian, d'Humières & Lallemand Reference Qian, d'Humières and Lallemand1992). The symmetric and antisymmetric parts of 
$f_l$ and 
$f_l^{eq}$ are expressed as
\begin{equation}
\left\{\begin{array}{@{}ll@{}}
f_l^+{=}\dfrac{f_l+f_{\bar{l}}}{2}, &
f_l^-{=}\dfrac{f_l-f_{\bar{l}}}{2},\\
f_l^{eq+}=\dfrac{f_l^{eq}+f_{\bar{l}}^{eq}}{2}, &
f_l^{eq-}=\dfrac{f_l^{eq}-f_{\bar{l}}^{eq}}{2},
\end{array}\right.
\end{equation}
where the index 
$\bar l$ is defined such that 
$\boldsymbol {c}_{\bar {\boldsymbol {l}}}=-\boldsymbol {c}_{\boldsymbol {l}}$. The external body force term 
$S^*_l$ is expressed as
\begin{equation} S^*_l=\left(1-\frac{1}{2\tau^+}\right) S^+_l + \left(1-\frac{1}{2\tau^-}\right) S^-_l, \end{equation}
where 
$S^+_l=(S_l+S_{\bar {l}})/2$ and 
$S^-_l=(S_l-S_{\bar {l}})/2$ are the symmetric and antisymmetric parts of 
$S_l$, which is given as
\begin{equation} S_l=w_l\left[ \frac{\boldsymbol{e}_l-\boldsymbol{u}}{c_s^2}+ \frac{(\boldsymbol{e}_l \boldsymbol{\cdot} \boldsymbol{u}) \boldsymbol{e}_l}{c_s^4}\right] \boldsymbol{F}. \end{equation} The macroscopic quantities 
$\rho$ and 
$\boldsymbol {u}$ are moments of the particle functions in the velocity space (Krúger et al. Reference Krúger, Kusumaatmaja, Kuzmin, Shardt, Silva and Viggen2017), where 
$\rho$ is expressed as
\begin{equation} \rho=\sum_{l=0}^8 f_l. \end{equation}The flow momentum corrected by the external forcing is
\begin{equation} \rho\boldsymbol{u}=\sum_{l=0}^8 f_l \boldsymbol{e}_l + \dfrac{1}{2}\,\boldsymbol{F}, \end{equation}
where the forcing 
$\boldsymbol {F}$ is the sum of the force 
$\boldsymbol {F}_c$ exerted by the cilia and the frictional force 
$\boldsymbol {F}_\nu$ generated by the PCL. We impose 
$\boldsymbol {F}_c$ on the ciliated nodes only. In the present study, we are interested in the long-term dynamics of the flow, i.e. in time scales that are much larger than the ciliary beating period. Therefore, time-dependent beating is simplified to a point force. Its magnitude is the same for all ciliated nodes and is constant over time, while its orientation is determined by the alignment rule during the simulation. In addition, 
$\boldsymbol {F}_c$ is assumed to be independent of mucus properties, e.g. viscosity of the mucus. The orientation of 
$\boldsymbol {F}_c$ is the same for the ciliated nodes in the same hexagonal element. In the present study, the magnitude of 
$\boldsymbol {F}_c$ is set to be the same as that of 
$\boldsymbol {F}_\nu$ during the initialization. The frictional force is proportional to the fluid velocity (
$\boldsymbol {F}_\nu =-\kappa \boldsymbol {u}$), and the PCL is assumed to be a Newtonian fluid, where 
$\kappa$ is the PCL friction coefficient. The frictional force is treated implicitly, and (2.10) becomes (Gsell et al. Reference Gsell, Loiseau, D'Ortona, Viallat and Favier2020)
\begin{equation} \rho\boldsymbol{u}=\dfrac{\displaystyle \sum_{l=0}^8f_l\boldsymbol{e}_l+\frac{1}{2}\,\boldsymbol{F}_c}{1+\kappa/2\rho}. \end{equation} Recall that the orientation 
$\theta _c^j$ of 
$\boldsymbol {F}_c$ on the 
$j$th ciliated cell is determined by the alignment rule, which was inspired by certain experimental observations. The daily variations of the flow pattern of cilia-driven cerebrospinal fluid have been observed in in vivo mouse brain ventricles (Faubel et al. Reference Faubel, Westendorf, Bodenschatz and Eichele2016). In addition, Guirao et al. (Reference Guirao2010) showed that ciliary-beat orientations on cell cultures issued from the subventrical zone of newborn mice could be changed drastically by applying an external flow. The directional collective order of ciliary beats on the multiciliated skin cells of the Xenopus embryo could also be refined by applying an external flow to skin explants (Mitchell et al. Reference Mitchell, Jacobs, Li, Chien and Kintner2007). Recently, it has been found that the directions of ciliary beating in human airways tend to align progressively along mucus streamlines (Gsell et al. Reference Gsell, Loiseau, D'Ortona, Viallat and Favier2020; Loiseau et al. Reference Loiseau, Gsell, Nommick, Jomard, Gras, Chanez, D'Ortona, Kodjabachian, Favier and Viallat2020). A maximum angle reorientation of 35
$^{\circ }$–40
$^{\circ }$ was shown. The direction of ciliary beating is stable over time and does not show reorientation when mucus was washed out. The reorientation of the cilia is not related to the motility of the cells because the tissue is jammed and the turnover of epithelial cells is very slow. These phenomena strongly suggest the existence of a coupling between hydrodynamics and long-range ciliary-beat orientation, inspiring the present alignment rule. Here, 
$\Delta \theta =\theta _f^j-\theta _c^j$ represents the angle difference between the local flow (
$\theta _f^j$) and the ciliary beating (
$\theta _c^j$). The flow velocity is averaged over the 
$j$th ciliated cell. The alignment rule is expressed as
\begin{equation} \theta_c^j(t+\Delta
t)=\left\{\begin{array}{@{}ll@{}}
\theta_c^j(t)+\varOmega\,\dfrac{\Delta\theta^j(t)}{|\Delta\theta^j(t)|}\,\Delta
t, & \Delta\theta^j(t)>\theta_0,\\
\theta_c^j(t), &
\Delta\theta^j(t) \leq \theta_0, \end{array}\right.
