## 3 Results and Discussion
### Parametric Investigations
Two distinctive beat patters have been reported for cilia from PCD disease subjects: stiff and wiper planar motions [12; 13]. The earlier is constructed by assuming a stiff cilium beats within a range of 30\({}^{\circ}\)\(\sim\)150\({}^{\circ}\) with twelve equally spaced phase numbers over one cycle. The later motion is constructed in the same manner but the range of motion is assumed 50\({}^{\circ}\)\(\sim\)130\({}^{\circ}\), where a straight line pointing from the cilium base to the cilium tip makes the angle with respect to the bottom wall. In addition to these ciliary abnormalities, two-dimensional beating kinematics (planar) is also incorporated into the model and compared against the three-dimensional ciliary motion (standard beat). This motion is constructed by setting the \(y\)-component of 3D beat profile equal to zero. Comparing the standard CBP to planar CBP, the flow rate \(\langle Q\rangle\) was almost identical. This agrees well with the results of [PERSON] et al. [57], who concluded that both 2D and 3D ciliary kinematics show qualitatively similar transport due to the metachronal waves. In the stiff planar and wiper cases, the fluid flow rate drops significantly: by 70% in planar and by 94% in wiper. _Parameter (IV): ciliary pack_. The spatial relationships of cilia on the epithelial surface is not very clear because cilia are tightly packed on the epithelia which makes it difficult to visualise them by means of scanning electron microscopy (SEM). [PERSON] and [PERSON] [3] interpreted an area of ciliary activity on cultured rabbit tracheal epithelium by a model in which cilia had equilateral triangular packing (Pack 1; Figure 9a). Here, we suggest two other possible packing types (see Figure 9a):  
Figure 8: Time-averaged PCL flow rate. 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). isosceles triangle (Pack 2)--which is initially employed in the present work for the cilia-driven fluid flow analyses and then for the first three parametric studies; and square pack (Pack 3)--which has been commonly used by researchers [7; 45; 54; 57]. To find out which packing type could be more effective in the fluid transport, the three mentioned packings are compared with one another. Note that the ciliary density (i.e., cilium/area taken up for each cilium) is kept constant at 0.055 cilium/\(\upmu\)m\({}^{2}\) for this comparison. That is to say, only 64, 68, and 72 cilia could be fitted within the ciliary domain using Pack 3, Pack 1, and Pack 2, respectively. The results show that Pack 1 provides the highest flow rate which is slightly larger than that of Pack 1, revealing that although pack 1 has a fewer number of driving ciliary forces, it is capable to push the fluid more efficiently. However, the square pack (Pack 3) decreases the forward motion of fluid up to nearly 40%. These findings could play an important role in the optimal design of cilia-on-chip systems. Parameter (V): missing ciliary patchExamination of the airway ciliated cells of smokers and CF patients shows loss of cilia [10; 14]. Exposure to airborne pollutants is another reason of loss of ciliated cells [17]. In order to study the regions of missing cilia, three patches of different sizes are considered using Pack 3, as depicted in Figure 9b. As seen in Figure 8e, \(\langle Q\rangle\) is inversely proportional to the patch size. Parameter (VI): ciliary densityDecrease of ciliary density on the nasal epithelium is another respiratory impairment in COPD subjects [58]. To explore the effect of ciliary density, 36-576 equally spaced cilia are distributed (using Pack3) over the ciliary domain (33 \(\upmu\)m\({}^{2}\)) in the \(x\)- and \(y\)-directions which correspond to inter-ciliary spacing range of 1.43-6.6 \(\upmu\)m within the same computational domain.