The Velocity of PCL Fluid in Human Lungs with Beaver and Joseph Boundary Condition by Using Asymptotic Expansion Method

Abstract

Humans breathe air into the respiratory system through the trachea, but with all the pollutants in our environment (both outside and inside), the air we breathe may not be clean. When that is so, the respiratory system secretes mucus to trap dirt that is inhaled through the nostrils. The respiratory tract contains hair-like structures in the epithelial tissue, called cilia: These wave back and forth to help expel particles of dust, dirt, mucus, and contaminants from the body. Cilia are found in this layer (a porous medium) and the fluid in this layer is called the periciliary layer (PCL). This study aims to determine the velocity of the PCL fluid flow in motile cilia. Usually, fluids move due to pressure changes, but in this study, the velocity of solids or of the cilia moves the PCL fluid. Stokes‐Brinkman equations are used to determine the velocity of PCL fluid flow when cilia form an angle with the horizontal plane. The Beavers and Joseph boundary condition is applied in this study. The asymptotic expansion method is adapted in order to determine the velocity of PCL from the movement of the cilia.

On the Asymptotic Boundary Condition at the Free-Fluid/Porous-Medium Interface in Periciliary Layer due to the Ciliary Movement

“…erefore ( 1) is called Stokes-Brinkman equations. e Brinkman equation used in ω 2 has been normalized in [11], which is…”

“…e approximation of other parts of the domain ω is named outer solution. en the solutions are combined through matching conditions provided in Section 3.3. e inner and outer solutions, derived in [11], are briefly introduced in Sections 3.1 and 3.2. e zeroth-order results after applying the matching conditions are provided in Section 3.4.…”

“…e equation applied in the free-fluid region was described by the Stokes equation or Navier-Stokes equations, while flow in the porous-medium region was usually modeled by using Darcy's Law or Brinkman equations [6,[8][9][10][11][12][13]. In general, the Brinkman model is used for the porous media with high porosity [14].…”

“…For example, Chandesris and Jamet [9] studied the Poiseuille flow over a permeable block by using the method of matched asymptotic expansion and determined the boundary conditions at the top of the permeable block. Poopra and Wuttanachamsri [11] calculated the velocity of the PCL fluid with the BJ boundary condition when cilia formed angles with the horizontal plane by using the asymptotic expansion method.…”

“…In our study, on the contrary, cilia in the porous domain are mobile. Although Poopra and Wuttanachamsri [11] studied the PCL with self-propelled solid phases, they did not seek for matching conditions to match the solutions at the transition zone at the free-fluid/porous-medium interface and they did not provide a proper boundary condition at the interface, which was practical to the problem. us, this research has been conducted to figure out the boundary conditions suiting for the problem.…”