\end{equation}
where 
$\varOmega$ is a fixed angular velocity used to drive the reorientation of the ciliary beating, which does not affect the final steady solution (Gsell et al. Reference Gsell, Loiseau, D'Ortona, Viallat and Favier2020). Its value is 
$\varOmega =U_0/D$, where 
$U_0=0.01$ is the reference velocity (in lattice units). Here, 
$\theta _0$ is the angle threshold set to allow the steady solutions, i.e. 
$\theta _c^j(t+\Delta t)=\theta _c^j(t)$ when 
$\Delta \theta ^j(t) \leq \theta _0$. The value of 
$\theta _0$ is very small (
$\theta _0=2\varOmega \,\Delta t=0.004$) to ensure a negligible influence on the final solutions. In summary, a two-way hydrodynamic coupling between the ciliary-beat orientation and the mucus motion is realized by the external forcing scheme and the alignment rule.
The Herschel–Bulkley model is employed to simulate non-Newtonian flows. The dynamic fluid viscosity 
$\mu$ is shear-dependent and is expressed as
\begin{equation} \mu = \dfrac{\sigma_0}{\dot \gamma} + K \dot \gamma^{n-1}, \end{equation}
where 
$\sigma _0$ is the yield stress, 
$K$ is the flow consistency, and 
$n$ is the flow index. When 
$n<1$, the viscosity decreases with increasing shear rate (shear thinning behaviour). When 
$n>1$, the viscosity increases with increasing shear rate (shear thickening behaviour). In the present study, only the shear thinning behaviour is investigated. Here, 
$\dot \gamma$ is the local shear-rate magnitude, which is expressed as
\begin{equation} \dot\gamma=\sqrt{2(S_{11}^2+2S_{12}^2+S_{22}^2)}, \end{equation}
where 
$S_{\alpha \beta }$ is the local shear-rate tensor, expressed as
\begin{equation} S_{\alpha \beta}={-}\dfrac{1}{2\rho c_s^2\tau^+} \left(\sum_{l=0}^8 (f_l-f_l^{eq})e_{l\alpha}e_{l\beta}+\dfrac{1}{2}\,(u_\alpha F_\beta+u_\beta F_\alpha)\right), \end{equation}
where 
$\tau ^+$ is time-dependent in non-Newtonian simulations. According to 
$\nu =c_s^2(\tau ^+ - 1/2)$ and (2.13), 
$\tau ^+$ is updated by
\begin{equation} \tau^+{=} \dfrac{\sigma_0\dot\gamma^{{-}1}+K\dot\gamma^{n-1}}{\rho c_s^2}+\dfrac{1}{2}. \end{equation} To avoid excessive viscosities during simulations, and to improve the numerical stability, the Herschel–Bulkley law is truncated. The maximum value of relaxation time 
$\tau _{max}^+$ is 50, and the viscosity ratio 
$\mu _{max}/\mu _{min}$ is 1000. A minimum 
$\dot \gamma$ is set as 
$10^{-14}$ to avoid zero 
$\dot \gamma$ in the simulation. As this threshold may seem arbitrary, a larger threshold (
$10^{-5}$) was tested. The contours of viscosity and shear rate are very similar to those when the threshold is 
$10^{-14}$, and the conclusions are unchanged. The viscosity will diverge as the shear rate approaches zero. However, this is reasonable since the mucus is a yield stress fluid. The reader is referred to Gsell et al. (Reference Gsell, D'Ortona and Favier2021) and Galko et al. (Reference Galko, Gsell, D'Ortona, Morin and Favier2022) for more details on the Herschel–Bulkley model.
The present model relies on five non-dimensional physical parameters, i.e. the ciliary density 
$\phi$, the interaction length 
$\lambda$, the Reynolds number 
$Re={\rho }U_0D/\mu _0$ (where 
$\mu _0$ is the reference viscosity), the Bingham number 
$Bn$, and the flow index 
$n$. Here, 
$\lambda$ is defined as
\begin{equation} \lambda = \dfrac{\sqrt{\mu_0/\kappa}}{D}. \end{equation} A high mucus viscosity favours the diffusion of momentum caused by the ciliary beating while a high PCL friction coefficient 
$\kappa$ prevents it. Thus 
$\lambda$ represents the typical range of influence of the ciliated cells (Gsell et al. Reference Gsell, Loiseau, D'Ortona, Viallat and Favier2020). The flow of mucus is caused by the momentum transferred from the tip of the cilia. The momentum diffuses over a larger fluid region when 
$\lambda$ is high. As mentioned previously, mucus flow is almost uniform and parallel to the epithelium. It can be assumed reasonably that no shear exists in the vertical direction for the present model. A reference shear rate can be defined as 
$\dot \gamma _0=U_0/D$. The reference viscosity is 
$\mu _0=K\dot \gamma _0^{n-1}$. The general definition of 
$Re$ becomes (Gsell et al. Reference Gsell, D'Ortona and Favier2021)
\begin{equation} Re = \dfrac{{\rho}U_0^{2-n}D^n}{K}, \end{equation}
where 
$Re=0.1$ is fixed to prevent inertial effects. Here, 
$K$ can be obtained from (2.18), and 
$Bn$ is defined as
\begin{equation} Bn = \dfrac{\sigma_0}{K}\left(\frac{D}{U_0}\right)^n, \end{equation}
where the value of 
$\sigma _0$ is determined by 
$Bn$ and can be obtained from (2.19).
