Motion of a deformable capsule through a hyperbolic constriction

Abstract

A model for the low-Reynolds-number flow of a capsule through a constriction is developed for either constant-flow-rate or constant-pressure-drop conditions. Such a model is necessary to infer quantitative information on the intrinsic properties of capsules from filtration experiments conducted on a dilute suspension of such particles. A spherical capsule, surrounded by an infinitely thin Mooney-Rivlin membrane, is suspended on the axis of a hyperbolic constriction. This configuration is fully axisymmetric and allows the entry and exit phenomena through the pore to be modelled. An integral formulation of the Stokes equations describing the flow in the internal and external domains is developed. It provides a representation of the velocity at any location in the flow as a function of the unknown forces exerted by the boundaries on the fluids. The problem is solved by a collocation technique in the case where the internal and external viscosities are equal. Microscopic quantities (instantaneous geometry, centre of mass velocity, elastic tensions in the membrane) as well as macroscopic quantities (entry time, additional pressure drop or flow rate reduction) are predicted as a function of the capsule intrinsic properties and flow characteristics. The results obtained for a capsule whose initial diameter is larger than that of the constriction throat show that the maximum energy expenditure occurs when the particle centre of mass is still upstream of the throat (typically 1 diameter away), and is thus due to the entry process. For large enough or rigid enough capsules, the model predicts entrance or exit plugging, in agreement with experimental observations. It is then possible to correlate the variation of the pore hydraulic resistance to the flow capillary number (ratio of viscous to elastic forces) and to the size ratio between the pore and the capsule.