| This dissertation explores the bulk (volume averaged) properties of suspensions of microswimmers in a fluid. A microswimmer is a microscopic object that propels itself through a fluid. A common example of a microswimmer is a bacterium, such as Bacillus subtilis. Our particular interest is the bulk rheological properties of suspensions of bacteria -- that is, studying how such a suspension deforms under the application of an external force. In the simplest case, the rheology of a fluid can be described by a scalar effective viscosity. The goal of this dissertation is to find explicit formulae for the effective viscosity in terms of known geometric and physical parameters characterizing bacteria and use them to explain experimental observations. Throughout the dissertation, we consider bacterial suspensions in the dilute limit, where bacteria are assumed to be so far apart that interactions between them are negligible. This simplifies calculations significantly and is the regime in which the most striking experimental results have been observed.;We first study suspensions of self-propelled particles using a two-dimensional (2D) Partial Differential Equation (PDE) model. A bacterium is modeled as a disk in 2D with self-propulsion provided by a point force in the fluid. A formula is obtained for the effective viscosity of such suspensions in the dilute limit. This formula includes the two terms that are found in the 2D version of Einstein's classical result for a passive suspension of spheres. To this, our main contribution is added, an additional term due to self-propulsion which depends on the physical and geometric properties of the suspension. This work demonstrates how bacterial self-propulsion can alter the viscosity of a fluid and highlights the importance of bacterial orientation.;Next, we present a more realistic PDE model for dilute suspensions of swimming bacteria in a three-dimensional fluid. In this work, a bacterium is modeled as a prolate spheroid with self-propulsion once more provided by a point force. Furthermore, the bacterium is subject to a random torque in order to model tumbling (random reorientation). This model is used to calculate the effective viscosity of the suspension from the microscopic details of the interaction of an elongated body with a prescribed background flow, once more in the dilute limit. Due to a bacterium's asymmetric shape (in particular, unlike the case of rotationally symmetric bacteria used in the first model), interactions with generic planar background flows cause the bacterium to preferentially align in certain directions. Due to the random torque, the steady-state distribution of orientations is unique for a given background flow. Under this distribution of orientations, self-propulsion produces a reduction in the effective viscosity. For sufficiently weak background flows, the effect of self-propulsion on the effective viscosity dominates all other contributions, leading to an effective viscosity of the suspension that is lower than the viscosity of the ambient fluid. This is in qualitative agreement with recent experiments on suspensions of Bacillus subtilis.;Finally, we present a method that can be used to rigorously justify our effective viscosity formulae. In particular, we present a mathematical proof of Einstein's formula for the effective viscosity of a dilute suspension of spheres when the spheres are centered on the vertices of a cubic lattice. This proof admits a generalization to other particle shapes and the inclusion of self-propulsion. We keep the size of the container finite in the dilute limit and consider boundary effects. Einstein's formula is recovered as a first-order asymptotic expansion of the effective viscosity in the volume fraction o. To rigorously justify this expansion, we obtain an explicit upper and lower bound on the effective viscosity and show that they match to order o3/2. |