The ranges of 
$\phi$, 
$\lambda$, 
$Bn$ and 
$n$ are physiological and inspired by experimental measurements. Table 1 shows the experimental measurements of physical properties of the mucus. During the ciliogenesis in experiments (Loiseau et al. Reference Loiseau, Gsell, Nommick, Jomard, Gras, Chanez, D'Ortona, Kodjabachian, Favier and Viallat2020), ciliary density increases from 0 to a value approximately 0.7–0.8 (normal ciliary density in the airway (Staudt et al. Reference Staudt, Rogalski, Tilley, Kaner, Harvey and Crystal2014)). In the present study, 
$\phi$ varies in the range 
$0.1\leq \phi \leq 0.7$. From dimensional analysis, the PCL friction coefficient is 
$\kappa \approx \mu _p/\delta _p^2$, where 
$\mu _p$ and 
$\delta _p$ are the dynamic viscosity and thickness of the PCL; 
$\mu _p \approx 10^{-3}$ Pa s and 
$\delta _p \approx 10^{-5}$ m are obtained from Button et al. (Reference Button, Cai, Ehre, Kesimer, Hill, Sheehan, Boucher and Rubinstein2012). Also, 
$D \approx 2\times 10^{-5}$ m and 
$\mu \approx 5\times 10^{-3} \unicode{x2013} 5\times 10^{-2}$ Pa s (the viscosity of the healthy mucus) are obtained from Loiseau et al. (Reference Loiseau, Gsell, Nommick, Jomard, Gras, Chanez, D'Ortona, Kodjabachian, Favier and Viallat2020). Therefore, 
$\lambda$ varies approximately in the range 
$1\leq \lambda \leq 4$. Yield stress 
$\sigma _0 \approx 0.05$ Pa, averaged flow index 
$n \approx 0.15$ and flow consistency 
$K \approx 0.28$ are obtained and derived from Jory et al. (Reference Jory, Donnarumma, Blanc, Bellouma, Fort, Vachier, Casanellas, Bourdin and Massiera2022). The normal mucus velocity 
$U_0$ is approximately 
$1.783 \times 10^{-4}$ m s
$^{-1}$ (Morgan et al. Reference Morgan, Pearson, De Iongh, Mackey, Van der Wall, Peters and Rutland2004). Therefore, 
$Bn$ is approximately 0.128, which is very close to the critical 
$Bn$ (0.15) in the present study for the transition to the FU regime when 
$\phi =0.7$. In the present study, 
$Bn$ varies in the range 
$0\leq Bn \leq 0.3$, enabling the observation of the transition to the FU regime for a very low ciliary density (
$\phi =0.1$). In the present study, 
$n$ varies in the range 
$0.3\leq n \leq 1$, which is sufficient to observe the transition to the FU regime for a very low ciliary density. Details of the regime transition will be discussed latter. Values 
$Bn=0$ and 
$n=1$ represent a Newtonian case.
Experimental measurements of physical properties.
Approximately 4500 simulations were performed to produce the results. The general procedure of the present numerical algorithm for the simulation of a coupled cilia–mucus system in Herschel–Bulkley flows can be summarized as follows (where the time march loop is performed until the steady solution is obtained). (i) At the 
$n$th time step, calculate the angular difference 
$\Delta \theta ^j$ and update the orientation 
$\theta _c^j$ of 
$\boldsymbol {F}_c$ by (2.12). (ii) Perform the collision step on the right-hand side of (2.3). Update the value of the relaxation time 
$\tau ^+$ by (2.16). (iii) Perform the streaming step on the left- side of (2.3) to obtain the new 
$f_l$. (iv) Calculate the new mucus density 
$\rho$ and mucus velocity 
$\boldsymbol u$ by (2.9) and (2.11). Update the value of the frictional force 
$\boldsymbol {F}_\nu$.
3. Results and discussion
3.1. Mucus flow regimes of the cilia–mucus system
In the present study, three distinct mucus flow regimes are observed: a PO regime, an S regime and an FU regime. Figure 2(a) shows the contours of non-dimensional vorticity (
$\omega _z=(D/U_0)\,|\boldsymbol {\nabla }\times \boldsymbol u|$) of the three regimes for different 
$\lambda$ and 
$\phi$ for a Newtonian fluid (
$Bn=0$, 
$n=1$), where vectors indicate the direction of ciliary beating (local flow). The results in figures 2(a) and 4 obtained by the present model are almost the same as those obtained by Gsell et al. (Reference Gsell, Loiseau, D'Ortona, Viallat and Favier2020). The PO regime is characterized by short-range coordination between adjacent cilia without the appearance of large-scale flow structures. The S regime is characterized by long-range coordination of cilia with the formation of obvious mucus swirls. In addition, the FU regime is characterized by long-range coordination of cilia with almost unidirectional flows. These three regimes have been observed in experiments of Loiseau et al. (Reference Loiseau, Gsell, Nommick, Jomard, Gras, Chanez, D'Ortona, Kodjabachian, Favier and Viallat2020). The PO regime corresponds to the pattern in their figures 1(c–e). The S regime corresponds to the pattern characterized by a small swirl (their figures 1f–h). The FU regime corresponds to the pattern characterized by a large swirl that occupies the entire culture chamber (their figure 1i), which is caused by the closed culture chamber. The appearance of the unidirectional flow in the present FU regime is attributed mainly to the periodic boundary condition.
Final steady contours of non-dimensional vorticity 
$\omega _z$ (vectors indicate ciliary beating direction, and the colour bar indicates the magnitude of the vorticity) for (a) Newtonian fluid (Bingham number 
$Bn=0$, flow index 
$n=1$), and (b) non-Newtonian fluid (
$Bn=0.05$, 
$n=0.9$). Three different mucus flow regimes are shown: PO regime, S regime and FU regime. Part of the computational domain is shown, and the scale bars correspond to 
$10D$.
