## 4 Conclusions  
By using a finite-difference fluid solver coupled to an IB method, we first examined cilia-driven fluid flows, and then thorough parametric studies were performed to better understand the fluid flow characteristics of PCL in humans with ciliary disorders. Our results can be summarized as follows:  
Figure 11: Time-averaged wall shear stress. Effect of (**a**) beat frequency, (**b**) length of cilia, (**c**) ciliary beat pattern, (**d**) ciliary pack, (**e**) missing ciliary patch, and (**f**) ciliary density (number of cilia over the ciliated domain). (1) A complex region was generated by the ciliary carpet, and a substantial amount of flow is transferred in the direction parallel to effective stroke. Further, the visualization of the velocity field, particularly above the cilia tips, showed unidirectional transport of the ASL over time. This is in accord with the actual muco-ciliary mechanism. It was also found that vortex-like structures are formed above the cilia tips once they are moving backward during their recovery stroke. However, when the cilia move forward during their effective stroke, small vortices appear below the cilia tip. (2) Our flow rate and surface velocity results imply the significant role of ciliary beat frequency, suggesting that the discovery of novel pharmaceuticals that can target cilia and increase their frequency of beating could enhance the PCL transport, and as a result, better transport of the ML under diseased conditions, such as asthma, or in cigarette smokers. It should be noted that increasing the beat frequency results in a high shear stress at the epithelium which may adversely affect the epithelial functions. (3) It was shown that decreasing the ciliary length, which is the case in COPD patients and smokers, hinders the PCL flow rate and the surface mean velocity. It is also found that if the length of cilia is increased, it is less likely to see any significant changes in the PCL flow rate, while the time-averaged surface velocity would increase. (4) In PCD patients with stiff planar and wiper ciliary beat patterns, pathogens are not effectively cleared from the respiratory tracts due to the large reduction of PCL flow rate, as well as the surface velocity. Nevertheless, the impaired ciliary beat patterns would not significantly change the shear stress. (5) Generally, the triangular ciliary packs could outperform the square pack, thus increasing the pumping performance of engineered ciliated tissues in microfluidic chips. Note that the case of the equilateral triangle pack showed a higher fluid transport rate than the isosceles triangle pack, but less wall shear stress. (6) If the patch size is increased, the flow rate, surface velocity, and the shear stress of fluid will be reduced accordingly. Thus, CF patients and cigarette smokers are more prone to infection from the loss of the cilia. (7) It was found that the flow rate and surface velocity first increase sharply with increasing ciliary density (or number of cilia within the same computational domain) and then reach a plateau. On the other hand, the changes of shear stress depend highly on the number of cilia and did not level off at the higher ciliary density. In the future work, the present model can be extended to study how drugs dissolved into the PCL would be absorbed by the epithelial cells, or how chemical species secreted by epithelial cells would be dispersed by PCL flows. [PERSON] and [PERSON] implemented the numerical model. [PERSON] performed the numerical simulations and the model output postprocessing. [PERSON] wrote the first draft of the manuscript. [PERSON], [PERSON] supervised and critically reviewed and changed the first draft and suggested additional simulations. [PERSON] supervised and critically reviewed and revised the subsequent drafts. All authors approved the final version of the manuscript. This research received no external funding. [PERSON] would like to acknowledge the QUTPRA scholarship scheme for financial support. The authors declare no conflict of interest.