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Metachronal propulsion of non-Newtonian viscoelastic mucus in an axisymmetric tube with ciliated walls

Cilia-induced flow of viscoelastic mucus through an idealized two-dimensional model of the human trachea is presented. The cilia motion is simulated by a metachronal wave pattern which enables the mobilization of highly viscous mucus even at nonzero Reynolds numbers. The viscoelastic mucus is analyzed with the upper convected Maxwell viscoelastic formulation which features a relaxation time and accurately captures normal stress generation in shear flows. The governing equations are transformed from fixed to wave (laboratory) frame with appropriate variables and resulting differential equations are perturbed about wave number. The trachea is treated as an axisymmetric ciliated tube. Radial and axial distributions in axial velocity are calculated via the regular perturbation method and pressure rise is computed with numerical integration using symbolic software MATHEMATICA‘TM’. The influence of selected parameters which is cilia length, and Maxwell viscoelastic material parameter i.e. relaxation time for prescribed values of wave number are visualized graphically. Pressure rise is observed to increase considerably with elevation in both cilia length and relaxation time whereas the axial velocity is markedly decelerated. The simulations provide some insight into viscous-dominated cilia propulsion of rheological mucus and also serve as a benchmark for more advanced modeling.