Modeling problems in mucus viscoelasticity and mucociliary clearance

From the common cold and allergies to severe chronic and acute respiratory impairments, the function of the body\u27s mucociliary clearance mechanism plays a primary defense role. Mucus demonstrates numerous non-Newtonian behaviors which set it apart from viscous fluids. Among them: Bingham plastic behavior, shear-thinning, and elasticity on short time scales due to relaxation time. Experimental evidence suggests that certain rheologies promote effective transport. In an effort to reveal the mechanisms controlling transport, models are developed. Firstly, a steady state model which idealizes the mucus as a rigid body is created in order to bring together disparate bodies of experimental work from the literature. The force balance reveals that the force cilia are capable of exerting cannot be related, simply, to the velocity of mucus. That is, only a fraction of the force cilia are capable of exerting is required to steadily transport mucus at the velocities observed experimentally. Likewise, the velocities estimated by this model when cilia force is the input are overestimated by one to two orders of magnitude. This incongruity formally motivates the inclusion of one of mucus\u27s rheological behaviors, stress relaxation. The first viscoelastic problem considered is the response of the linear Maxwell fluid to an oscillating plate. Though a problem commonly discussed in textbooks on theoretical viscoelasticity, the complete analytical solutions are not provided. Here, solutions are found and graphed in terms of the phase and amplitude of the velocity field resultant from the oscillations of the plate; all derivations are shown in their entirety. The effects of stress relaxation (sometimes referred to as memory) and inertia on phase and amplitude are shown to have frequency dependence. Furthermore, it is shown that oscillatory shear perturbations to a viscoelastic Maxwell fluid always travel further and faster away from the source as Deborah number (a dimensionless parameter governing the importance of viscoelastic forces, De=0 corresponds to a Newtonian fluid) is increased. The limitation of the linear Maxwell fluid is illustrated by attempting to apply the constitutive equation to a thin film flow problem. It is found that the stress field of the solution only differs from the viscous case if the boundary conditions are transient; that is, the constitutive equation cannot account for the changes in stress that occur over space. The time derivative must be replaced by a Convected Derivative to achieve the proper Lagrangian to Eulerian coordinate transformation and is considered in a final set of problems. Three problems were completed using the Upper Convected Maxwell model for viscoelasticity. The first considers a purely unidirectional shear flow which, unlike a viscous fluid, possesses tensile stresses along streamlines. The model posits that these additional stresses are essential for transport by allowing regions which are actively sheared, to hold up inactive regions. A novel relationship between applied stress and relaxation time is developed; the model shows that increasing the relaxation time of mucus decreases the amount of stress that must be imparted by cilia. In the second two problems, the UCM equations are simplified via a perturbation series expansion for small Weissenberg number (also a dimensionless group governing the importance of viscoelastic forces). This technique allows the analytically solvable viscous (also referred to as the unperturbed or order one) solutions to be used to estimate the impact of small amounts of stress memory. It is found that elasticity increases the developing region of a viscous flow; all stress components are convected downstream due to flow memory. Likewise, in the sinusoidally varying stress case, the velocity field is always shifted further away from the phase of the applied stress as viscoelastic forces are increased. It is also found that the departure from the viscous solution is dramatically reduced if the stress distribution is moving at the same velocity as the mucal flow. This shows the benefit of an antiplectic wave speed (the physiologically relevant case in which the phase of the cilial beat is moving opposite to transport) as there is no danger that these two can be in phase with one another. Model restrictions prevent quantitative gauges of transport efficiency as a function of metachronal wave parameters and relaxation time to be made. Several additional problems are proposed to address unanswered modeling questions and experimental solutions for the lack of rheological data on tracheal mucus are suggested