In this dissertation, we study the behavior of microscopic organisms utilizing lateral and spiral bending waves to swim in a fluid. More specifically, spermatozoa encounter different fluid environments filled with mucus, cells, hormones, and other large proteins. These networks of proteins and cells are assumed to be stationary and of low volume fraction. They act as friction, possibly preventing or enhancing forward progression of the swimmers. The flow in the medium is described as a viscous fluid with a resistance term known as a Brinkman fluid. It depends on the Darcy permeability parameter affecting the swimming patterns of the flagella. To further understand these effects we study the asymptotic swimming speeds of an infinite-length swimmer propagating planar or spiral bending waves in a Brinkman fluid. We find that, up to the second order expansion, the swimming speeds are enhanced as the resistance increases. The work to maintain the planar bending and the torque exerted on the fluid are also examined. The Stokes limits of the swimming speeds, the work and the torque are recovered as resistance goes to zero. The analytical solutions are compared with numerical results of finite-length swimmers obtained from the method of Regularized Brinkmanlets (MRB). The study gives insight on the effects of the permeability, the length and the radius of the cylinder on the performance of the swimmers. In addition, we develop a grid-free numerical method to study the bend and twist of an elastic rod immersed in a Brinkman fluid. The rod is discretized using a Kirchhoff Rod (KR) model. The linear and angular velocity of the rod are derived using the MRB. The method is validated through a couple of benchmark examples including the dynamics of an elastic rod, and the planar bending of a flagellum in a Brinkman fluid. The studies show how the permeability and stiffness coefficients affect the waveforms, the energy, and the swimming speeds of the swimmers. Also, the beating pattern of the spermatozoa flagellum depends on the intracellular concentrations of calcium ([Ca2+]). An increase of [Ca2+] is linked to hyperactivated motility. This is characterized by highly asymmetrical beating, which allows spermatozoa to reach the oocyte (egg) or navigate along the female reproductive tract. Here, we couple the [Ca2+] to the bending model of a swimmer in a Brinkman fluid. This computational framework is used to understand how internal flagellar [Ca2+] and fluid resistance in a Brinkman fluid alter swimming trajectories and flagellar bending.
Worcester Polytechnic Institute
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Ho, N. (2016). Swimming Filaments in a Viscous Fluid with Resistance. Retrieved from https://digitalcommons.wpi.edu/etd-dissertations/211
Brinkman equations, Asymptotic Analysis, Numerical Methods