Pulsatile parallel flow of air and a viscoelastic fluid with multiple characteristic times. An application to mucus in the trachea and the frequency of cough.

We study the dynamics of a binary fluid, where the two fluids are flowing parallel to each other in a cylindrical geometry, and driven by a pulsatile pressure gradient. One of the fluids is a low viscosity Newtonian fluid, the other one is viscoelastic. In order to be able to apply the model to different biofluids, we consider that the viscoelastic fluid has several characteristic times. We characterize the dynamics of the fluids as generalized Darcy's laws, with linear response functions to pulsatile pressure gradients, whose parameters are coupled for both fluids through the fluid-fluid boundary conditions. We apply our results to the dynamics of mucus and air in the trachea and find that the frequency that allows for a larger movement of the mucus, coincides with the experimental frequency of cough. This allows us to propose a plausible explanation for the frequency of cough in healthy individuals, a mechanical process to expel noxious substances from the respiratory system.