Viscous motion of disk-shaped particles through parallel-sided channels with near-minimal widths

Abstract

The motion of a rigid disk-shaped particle with rounded edges, which fits closely in the space between two parallel flat plates, and which is suspended in a viscous fluid subject to an imposed pressure gradient, is analysed. This problem is relevant to the squeezing of red blood cells through narrow slot-like channels which are found in certain tissues. Mammalian red cells, although highly flexible, conserve volume and surface area as they deform. Consequently, a red cell cannot pass intact through a channel which is narrower than some minimum width. In channels that are just wide enough to permit cell passage, the cell is deformed into its ‘critical’ shape: a disk with rounded edges. In this paper, the fluid mechanical aspects of such motions are considered, and the particle is assumed to be rigid with the critical shape. The channel cross-section is assumed to be rectangular. The flow of the suspending fluid is described using lubrication theory. Use of lubrication theory is justified by considering the motion of a circular cylinder between parallel plates. For disk-shaped particles, approximate solutions are obtained by applying lubrication equations throughout the flow domain. In the region beyond the particle, this is equivalent to assuming a Hele-Shaw flow. More accurate solutions are obtained by including effects of boundary layers around the particle and at the sides of the channel. Pressure distributions and particle velocities are computed as functions of geometrical parameters, and it is shown that the particle may move faster or slower than the mean velocity of the surrounding fluid, depending on the channel dimensions.