The inertialess motion of a vesicle tightly fitting inside a tube is investigated theoretically. The lipid bilayer membrane is modeled as a two-dimensional, incompressible fluid that admits resistance to bending deformation. Assuming symmetry about the tube axis, lubrication theory is used to determine the steady-state behavior over a range of shapes, sizes, flow rates. Of particular interest are the excess speed of the vesicle over the mean velocity of the far-field Poiseuille flow, the tension development in the bilayer, and the apparent viscosity of the fluid, the latter of which is determined from the relationship between the pressure drop across the vesicle and the flow rate. The deformed shape of the vesicles is determined at different flow rates; in particular, a branch of unstable shapes, not accessible by simulation techniques, is produced by the lubrication theory.

In very weak flows, the vesicle behaves like a rigid particle, and the apparent viscosity remains approximately constant with flow rate. The constraints of area- and volume-incompressibility result in deformed shapes not observed for drops or elastic capsules. Vesicles that are nearly spherical experience little deformation, with a pressure distribution that mirrors that of weakly elastic, solid, spherical particles. Increasing the flow strength results in enhanced deformation, and, consequently, nonlinear dependence of both the excess speed of the vesicle and the apparent viscosity on the flow rate. This effect is pronounced for large, prolate vesicles, i.e., vesicles that are far from spherical and fit snugly inside the tube. Furthermore, the meridionally averaged excess pressure outside the vesicle decreases with increasing flow rate, and falls more dramatically for nearly spherical vesicles. The theoretical predictions are complemented by results from direct numerical simulation of the vesicle motion using a Stokes-flow boundary integral equation method.

The findings of this work may be important for understanding the motion of vesicles through constrictions and other highly confined flow geometries. Suggestions for possible experiments, which are scarce in the current literature, are presented.

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