## 3 Results and Discussion
### Parametric Investigations  
To achieve the goal of this paper, we perform thorough parametric investigations to evaluate the effects of parameters including ciliary beat frequency \(CBF\), length of cilia \(L\), ciliary beat pattern (CBP), ciliary pack, ciliary density, and missing ciliary patch on the simulation results. It will also be explained how changes in these parameters could represent some of the abnormalities listed in Table 1.  
Figure 6: Snapshots of the streamlines and fluid velocity profiles in the \(x\)-direction (upper-right panel) taken at five different instants (i.e., (**b**) \(t=0\) s, (**b**) \(t=0.02\) s, (**c**) \(t=0.04\) s, (**d**) \(t=0.06\) s, and (**e**) \(t=0.08\) s) within the stable beating cycle (T = 0.1 s) to highlight the cilia-driven flow mechanism and the fluid mixing. In case (**b**), cilium 1 stands up to perform its effective stroke that continues up to the instant \(t=0.04\) s (case (**c**)), after which, the recovery phase (case (**d**)) begins. This motion continues and the cilium arrives to its initial position (case (**a**)). Left and right panels show the top and side views of the computational domain, respectively. Note that for better clarity of the velocity profiles, the \(x\)-axis in each plot has a different scale. Figure 7: Transients of PCL flow rate \(Q\), surface mean velocity \(u_{surf}\), wall shear stress \(\tau\), and volume-averaged vorticity magnitude \(|\omega|\) for the standard case over the third cycle. The results of our parametric study are assessed in terms of three metrics: (i) Time-averaged flow rate as  
\[\langle Q\rangle=(1/T)\int Qdt, \tag{17}\]  
(ii) time-averaged surface mean velocity as  
\[\langle u_{surf}\rangle=(1/T)\int u_{surf}dt, \tag{18}\]  
and (iii) time-averaged wall shear stress as  
\[\langle\tau_{w}\rangle=(1/T)\int\tau_{w}dt. \tag{19}\]  
It should be noted that in all of the following results, the numerical setting is the same as the standard setting given in Table 3, except when stated differently. Figure 8 shows the variation of time-averaged flow rate \(\langle Q\rangle\) generated by the ciliary action for six different parameters. _Parameter (I): Ciliary beat frequency (\(CBF\))_. In the healthy human lung, cilia beat rapidly at about 8-15 Hz [55]. However, experimental findings in PCD shows a considerable \(CBF\) variability from 1.4-24.9 Hz due to gene mutations [56]. Further, in acquired disorders, such as cigarette smoking, environmental pollutants, and asthma, \(CBF\) tends to decrease [9; 10; 19]. To study the effect of \(CBF\) on the flow rate, the beat frequency is varied in the range of 5-25 Hz. The flow rate varies linearly with \(CBF\), as shown in Figure 8a. The effect of \(CBF\) on the results is significant--a decrease in \(CBF\) from 10 to 5 Hz, due to diseased conditions for example, reduces the flow rate \(\langle Q\rangle\) by 50%. Note that 10 Hz is considered as the standard or healthy beat frequency throughout this work. _Parameter (II): ciliary length (\(L\))_. Clinical findings reveal that cigarette smokers, people exposed to environmental pollutants and patients suffering from COPD have shorter cilia [15; 16; 21]. As such, it is important to investigate the PCL locomotion for abnormal ciliary length. It was found that the shorter the cilia are, the more the PCL flow rate is hindered. Compared to the standard length \(L=6\)\(\upmu\)m, the ciliary lengths of 5 \(\upmu\)m and 4 \(\upmu\)m led to 25% and 48% reduction in the fluid flow rate, respectively. However, increasing the length to 7 \(\upmu\)m could not alter the result significantly--only 2% increase was observed. This may be related to the cilia tips getting closer to the PCL surface and resulting interactions, but the exact cause is left for future investigation. _Parameter (III): CBP_. Studies of CBP in individuals with asthma and COPD showed impaired CBPs in bronchial epithelial [20] and nasal cilia [22], respectively.