Dispersion of solute released from a sphere flowing in a microchannel

Abstract

A solute is released from the surface of a sphere flowing freely in a cylindrical channel mimicking a modern drug delivery agent in a blood vessel. The solute then disperses by the combined action of advection and diffusion. We consider reflecting boundary conditions on the sphere and absorbing boundary conditions on the channel surface mimicking a biochemical reaction between the drug and endothelial cells on the vessel surface. The drug is released either instantaneously or continuously in time. The two key observables are the mean residence time in the flow before the drug is absorbed and the width over which it is spread on the vessel surface upon reaction. We numerically solve the Fokker–Planck equation for the time-dependent substance concentration combined with an analytical solution of the flow field. As expected, we find that the presence of the sphere leads to a substantial reduction in mean residence time and reaction width. Surprisingly, however, even in the limit of very large Péclet numbers (high velocities) the sphere-free case is not generally recovered. This observation can be attributed mainly to the small, but non-negligible radial flow component induced by the moving sphere. We further identify a strong influence of the release position which sharply separates two qualitatively different regimes. If the release position is between $\unicode[STIX]{x1D703}_{0}=0$ (front) and a critical $\unicode[STIX]{x1D703}_{c}$ the substance is quickly advected away from the sphere and its overall behaviour is similar to free diffusion in an empty channel. For release between $\unicode[STIX]{x1D703}_{c}$ and $\unicode[STIX]{x1D703}_{0}=\unicode[STIX]{x03C0}$ (tail), on the other hand, the substance is pushed towards the sphere leading to behaviour reminiscent of confined diffusion between two infinitely long cylinders. The critical position $\unicode[STIX]{x1D703}_{c}$ is generally smaller than $\unicode[STIX]{x03C0}/2$ which would correspond to an equatorial release position.