Microhydrodynamics of vesicles in channel flow
Keywords: Chemical Engineering.

The hydrodynamic resistance of fluid vesicles suspended in channel flow is studied theoretically as a model system for single-cell microfluidic measurements. When freely suspended in fluid flow, particles (e.g., rigid and soft particles, droplets, vesicles, and biological cells) act as ``resistors'' in a fluidic circuit. The relationship between the hydrodynamic resistance generated by even a single particle is complicated when the particle can deform, due to the nonlinear coupling between fluid flow and shape deformation. This coupling is markedly more complex for vesicles than for droplets, due to the diverse shape space promoted by the constraint that the surface of a vesicle remains incompressible. In this thesis, we tackle the challenging problem of determining the motion of vesicles -- both single vesicles and vesicle trains -- in channel flow using a variety of tools, drawing from perturbation techniques, coarse-grained theory, and direct numerical simulations. The problem investigated in this thesis can be defined by three regimes of vesicle confinement. Under high confinement, the vesicle is lubricated by a very thin film and the flow is well described by lubrication theory. We investigate this regime for circular Poiseuille flow by means of narrow-gap analysis, wherein the clearance is defined as a small parameter and a formal perturbation series is constructed. We find that the vesicle length plays a crucial role in determining the scaling of the hydrodynamic figures of merit -- namely, the relative velocity of the vesicle compared to the mean flow and the extra pressure drop required to push it through the channel -- with respect to the clearance. Membrane bending elasticity is included as a parameter in the theoretical result, and is shown to break the linear dependence of the vesicle velocity and extra pressure drop on the applied mean flow. Under moderate confinement, the lack of a separation of length scales precludes the use of narrow-gap analysis. One must resort to direct numerical calculations, which have proven to be challenging despite the rapid evolution of modern computers. We tackle this challenging regime for both circular tubes and square channels using a combination of boundary element simulations and lubrication theory. Our numerical results were verified using the aforementioned narrow-gap analysis. It is shown that relaxing the confinement greatly expands the variety of accessible vesicle shape configurations. Significantly, as the vesicle aspect ratio increases, the centerline shape becomes unstable and breaks symmetry. At high confinement, this effect is dampened. Bending elasticity qualitatively changes both the vesicle shape configurations and hydrodynamical integral quantities, with increasing sensitivity as the vesicle becomes less confined. For square channels, an interesting new phenomenon is observed wherein the vesicle can move slower than the mean fluid velocity. This observation is validated by microfluidic experiments, in collaboration with our colleagues at Texas Tech University. Finally, under low confinement the disturbance produced by a vesicle is relatively weak. It is challenging to apply direct numerical calculations to this regime due to the high degree of resolution needed to compute the extra pressure drop. Numerically, this arises as a limitation in discretization of the channel wall, because the extra pressure drop is determined from an integral of the wall surface tractions. If these tractions are weak, then a high density of surface elements is required in order to accurately compute the force integral. An alternative approach, which has become popular in the literature, is to assume that the vesicle is in an unbounded flow to leading order. It is then possible to determine the theoretical limit of the relative velocity and extra pressure drop at low confinement, by use of the reciprocal theorem. We tackle this problem for a restricted subset of vesicle shapes (small, quasi-spherical vesicles) and show how the extra pressure drop couples to the shape deformation in this regime. In doing so, we also generalize previous theories of vesicle motion, which have hitherto focused on specific flow types, to an arbitrary, quadratic flow field.