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.

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Cilia on the surface of ciliated cells in the respiratory system are organelles that beat forward and backward to generate metachronal waves to propel mucus out of lungs. The layer that contains the cilia, coating the interior epithelial surface of the bronchi and bronchiolesis, is called the periciliary layer (PCL). With fluid nourishment, cilia can move efficiently. The fluid in this region is named the PCL fluid and is considered to be an incompressible, viscous, Newtonian fluid. We propose there to be a free boundary at the tips of cilia underlining a gas phase while the cilia are moving forward. The Brinkman equation on a macroscopic scale, in which bundles of cilia are considered rather than individuals, with the Stefan condition was used in the PCL to determine the velocity of the PCL fluid and the height/shape of the free boundary. Regarding the numerical methods, the boundary immobilization technique was applied to immobilize the moving boundaries using coordinate transformation (working with a fixed domain). A finite element method was employed to discretize the mathematical model and a finite difference approach was applied to the Stefan problem to determine the free interface. In this study, an effective stroke is assumed to start when the cilia make a 140∘ angle to the horizontal plane and the velocitiesof cilia increase until the cilia are perpendicular to the horizontal plane. Then, the velocities of the cilia decrease until the cilia make a 40∘ angle with the horizontal plane. From the numerical results, we can see that although the velocities of the cilia increase and then decrease, the free interface at the tips of the cilia continues increasing for the full forward phase. The numerical results are verified and compared with an exact solution and experimental data from the literature. Regarding the fixed boundary, the numerical results converge to the exact solution. Regarding the free interface, the numerical solutions were compared with the average height of the PCL in non-cystic fibrosis (CF) human tissues and were in excellent agreement. This research also proposes possible values of parameters in the mathematical model in order to determine the free interface. Applications of these fluid flows include animal hair, fibers and filter pads, and rice fields.

The purpose of this study is to study the movement of the periciliary (PCL) fluid due to the ciliary locomotion. In this research, because the bundle of cilia is considered instead of a single cilium, Stokes–Brinkman equations in a macroscopic scale are employed to find the velocity of the PCL fluid. When the cilia are perpendicular to the horizontal plane, the PCL consists of only the cilia. The inclination of cilia (cilia make an angle θ θ < 90 ∘ to the horizontal plane) results in two different domains in the PCL, the regions comprising and not comprising cilia. The main objective of this study is to determine the appropriate boundary conditions of the velocity between these two regions where the PCL fluid is moved by self-propelled cilia rather than the pressure gradient. A matched asymptotic expansion method is applied to the Stokes–Brinkman equations to determine the constraints. Two boundary conditions at the interface are obtained and the velocity of the PCL fluid at the top of PCL can be used to be the boundary conditions at the bottom of the mucus layer to determine the velocity of mucus. This model can help engineers to build devices to treat patients who have problem with the respiratory system. Applications include modeling fluid flow through filters such as engine filter and rice fields.