To examine the formation of the three regimes for different 
$\lambda$ and 
$\phi$ (
$Bn=0$, 
$n=1$), the sequential processes of regime evolution from the initial state to the final steady state are plotted in figure 3. The contours are coloured by the non-dimensional flow velocity 
$U/U_0$. Figure 3(a) shows the formation of the PO regime when 
$\lambda$ and 
$\phi$ are very small (
$\lambda =1$ and 
$\phi =0.1$). At instant 
$a_1$, the domain is initialized with zero flow velocity, random ciliary-beat orientation and random cilia distribution. Mucus flow around the cilia is driven by the ciliary beating, visible at instant 
$a_2$. The momentum caused by a ciliated element decays rapidly in space due to the small 
$\lambda$. The ciliated elements are scattered with large distances due to the small 
$\phi$. Accordingly, the mucus flow caused by different cilia can interact with each other only if they are adjacent, resulting in several local flows without a typical flow structure at instant 
$a_3$. The flow velocity in the PO regime is very low.
The sequential processes of evolution of the different mucus flow regimes (
$Bn=0$, 
$n=1$, 
$T=250\,\Delta t$): (a) PO regime for 
$\lambda =1$ (interaction length) and 
$\phi =0.1$; (b) S regime for 
$\lambda =1$ and 
$\phi =0.5$; and (c) FU regime for 
$\lambda =4$ and 
$\phi =0.5$. Instantaneous contours of the non-dimensional velocity 
$U/U_0$ are shown. Part of the computational domain is shown, and the scale bars correspond to 
$10D$.
The S regime is formed when 
$\phi$ is large, as shown in figure 3(b). Mucus flows induced by ciliated elements have the same extension (
$\lambda =1$). However, as the ciliary density is higher, the coordination with neighbouring ciliated elements is improved, and the flow is organized over a greater distance. Several high-velocity regions are observed at instant 
$b_2$ due to the constructive interaction between the adjacent mucus flows. A uniform flow is not formed due to the low 
$\lambda$. At instant 
$b_3$, swirls appear after a longer period of coordination than the local flows in the PO regime.
The FU regime is obtained when 
$\lambda$ is increased, as shown in figure 3(c). The momentum generated by a ciliated element can propagate over a much greater distance. First, at instant 
$c_2$, swirls are formed quickly due to the rapidly diffused mucus flows. Beyond 
$c_2$, the further diffused mucus flows influence the ciliary beating over a larger area. After a long period of coordination, the cilia are almost aligned in the same direction, inducing a unidirectional and uniform flow at instant 
$c_3$.
3.2. Effects of ciliary density and interaction length
To quantitatively identify the PO, S and FU regimes in a wide range of parameters, two physical quantities are employed according to the characteristics of the three regimes. The first quantity is the polarization 
$P$ used to identify the FU regimes, which is the spatial averaging of the unitary velocity vectors, expressed as
\begin{equation} P = \left|\overline{\left(\frac {\boldsymbol{u}}{|\boldsymbol{u}|}\right)}\right|, \end{equation}
where 
$P \approx 1$ represents a unidirectional flow. In the present study, 
$P \geq 0.9$ indicates the FU regime. This critical value is selected based on the observation that 
$P$ increases sharply from a value below 0.6 to a value above 0.9 when the FU regime appears. This will be discussed in more detail later. The second quantity is the non-dimensional integral length 
$\varLambda$ normalized by the dimensional interaction length 
$\sqrt {\mu /\kappa }$:
\begin{equation} \varLambda=\sqrt{\frac{\kappa}{\mu}}\int_{0}^{L/2}\frac{R_{x}(\tau)+R_{y}(\tau)}{2}\,{\rm d}\tau, \end{equation}
where 
$L$ is the length of the computational domain, and 
$R_x(\tau )$ and 
$R_y(\tau )$ are the 
$x$ and 
$y$ components of the autocorrelation functions of the vorticity, respectively:
$$\begin{gather} R_x(\tau)=\frac{\overline{\omega_z(x,y)\,\omega_z(x+\tau,y)}}{\overline{\omega_z^2}}, \end{gather}$$
$$\begin{gather}R_y(\tau)=\frac{\overline{\omega_z(x,y)\,\omega_z(x,y+\tau)}}{\overline{\omega_z^2}}. \end{gather}$$ For 
$\tau > L/2$, the values of 
$R_x(\tau )$ and 
$R_y(\tau )$ are very small, except when 
$\tau$ approaches 
$L$ because of the periodic boundary condition. Therefore, the domain of integration is 
$\tau \in [0,L/2]$ in (3.2). Here, 
$\varLambda$ represents the length scale of flow structures. In particular, the length scale of flow structures is equivalent to the range of influence of the ciliated cells when 
$\varLambda =1$. Small and large 
$\varLambda$ indicate the PO and S regimes, respectively. The increase in 
$\varLambda$ is relatively smooth as the PO regime transitions to the S regime. The critical value 
$\varLambda =1.5$ is selected based on the observation of the flow regime from numerous simulations. In summary, 
$P < 0.9$ and 
$\varLambda < 1.5$ indicate the PO regime, 
$P < 0.9$ and 
$\varLambda \geq 1.5$ indicate the S regime, and 
$P \geq 0.9$ indicates the FU regime.
