Brown, Cora2021-01-252021-01-252020-11https://hdl.handle.net/11299/218052University of Minnesota Ph.D. dissertation. November 2020. Major: Mathematics. Advisor: Maria-Carme Calderer. 1 computer file (PDF); 166 pages.Experiments in living liquid crystals, the mixture of bacteria and an underlying nematic liquid crystal, present remarkable features, including the ability of bacteria to transport cargoes and the formation of patterns in the system to direct the motion of the bacteria. We present a fully variational model to describe such systems, and explore various geometries related to the experiments. The continuum mixture model derives elements from several mathematical sources, including the nematic flow theory of Ericksen and Leslie, the mixture theory of Flory and Huggins, and the anisotropic active theory of Ramaswamy. These elements are combined through the Principle of Minimum Energy dissipation into a model which accurately reproduces experimental results.The energy is provided to the system by oxygen externally supplied to the aerobic bacteria. We consider two types of confined planar flow: those in a thin-channel geometry, of shear type, and those in which the confinement occurs in the direction perpendicular to the plane. In the thin-gap problems, we determine several types of patterned solutions including the zero solution. We perform a stability analysis on the zero solution resulting in an eigenvalue problem which relates the stability of the problem to the amount of activity. This is significant in that it shows that motion is triggered by the activity source rather than the mechanical input required to trigger flow of passive liquid crystals. The threshold value of activity required for instability is related to the viscosity and to the entropic energy, proving that the zero solution is stable until the amount of activity overtakes the tendency of the liquid crystal to mix with the bacteria. In the second geometry, representing a slice through the experimental cell, we use computational methods to accurately reproduce experimental results regarding the relationship between the gap-width, the concentration of bacteria, the activity number, and the resulting period of director distortion. In particular, we show that at some range of the activity parameter, the flow is structured into channel domains for the traveling bacteria, with direction reversal across the domain boundary, and change in the angular alignment of the bacteria. The frequency of the channel pattern is found to increase with the value of the activity parameter. Finally, we introduce a new unit length relaxation approach to the Ericksen-Leslie system that overcomes the physical challenges presented by the standard, variable length setting. We conclude the dissertation pointing to future directions for analyzing the well-posedness of such mixture models, as well as exciting applications of new machine learning techniques in the field of defect and pattern detection in liquid crystals and living liquid crystals. The model we present acts as a foundational model for any system in which two or more anisotropic components are present, enabling future work in any such system, especially in the context of biology.enThesis or Dissertation