Free Interfaces at the Tips of the Cilia in the One-Dimensional Periciliary Layer
Abstract
:1. Introduction
2. Mathematical Model
3. Boundary Immobilization Technique
4. Model Discretization
5. Numerical Validation
6. Numerical Results
7. Conclusions
- To avoid the numerical grid adjustment for every change in the the angle , we adopted the boundary immobilization technique to the mathematical model;
- The one-dimensional isotropic finite element method was applied to the Brinkman equation and a finite different scheme was employed to the Stefan problem to calculate the numerical solutions;
- The numerical results were verified by comparing them with an exact solution with a fixed boundary condition at when . The results converge to the exact solution, where the -norm errors are provided in Table 1;
- The numerical results of each angle are provided in Figure 6, Figure 7 and Figure 8 for the forward stroke from the angles to . Even if the velocity of the cilia is highest at (Figure 5 in [41]), the highest velocity of the PCL fluid is found at . That is, the velocities of the PCL fluid increase from the starting angle and decrease from the angles to ;
- The free surfaces due to the movement of cilia are shown in Figure 10, Figure 11 and Figure 12. We discovered that the height of the surface increases for the full forward stroke although the velocities of the PCL fluid increase in the first half of the effective stroke and decrease for the second half. Moreover, we found that the initial value c does not change the shape of the free surface, but the different functions r cause the change in the shape of the surface as illustrated in Figure 11 and Figure 12;
- To the author’s knowledge, no experimental data in the literature provide the shape of the free surface, only the average value is given. The numerical results were compared with the available experimental data. For the non-CF tissues, the average heights of the PCL from our results and the experimental data were and m, respectively. The difference between and is , which is small. This is one of the justifications that shows that our mathematical model is suitable for this problem.
Funding
Acknowledgments
Conflicts of Interest
References
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Number of Nodes along z-Axis | -Norm Errors |
---|---|
10 | 4.0837 |
20 | 1.3711 |
100 | 0.1184 |
400 | 0.0148 |
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Wuttanachamsri, K. Free Interfaces at the Tips of the Cilia in the One-Dimensional Periciliary Layer. Mathematics 2020, 8, 1961. https://doi.org/10.3390/math8111961
Wuttanachamsri K. Free Interfaces at the Tips of the Cilia in the One-Dimensional Periciliary Layer. Mathematics. 2020; 8(11):1961. https://doi.org/10.3390/math8111961
Chicago/Turabian StyleWuttanachamsri, Kanognudge. 2020. "Free Interfaces at the Tips of the Cilia in the One-Dimensional Periciliary Layer" Mathematics 8, no. 11: 1961. https://doi.org/10.3390/math8111961
APA StyleWuttanachamsri, K. (2020). Free Interfaces at the Tips of the Cilia in the One-Dimensional Periciliary Layer. Mathematics, 8(11), 1961. https://doi.org/10.3390/math8111961