For comparison, we first examine the effects of 
$\lambda$ and 
$\phi$ on the formation of the mucus flow regime in the Newtonian case (
$Bn=0$, 
$n=1$). Figure 4 shows a phase diagram in the ranges 
$1 \leq \lambda \leq 4$ and 
$0.1 \leq \phi \leq 0.7$. Random initialization can result in different flow regimes under certain conditions. Therefore, for each case in the diagram, 20 randomly initialized simulations were performed. Here, 
$f$ is the occurrence frequency of the most frequent flow regime over a set of 20 simulations, and 
$\bar {P}$ and 
$\bar {\varLambda }$ are the averaged polarization and integral length calculated by the simulations converged to the most frequent flow regime. In figure 4(a), symbols are coloured by the value of 
$\bar {P}$, which increases with increasing 
$\lambda$ and 
$\phi$. In figure 4(b), symbols are coloured by the value of 
$\bar {\varLambda }$. For 
$\bar {P} \geq 0.9$ (FU regime), 
$\bar {\varLambda }$ is set to be 1 and has no physical meaning. We do not discuss the length scale of the flow structures for the FU regime because it is theoretically infinite; 
$\bar \varLambda$ increases with 
$\phi$. The effect of 
$\lambda$ is less clear, but a small tendency of increasing 
$\bar \varLambda$ with decreasing 
$\lambda$ may be observed. In figure 4(c), symbols are coloured by the value of 
$f$. The diagram is divided into three regions by dashed lines according to the maps of 
$\bar {P}$ and 
$\bar {\varLambda }$. The PO regime appears in the region with low 
$\lambda$ and 
$\phi$. The S regime appears in the region with low 
$\lambda$ and high 
$\phi$. The FU regime appears in the region with high 
$\lambda$ and high 
$\phi$. As mentioned for figure 3, these are determined mainly by the range of influence of the ciliated cells and the interaction of the mucus flows caused by adjacent cilia; 
$f$ is relatively small for the points near the regime boundary.
Mucus flow regime diagram depending on 
$\lambda$ and 
$\phi$ for 
$Bn=0$ and 
$n=1$. Symbols are coloured using the values of (a) averaged polarization 
$\bar {P}$, (b) averaged integral length 
$\bar {\varLambda }$, and (c) occurrence frequency 
$f$ of the mucus flow regime. The triangle, circle and square symbols correspond to PO, S and FU regimes, respectively.
3.3. Effects of yield stress and shear thinning properties
For studying the effects of non-Newtonian properties on the flow regime formation, a phase diagram in the ranges 
$1 \leq \lambda \leq 4$ and 
$0.1 \leq \phi \leq 0.7$ is shown in figure 5. The simulated mucus is shear thinning, 
$n=0.9$, and has a yield stress 
$Bn=0.05$. The regions with high 
$\bar {P}$ and low 
$\bar {\varLambda }$ are significantly enlarged compared to those for 
$Bn=0$ and 
$n=1$, as shown in figures 5(a,b). Figure 5(c) clearly shows the displacement of the regime boundary, where the red and black dashed lines represent the boundaries of the non-Newtonian cases (
$Bn=0.05$, 
$n=0.9$) and the Newtonian cases (
$Bn=0$, 
$n=1$), respectively. In general, the regions of the PO and S regimes are reduced, and the region of the FU regime is increased. The PO regime appears only when 
$\lambda$ and 
$\phi$ are very low. The S regime is obtained at lower 
$\lambda$, while it appears in a wider range of 
$\phi$. A lower 
$\lambda$ allows the activation of the FU regime. The boundary between the PO and FU regimes is significantly changed. These indicate that the flow regimes of all cases can be converted to the FU regime by varying 
$Bn$ and 
$n$. The point with 
$\phi =0.1$ and 
$\lambda =1$ would be the last case to complete this conversion. The flow regime of three non-Newtonian cases is shown in figure 2(b), which shows a significant regime transition when compared with the Newtonian cases in figure 2(a).
Mucus flow regime diagram depending on 
$\lambda$ and 
$\phi$ for 
$Bn=0.05$ and 
$n=0.9$. Symbols are coloured using the values of (a) 
$\bar {P}$, (b) 
$\bar {\varLambda }$, and (c) 
$f$. The red and black dashed lines represent the boundaries of the non-Newtonian cases (
$Bn=0.05$, 
$n=0.9$) and the Newtonian cases (
$Bn=0$, 
$n=1$), respectively.
Here, we further examine the effects of 
$Bn$ and 
$n$ on the flow regime formation in detail. First, 
$n=1$ is fixed to explore the effect of 
$Bn$ independently, i.e. a Herschel–Bulkley fluid reduces to a Bingham fluid. In figure 6(a), the value of 
$\bar P$ is presented with respect to the Bingham number 
$Bn$. Recall that to obtain the value of 
$\bar P$, only the simulations in the most frequent regime have been used. Thus each curve represents two sets of data that can be considered independently, and the sharp transition indicates the critical value of 
$Bn$ that induces a transition to the FU regime. In the PO & S regime, 
$\bar P$ increases monotonously with 
$Bn$, except for the case 
$\phi =0.7$, where the variation is less clear. After the transition to the FU regime, 
$\bar P$ remains almost constant. In general, a larger 
$\phi$ leads to a larger 
$\bar {P}$, confirming the results in figures 4(a) and 5(a). The critical 
$Bn$ increases and then decreases with increasing 
$\phi$, resulting in a maximum critical 
$Bn$ at 
$\phi =0.3$. In figure 6(b), the variation of 
$\bar {\varLambda }$ as a function of 
$Bn$ is shown. The FU regime is not included due to its theoretically infinite length scale of the flow structures. 
$\bar {\varLambda }$ increases with increasing 
$Bn$ for different 
$\phi$. A larger 
$\phi$ leads to a larger 
$\bar {\varLambda }$, confirming the results in figures 4(b) and 5(b). For 
$\phi =0.1$, a transition from the PO regime to the S regime is observed by increasing 
$Bn$. For 
$\phi > 0.1$, only the S regime is obtained. The above results suggest that the range of influence of the ciliated cells is increased by increasing 
$Bn$.
Values of (a) 
$\bar {P}$ and (b) 
$\bar {\varLambda }$ as functions of 
$Bn$, for different 
$\phi$ (
$\lambda =1$, 
$n=1$).
Figure 7 shows the variations of 
$\bar {P}$ and 
$\bar {\varLambda }$ as functions of 
$n$ for different 
$\phi$, with 
$\lambda =1$ and 
$Bn=0$ fixed. The sharp increase of 
$\bar P$ in figure 7(a) indicates the critical value of 
$n$ that leads to a transition to the FU regime. The critical 
$n$ increases with increasing 
$\phi$. In contrast to the effect of 
$Bn$, 
$\bar P$ increases monotonously with decreasing 
$n$ in the PO & S regime. After the transition to the FU regime, 
$\bar P$ remains almost constant. In figure 7(b), the decrease of 
$n$ can also lead to a transition from the PO regime to the S regime when the ciliary density is low (
$\phi =0.1$). The above results suggest that the range of influence of the ciliated cells is increased by decreasing 
$n$.
