Mathematical modelling of cilia-driven transport of biological fluids

Cilia-driven flow occurs in the airway surface liquid, in the female and male reproductive tracts and enables symmetry-breaking in the embryonic node. Viscoelastic rheology is found in healthy states in some systems, whereas in others may characterise disease, motivating the development of mathematical models that take this effect into account. We derive the fundamental solution for linear viscoelastic flow, which is subsequently used as a basis for slender-body theory. Our numerical algorithm allows efficient computation of three-dimensional time-dependent flow, bending moments, power and particle transport. We apply the model to the large-amplitude motion of a single cilium in a linear Maxwell liquid. A relatively short relaxation time of just 0.032 times the beat period significantly reduces forces, bending moments, power and particle transport, the last variable exhibiting exponential decay with relaxation time. A test particle is propelled approximately one-fifth as quickly along the direction of cilia beating for scaled relaxation time 0.032 as in the Newtonian case, and mean volume flow is abolished, emphasizing the sensitivity of cilia function to fluid rheology. These results may have implications for flow in the airways, where the transition from Newtonian to viscoelastic rheology in the peri-ciliary fluid may reduce clearance.