We derive the effective viscosity of dilute suspensions of swimming bacteria from the microscopic details of the interaction of an elongated body with the background flow. An individual bacterium propels itself forward by rotating its flagella and reorients itself randomly by tumbling. Due to the bacterium's asymmetric shape, interactions with a prescribed generic (such as planar shear or straining) background flow cause the bacteria to preferentially align in directions in which self-propulsion produces a significant reduction in the effective viscosity, in agreement with recent experiments on suspensions of Bacillus subtilis.

We have developed a new programming framework, called Sieve, to support parallel numerical PDE 1 algorithms operating over distributed meshes. We have also developed a reference implementation of Sieve in C++ as a library of generic algorithms operating on distributed containers conforming to the Sieve interface. Sieve makes instances of the incidence relation, or arrows, the conceptual first-class objects represented in the containers. Further, generic algorithms acting on this arrow container are systematically used to provide natural geometric operations on the topology and also, through duality, on the data. Finally, coverings and duality are used to encode not only individual meshes, but all types of hierarchies underlying PDE data structures, including multigrid and mesh partitions.In order to demonstrate the usefulness of the framework, we show how the mesh partition data can be represented and manipulated using the same fundamental mechanisms used to represent meshes. We 1 Partial differential equation(s). present the complete description of an algorithm to encode a mesh partition and then distribute a mesh, which is independent of the mesh dimension, element shape, or embedding. Moreover, data associated with the mesh can be similarly distributed with exactly the same algorithm. The use of a high level of abstraction within the Sieve leads to several benefits in terms of code reuse, simplicity, and extensibility. We discuss these benefits and compare our approach to other existing mesh libraries.

Suspensions of self-propelled particles are studied in the framework of two-dimensional (2D) Stokesean hydrodynamics. A formula is obtained for the effective viscosity of such suspensions in the limit of small concentrations. This formula includes the two terms that are found in the 2D version of Einstein's classical result for passive suspensions. To this, the main result of the paper is added, an additional term due to self-propulsion which depends on the physical and geometric properties of the active suspension. This term explains the experimental observation of a decrease in effective viscosity in active suspensions.

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