Values of (a) 
$\bar {P}$ and (b) 
$\bar {\varLambda }$ as functions of 
$n$, for different 
$\phi$ (
$\lambda =1$, 
$Bn=0$).
To visualize the transition from the PO regime to the S regime by increasing 
$Bn$, we further examine the instantaneous contours of the flow velocity 
$U/U_0$, the dynamic fluid viscosity 
$\mu$, and the local shear-rate magnitude 
$\dot \gamma$, by increasing 
$Bn$ from 0 to 0.15 in figure 8 (
$\lambda =1$, 
$\phi =0.1$, 
$n=1$). Note that the steady solution for 
$Bn=0$ (instant 
$a_1$) is used as the initial condition for the simulation with 
$Bn=0.15$. A steady solution for 
$Bn=0.15$ is obtained at instant 
$a_2$. In figure 8(a), swirls are more pronounced at instant 
$a_2$, corresponding to the increase in 
$\varLambda$ from 6.36 to 10.99. This is caused mainly by the evolution of the 
$\mu$ and 
$\dot \gamma$ distributions in figures 8(b,c). At instant 
$a_1$, 
$\mu$ is close to its reference value, and 
$\dot \gamma$ caused by the ciliary beating is high. At instant 
$a_2$, 
$\mu$ in the region with low shear rate significantly increases with increasing 
$Bn$ (yield stress). This can be verified by checking (2.13). The momentum diffuses farther when the viscosity is high. Accordingly, the mucus flow caused by a ciliated element affects the ciliary-beat orientation further away, which favours the coordination of the cilia, thereby resulting in a transition from the PO regime to the S regime. A reorientation of the cilia can be observed clearly in figure 8(a). However, the high viscosity makes the mucus difficult to shear, resulting in a decrease in 
$\dot \gamma$ in figure 8(c). At instant 
$a_2$, the low 
$\dot \gamma$ region (blue) corresponding to the high 
$\mu$ region and the non-ciliated region are in solid body rotation or in solid body motion. This solid body rotation has been observed experimentally in the core of swirl by Loiseau et al. (Reference Loiseau, Gsell, Nommick, Jomard, Gras, Chanez, D'Ortona, Kodjabachian, Favier and Viallat2020), visualized in their figures 1(h,i) and their supplementary movies 5 and 6 (available at https://www.biorxiv.org/content/10.1101/2019.12.16.878108v1.supplementary-material). The role of the yield stress in generating solid body rotation is that the effective viscosity diverges as the shear rate approaches zero. Here, we examine the distribution of longitudinal velocity 
$u_y$ in the core of swirls marked in figure 8(a) for 
$Bn=0$ and 0.15; 
$u_y$ is extracted along the red line as schematized in figure 9(a). The distributions of 
$u_y$ are shown in figures 9(b,c). Here, 
$u_y$ varies linearly in the radial direction when 
$Bn=0.15$, indicating a solid body rotation. This was not reproduced in the previous study (Gsell et al. Reference Gsell, Loiseau, D'Ortona, Viallat and Favier2020) since a Newtonian fluid was modelled. The addition of the yield stress property to the mucus allows a more accurate modelling of the experiments.
The sequential process of a transition from the PO regime (instant 
$a_1$, 
$Bn=0$) to the S regime (instant 
$a_2$, 
$Bn=0.15$) by increasing 
$Bn$. The steady solution for 
$Bn=0$ (instant 
$a_1$) is used as the initial condition for the simulation with 
$Bn=0.15$. A steady solution for 
$Bn=0.15$ is obtained at instant 
$a_2$. Instantaneous contours are coloured by (a) 
$U/U_0$, (b) dynamic fluid viscosity 
$\mu$ (whose value at instant 
$a_1$ corresponds to a Newtonian fluid), and (c) local shear-rate magnitude 
$\dot \gamma$ (
$\lambda =1$, 
$\phi =0.1$, 
$n=1$). Part of the computational domain is shown, and the scale bars correspond to 
$10D$.
(a) Schematic of velocity extraction in the core of a swirl. Distributions of the longitudinal velocity 
$u_y$ along the radial direction in the core of swirls marked in figure 8(a), for (b) 
$Bn=0$, and (c) 
$Bn=0.15$.
The instantaneous contours of 
$U/U_0$, 
$\mu$ and 
$\dot \gamma$ by increasing 
$Bn$ from 0.15 to 0.3 are shown in figure 10 (
$\lambda =1$, 
$\phi =0.1$, 
$n=1$). The steady solution for 
$Bn=0.15$ (instant 
$a_2$ in figure 8 or instant 
$a_1$ in figure 10) is used as the initial condition for the simulation with 
$Bn=0.3$. A steady solution for 
$Bn=0.3$ is obtained at instant 
$a_3$. The further increase in 
$Bn$ substantially increases 
$\mu$ at instant 
$a_2$ irrespective of the high 
$\dot \gamma$ region at instant 
$a_1$. This further enhances the diffusion of momentum and the coordination of different cilia, resulting in a larger swirl in figure 10(a) (instant 
$a_2$). The increase in 
$\mu$ decreases 
$\dot \gamma$, which in turn increases 
$\mu$ at instant 
$a_3$. Thus there is a positive feedback between the increased 
$\mu$ and the decreased 
$\dot \gamma$ for the flow with yield stress. The S regime gradually converts to the FU regime due to the further diffusion of momentum.
The sequential process of a transition from the S regime (instant 
$a_1$, 
$Bn=0.15$) to the FU regime (instant 
$a_3$, 
$Bn=0.3$) by increasing 
$Bn$: instantaneous contours of (a) 
$U/U_0$, (b) 
$\mu$, and (c) 
$\dot \gamma$ (
$\lambda =1$, 
$\phi =0.1$, 
$n=1$). Part of the computational domain is shown, and the scale bars correspond to 
$10D$.
