We study the propulsion matrix of bacterial flagella numerically using slender body theory and the regularized Stokeslet method in a biologically relevant parameter regime. All three independent elements of the matrix are measured by computing propulsive force and torque generated by a rotating flagellum, and the drag force on a translating flagellum. Numerical results are compared with the predictions of resistive force theory, which is often used to interpret micro-organism propulsion. Neglecting hydrodynamic interactions between different parts of a flagellum in resistive force theory leads to both qualitative and quantitative discrepancies between the theoretical prediction of resistive force theory and the numerical results. We improve the original theory by empirically incorporating the effects of hydrodynamic interactions and propose new expressions for propulsive matrix elements that are accurate over the parameter regime explored.
We systematically explore near equilibrium, flow-driven, and flow-activity coupled dynamics of polar active liquid crystals using a continuum model. Firstly, we re-derive the hydrodynamic model to ensure the thermodynamic laws are obeyed and elastic stresses and forces are consistently accounted. We then carry out a linear stability analysis about constant steady states to study near equilibrium dynamics around the steady states, revealing long-wave instability inherent in this model system and how active parameters in the model affect the instability. We then study model predictions for onedimensional (1D) spatial-temporal structures of active liquid crystals in a channel subject to physical boundary conditions. We discuss the model prediction in two selected regimes, one is the viscous stress dominated regime, also known as the flow-driven regime, while the other is the full regime, in which all active mechanisms are included. In the viscous stress dominated regime, the polarity vector is driven by the prescribed flow field. Dynamics depend sensitively on the physical boundary condition and the type of the driven flow field. Bulk-dominated temporal periodic states and spatially homogeneous states are possible under weak anchoring conditions while spatially inhomogeneous states exist under strong anchoring conditions. In the full model, flow-orientation interaction generates a host of planar as well as out-of-plane spatial-temporal structures related to the spontaneous flows due to the molecular self-propelled motion. These results provide contact with the recent literature on active nematic suspensions. In addition, symmetry breaking patterns emerge as the additional active viscous stress due to the polarity vector is included in the force balance. The inertia effect is found to limit the long-time survival of spatial structures to those with small wave numbers, i.e., an asymptotic coarsening to long wave structures. A rich set of mechanisms for generating and limiting the flow structures as well as the spatial-temporal structures predicted by the model are displayed.