Hydrodynamic instability of a suspension of spherical particles through a branching network of circular tubes

A numerical method is presented for computing the unsteady flow of a monodisperse suspension of spherical particles through a branching network of circular tubes. The particle motion and interparticle spacing in each tube are computed by integrating in time a one-dimensional convection equation using a finite-difference method. The particle fraction entering a descendent tube at a divergent bifurcation is related to the local and instantaneous flow rates through a partitioning law proposed by Klitzman and Johnson involving a dimensionless exponent, q ≥ 1. When q = 1, the particle stream is divided in proportion to the flow rate; as q → ∞, the particles are channeled into the tube with the highest flow rate. The simulations reveal that when the network involves two or more generations, a supercritical Hopf bifurcation occurs at a critical value of q, yielding spontaneous, self-sustained oscillations in the segment flow rates, pressure drop across the network, and particle spacing in each tube. A phase diagram is presented to establish conditions for unsteady flow. As found recently for blood flow in a capillary network, oscillations can be induced for a given network tree order by decreasing the ratio of the tube diameter from one generation to the next or by decreasing the diameter of the terminal segments. The instability is more prominent for rigid than deformable particles, such as drops, bubbles, and cells, due to strong lubrication forces between the tightly fitting particles and tube walls. Variations in the local particle spacing, therefore, have a more significant effect on the effective viscosity of the suspension in each tube and pressure drop required to drive a specified flow rate.