For the regime transition induced by varying 
$n$, the instantaneous contours of 
$U/U_0$, 
$\mu$ and 
$\dot \gamma$ by decreasing 
$n$ from 1 to 0.6 are shown in figure 11 (
$\lambda =1$, 
$\phi =0.1$, 
$Bn=0$). According to (2.13) and (2.18), the viscosity is 
$\mu =(\rho U_0^2 \dot \gamma ^{-1}\,Re^{-1})(D\dot \gamma /U_0)^n$. Here, 
$\mu$ is increased when 
$n$ decreases from 1 to 0.6 (
$D\dot \gamma /U_0<1$), enhancing the diffusion of momentum and reducing 
$\dot \gamma$. The decrease in 
$\dot \gamma$ in turn leads to an increase in 
$\mu$ due to the shear thinning behaviour. There is also a positive feedback between the increased 
$\mu$ and the decreased 
$\dot \gamma$ for the shear thinning flow. A transition from the PO regime to the S regime is observed at instant 
$a_2$ due to the enhanced diffusion of momentum, corresponding to the increase in 
$\varLambda$ from 6.36 to 10.15. The effect of shear thinning (
$n=0.6$ and 
$Bn=0$) on 
$\mu$ is weak compared to the effect of yield stress (
$n=1$ and 
$Bn=0.15$).
The sequential process of a transition from the PO regime (instant 
$a_1$, 
$n=1$) to the S regime (instant 
$a_2$, 
$n=0.6$) by decreasing 
$n$: instantaneous contours of (a) 
$U/U_0$, (b) 
$\mu$ (whose value at instant 
$a_1$ corresponds to a Newtonian fluid), and (c) 
$\dot \gamma$ (
$\lambda =1$, 
$\phi =0.1$, 
$Bn=0$). Part of the computational domain is shown, and the scale bars correspond to 
$10D$.
Figure 12 shows the instantaneous contours of 
$U/U_0$, 
$\mu$ and 
$\dot \gamma$ by further decreasing 
$n$ from 0.6 to 0.3 (
$\lambda =1$, 
$\phi =0.1$, 
$Bn=0$). As mentioned above, the decrease of 
$n$ leads to the increase of 
$\mu$. The substantial enhancement of the shear thinning effect significantly enhances the positive feedback between the increased 
$\mu$ and the decreased 
$\dot \gamma$. At instant 
$a_2$, the increase of 
$\mu$ results in the formation of large-scale swirls. At instant 
$a_3$, the further increase of 
$\mu$ and the full coordination of cilia and mucus induce a transition from the S regime to the FU regime. In summary, both the increase of 
$Bn$ and the decrease of 
$n$ lead to the successive appearance of PO, S and FU regimes. This is closely related to the increase of 
$\mu$ and the diffusion of momentum. Note that the flow velocities in the ciliated region are lower for the FU regime than for the PO (or S) regime. In the FU regime, the momentum diffuses into the non-ciliated region, and the flow velocity is averaged over ciliated nodes with beating force, and non-ciliated nodes with only friction. In the PO regime, the velocity remains localized above the ciliated region, which is ineffective for the mucus transport.
The sequential process of a transition from the S regime (instant 
$a_1$, 
$n=0.6$) to the FU regime by decreasing 
$n$ (instant 
$a_3$, 
$n=0.3$): instantaneous contours of (a) 
$U/U_0$, (b) 
$\mu$, and (c) 
$\dot \gamma$ (
$\lambda =1$, 
$\phi =0.1$, 
$Bn=0$). Part of the computational domain is shown, and the scale bars correspond to 
$10D$.
The variation of 
$\dot \gamma$ directly influences 
$\mu$ under the yield stress and shear thinning effects. Here, we calculate the spatially averaged shear-rate magnitude 
$\bar {\dot \gamma }$ for different cases. The variation of 
$\bar {\dot \gamma }$ as a function of 
$Bn$ (
$\lambda =1$, 
$n=1$) and 
$n$ (
$\lambda =1$, 
$Bn=0$) for different 
$\phi$ is shown in figure 13, where 
$\bar {\dot \gamma }$ decreases with increasing 
$Bn$ and decreasing 
$n$, indicating the increase in mucus viscosity. In addition, 
$\bar {\dot \gamma }$ increases with increasing 
$\phi$ until 
$\phi =0.5$ due to the increase in the number of cilia. The further increase in 
$\phi$ significantly enhances the coordination between different cilia, which tend to beat in the same direction and result in a lower shear. Beyond 
$\phi =0.3$, 
$\bar {\dot \gamma }$ is relatively insensitive to the increase in 
$\phi$.
Averaged shear-rate magnitude 
$\bar {\dot \gamma }$ as a function of (a) 
$Bn$ (
$\lambda =1$, 
$n=1$) and (b) 
$n$ (
$\lambda =1$, 
$Bn=0$), for different 
$\phi$.
