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Mathematical and Computational Models of Fluctuating Vesicles in Time-Varying Flows


Modeling vesicle dynamics involves a complicated moving boundary problem while the shapes of vesicles are determined dynamically from a balance between interfacial forces and fluid stresses. In this thesis, we investigate the dynamics of a two-dimensional fluctuating vesicle in a viscous fluid.

Firstly we develop a two-dimensional stochastic immersed boundary method (SIBM) and analyze thermal fluctuations by matching the numerical results with a theoretical solution. Then we apply the SIBM with fitted thermal fluctuations to study the long term dynamics of an impermeable vesicle in a periodically time-reversed flow. The wrinkling process contains three stages. In the first stage, high-order modes are excited by the negative surface tension and wrinkles appear. In the second stage, low Fourier modes increase, the high-order wrinkles decay, and the shapes of vesicles keep relatively stable. In the last stage, the second Fourier mode grows and dominates. The shapes of the vesicle are ellipse-like with inclination angle θ ≈ 45◦. Then by performing an asymptotic linear analysis of a quasi-circular vesicle, we derive and solve the deterministic and stochastic equations for the motion of membrane interface numerically. The linear theory also indicates this three stage process.

Finally, we investigate the nonlinear wrinkling dynamics of a permeable vesicle using an extension of the SIBM. We observe the vesicle shrinkage and the wrinkles on the membrane caused by a large osmosis pressure. We extend the linear theory to account for permeability and find a good agreement between linear and fully nonlinear vesicle dynamics.

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