Microscale Shear Stress Induced by Slow Flow in Fibrous Media

Tendon tissue engineering aims to grow functional tissue in vitro. One approach is to grow tendon cells on fibres that can be forced and supplied with nutrients by perfusing media. Motivated by understanding the relation between forcing and shear stress experienced by the cells, we present solutions to homogenised equations describing the interaction of slow Newtonian flow and aligned strings. Derived using multiscale asymptotics, fluid flow in the homogenised model is governed by a modified Darcy Law, coupled to string displacement via a homogenised force balance. We consider analytical solutions for the fluid flow induced by imposed oscillations of the string ends and an imposed flux. For oscillation of the string ends, we present scaling laws for the average shear stress in the limit of high and low frequency. When an inlet flux drives fluid flow, we find the average microscale shear stress for high porosity is predominantly set by parameters resulting from solution to the microscale problem obtained in the derivation of the homogenised model.