3.4. Effective interaction length
The mucus viscosity 
$\mu$ is the dominant parameter that affects the regime formation when varying 
$Bn$ and 
$n$ according to the above discussions. In the present study, the viscosity effect is included in 
$\lambda$, which is defined based on the reference mucus viscosity. To consider the variation of 
$\mu$, a spatially averaged viscosity 
$\bar {\mu }$ is calculated for the cases with different 
$Bn$ and 
$n$. The lattice nodes with maximum relaxation time are excluded due to the truncated Herschel–Bulkley law used in the present study. In fact, their viscosity should be considered almost infinite. An effective interaction length 
$\lambda ^*$ is defined based on 
$\bar {\mu }$ instead of 
$\mu$, and 
$\lambda ^*$ is found to be more suitable to represent the range of influence of the ciliated cells. Figure 14 shows the variation of 
$\lambda ^*$ as a function of 
$Bn$ (
$\lambda =1$, 
$n=1$) and 
$n$ (
$\lambda =1$, 
$Bn=0$) for different 
$\phi$. Here, 
$\lambda ^*$ increases with increasing 
$Bn$ and decreasing 
$n$, which favours the diffusion of momentum and the coordination between cilia and mucus, thereby resulting in the regime transition. This confirms the results shown in figures 8 and 10–12. For 
$\phi \geq 0.3$, 
$\lambda ^*$ is smaller than for 
$\phi =0.1$, and the curves of 
$\lambda ^*$ almost collapse onto a single curve. This corresponds to the variation of 
$\bar {\dot \gamma }$ in figure 13. Furthermore, the variation of 
$\lambda ^*$ is opposite to the variation of 
$\bar {\dot \gamma }$ due to the yield stress and shear thinning effects.
Effective interaction length 
$\lambda ^*$ as a function of (a) 
$Bn$ (
$\lambda =1$, 
$n=1$) and (b) 
$n$ (
$\lambda =1$, 
$Bn=0$), for different 
$\phi$.
To examine the dependence of the regime formation on 
$\lambda ^*$, we compare the variation of 
$\bar P$ as a function of 
$\lambda$ and 
$\lambda ^*$ for different 
$Bn$ (
$\phi =0.1$, 
$n=1$) in figure 15. Recall that to obtain the value of 
$\bar P$, only the simulations in the most frequent regime have been used. Here, 
$\bar P$ increases with increasing 
$\lambda$, and a rapid increase of 
$\bar P$ can be observed at the critical points for the appearance of the FU regime. The FU regime appears at a smaller 
$\lambda$ as 
$Bn$ is increased. Here, we consider only the critical condition of FU regime formation because the FU regime is the most efficient for mucus transport. The curves of 
$\bar P$ are scattered for different 
$Bn$ in figure 15(a). In figure 15(b), the curves collapse onto a single curve as a whole for 
$Bn > 0$. Here, 
$Bn=0$ deviates significantly from the collapsed curve; 
$\lambda ^*$ is not enough to predict the critical condition for different 
$Bn$. This may be related to the fact that the increase in 
$Bn$ leads to a sharp increase in 
$\mu$, which is truncated when the maximum value is reached.
The value of 
$\bar P$ as a function of (a) 
$\lambda$ and (b) 
$\lambda ^*$, for different 
$Bn$ (
$\phi =0.1$, 
$n=1$).
Figure 16 shows the variation of 
$\bar P$ as a function of 
$\lambda$ and 
$\lambda ^*$ for different 
$n$ (
$\phi =0.1$, 
$Bn=0$). In figure 16(a), the curves of 
$\bar P$ are also scattered, and the critical 
$\lambda$ decreases with decreasing 
$n$. After the rescaling is performed, the curves collapse onto a single curve as a whole. The critical 
$\lambda ^*$ is distributed in a narrow range 
$\lambda ^* \approx 3.8\unicode{x2013}5.1$. Here, 
$\lambda ^*$ is more suitable for predicting the critical condition of the FU regime formation for different 
$n$ than that for different 
$Bn$.
The value of 
$\bar P$ as a function of (a) 
$\lambda$ and (b) 
$\lambda ^*$, for different 
$n$ (
$\phi =0.1$, 
$Bn=0$).
4. Conclusions
In this work, the hydrodynamic coupling of a cilia–mucus system in Herschel–Bulkley flows was investigated numerically using a two-dimensional hydrodynamic model. The mucus flow was predicted based on the lattice Boltzmann method, and the interaction between the cilia and the mucus was handled by an alignment rule. Numerical simulations were performed in wide ranges of ciliary density (
$\phi$), interaction length (
$\lambda$), Bingham number (
$Bn$) and flow index (
$n$) to highlight the effects of yield stress and shear thinning properties on the mucus flow regime. For the effects of 
$\phi$ and 
$\lambda$, a poorly organized (PO) regime, a swirly (S) regime and a fully unidirectional (FU) regime were identified. The PO regime appears with low 
$\lambda$ and 
$\phi$. The S regime appears with low 
$\lambda$ and high 
$\phi$. The FU regime appears with high 
$\lambda$ and high 
$\phi$. These are determined by the range of influence of the ciliated cells (range of momentum diffusion) and the coordination between different cilia. For the effects of 
$Bn$ and 
$n$, the range of influence of the ciliated cells is increased by increasing 
$Bn$ and decreasing 
$n$, resulting in the activation of the S and FU regimes at lower 
$\phi$ and 
$\lambda$. Mucus viscosity is found to be the dominant parameter affecting the regime formation when varying 
$Bn$ and 
$n$. We define an effective interaction length 
$\lambda ^*$ based on the spatially averaged viscosity obtained from the final steady solution instead of the reference viscosity, which is more appropriate than 
$\lambda$ to represent the range of influence of the ciliated cells. This 
$\lambda ^*$ increases with increasing 
$Bn$ and decreasing 
$n$, explaining the regime formation upon introduction of Herschel–Bulkley flows. After rescaling, the critical 
$\lambda ^*$ values for the appearance of the FU regime are still scattered for different 
$Bn$, while the critical 
$\lambda ^*$ values are distributed in a narrow range for different 
$n$. So 
$\lambda ^*$ is more suitable to predict the critical condition of FU regime formation for different 
$n$ than that for different 
$Bn$. Furthermore, the present model is capable of reproducing the solid body rotation observed in experiments, showing a more precise prediction than that of a Newtonian model for the mucus.
Acknowledgements
Centre de Calcul Intensif d'Aix-Marseille University is acknowledged for granting access to its high-performance computing resources. The authors thank Dr S. Gsell for his kind assistance with the code.
Funding
This work was supported by the BonchoClogDrain project (ANR-22-CE30-0045) funded by the French National Research Agency (ANR).
Declaration of interests
The authors report no conflict of interest.
