Numerical Homogenization on Approach for Stokesian Suspensions
In this technical report we investigate efficient methods for numerical simulation of active suspensions. The prototypical system is a suspension of swimming bacteria in a Newtonian fluid. Rheological and other macroscopic properties of such suspensions can differ dramatically from the same properties of the suspending fluid alone or of suspensions of similar but inactive particles. Elongated bacteria, such as E. coli or B. subtilis, swim along their principal axis, propelling themselves with the help of flagella, attached at the anterior of the organism and pushing it forward in the manner of a propeller. They interact hydrodynamically with the surrounding fluid and, because of their asymmetrical shape, have the propensity to align with the local flow. This, along with the dipolar nature of bacteria (the two forces a bacterium exerts on a fluid - one due to self-propulsion and the other opposing drag - have equal magnitude and point in opposite directions), causes nearby bacteria to tend to align, resulting in a intermittent local ordering on the mesoscopic scale, which is between the microscopic scale of an individual bacterium and the macroscopic scale of the suspension (e.g., its container). The local ordering is sometimes called a collective mode or collective swimming. Thanks to self-propulsion, collective modes inject momentum into the fluid in a coherent way. This enhances the local strain rate without changing the macroscopic stress applied at the boundary of the container. The macroscopic effective viscosity of the suspension is defined roughly as the ratio of the applied stress to the bulk strain rate. If local alignment and therefore local strain-rate enhancement, are significant, the effective viscosity can be appreciably lower than that of the corresponding passive suspension or even of the surrounding fluid alone. Indeed, a sevenfold decrease in the effective viscosity was observed in experiments with B. subtilis. More generally, local collective swimming resulting from bacterial alignment can significantly alter other macroscopic properties of the suspension, such as the oxygen diffusivity and mixing rates. In order to understand the unique macroscopic properties of active suspensions the connection between microscopic swimming and alignment dynamics and the mesoscopic pattern formation must be clarified. This is difficult to do analytically in the fully general setting of moderately dense suspensions, because of the large number of bacteria involved (approx. 10 10 cm -3 in experiments) and the complex, time-dependent geometry of the system. Many reduced analytical models of bacterial have been proposed, but all of them require validation. While comparison with experiment is the ultimate test of a model's fidelity, it is difficult to conduct experiments matched to these models assumptions. Numerical simulation of the microscopic dynamics is an acceptable substitute, but it runs into the problem of having to discretize the fluid domain with a fine-grained boundary (the bacteria) and update the discretization as the domain evolves (bacteria move). This leads to a prohibitively high number of degrees of freedom and prohibitively high setup costs per timestep of simulation. In this technical report we propose numerical methods designed to alleviate these two difficulties. We indicate how to (1) construct an optimal discretization in terms of the number of degrees of freedom per digit of accuracy and (2) optimally update the discretization as the simulation evolves. The technical tool here is the derivation of rigorous error bounds on the error in the numerical solution when using our proposed discretization at the initial time as well as after a given elapsed simulation time. These error bounds should guide the construction of practical discretization schemes and update strategies. Our initial construction is carried out by using a theoretically convenient, but practically prohibitive spectral basis, which is a Galerkin basis of functions with global support. At the end of this report we propose localization techniques while maintaining acceptable error bounds. No numerical experiments were conducted as part of this study, but we envision that we may undertake such studies and further development of the method, jointly or individually.