Analytical solution of a 3D model for the airflow in a human oral cavity

Chatvarin Tasawang; Supachara Kongnuan

Authors

Chatvarin Tasawang Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Khlong Luang, Pathum Thani, 12121
Supachara Kongnuan Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Khlong Luang, Pathum Thani, 12121

Keywords:

three-dimensional mathematical model, oral cavity, human upper respiratory tract, Navier-Stokes equations, analytical solution

Abstract

Understanding characteristic of the airflow in a human respiratory tract is very important factor to treatment in the respiratory disease. In this paper, we propose a three-dimensional mathematical modelling for the airflow in a human oral cavity. The airflow is assumed to be axially symmetric flow and driven by the oscillating pressure gradient. The governing equations for describing the behavior of airflow are composed the Navier-Stokes equations and the continuity equation in a cylindrical coordinates system. To solve the model, we presented method of analytical solution for the airflow velocity. We obtained a solution in a Fourier-Bessel series form. Then, we simulated the airflow field on a three-dimensional geometry of the oral cavity area. The obtained results show that the characteristic of magnitude and direction of the airflow correspond to the fact of the airflow in human airway and the previous research works.

References

Emin, M., & Erdem, C. (2009). An analytical Solution of the Navier-Stokes Equation for Flow over a Moving Plate Bounded by Two Side Walls. Journal of Mechanical Engineering, 55(12), 749-754.

Gockenbach, M. S. (2011). Partial differential equations: analytical and numerical methods (2nd ed.). The United States of America: The Society for Industrial and Applied Mathematics.

Hranitz, M. J. (2009). APHNT: Respiratory Physiology Outlines. Retrieved from http://facstaff. bloomu.edu/jhranitz/Courses/APHNT/Outlines/ Respir

Kongnuan, S., & Pholuang, J. (2012). A Fourier Series-Based Analytical Solution for the oscillating Airflow in a Human Respiratory Tract. International journal of pure and applied mathematics, 78(5), 721-733.

Mohyuddin, M. R., Siddiqui, A. M., Hayat, T., Siddqui, J., & Asghar, S. (2008). Exact solutions of time-dependent Navier-Stokes equations by Hodograph-Legendre transformation method. Tamsui Oxford Journal of Mathematical Sciences, 24(3), 257-268.

Qingxing, X., Fong, Y. L., Chi-Hwa, W. (2009). Transport and deposition of inertial aerosols in bifurcated tubes under oscillatory flow. Chemical Engineering Science, 64, 830-846.

Otarod, O., & Otarod, D. (2006). Analytical solution for Navier-Stokes equations in two dimensions for laminar incompressible flow. Retrieved from http://arxiv.org/physics/0609186

Rosu, H., & Romero, J. L. (1999). Ermakov approach for the one-dimensional Helmholtz Hamiltonian. Nuovo Cimento, 114, 569-574.

Tsangaris, S., & Vlachakis, N. W. (2003). Exact solution of the Navier-Stokes equations for the oscillating flow in duct of a cross-section of right-angled isosceles triangle. Z. angew. Math. Phys., 54, 1094-1100.

Wang, K., Denney, T. S., Morrison, E. E., & Vodyanoy, V. J. (2005). Numerical simulation of air flow in the human nasal cavity. Retrieved from http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1615757

Zhang, Z., & Kleinstreuer, C. (2004). Airflow structures and nano-particle deposition in a human upper airway model. Journal of Computational Physics, 198, 178-210.