Abstract
A fast and accurate numerical technique is developed for solving the biharmonic equation in a multiply connected domain, in two dimensions. We apply the technique to the computation of slow viscous flow (Stokes flow) driven by multiple stirring rods. Previously, the technique has been restricted to stirring rods of circular cross section; we show here how the prior method fails for noncircular rods and how it may be adapted to accommodate general rod cross sections, provided only that for each there exists a conformal mapping to a circle. Corresponding simulations of the flow are described, and their stirring properties and energy requirements are discussed briefly. In particular the method allows an accurate calculation of the flow when flat paddles are used to stir a fluid chaotically.
Highlights
In this paper we investigate a two-dimensional model for a highly viscous Newtonian fluid, stirred in a vat by the motion of one or more stirring rods: a “batch stirring device” ͑or BSD.1 As one might expect from everyday experience, a fluid can readily be stirred effectively in such a device,1–5 even in the Stokes flow regime, as examined here
We emphasize that the fluid motion satisfies boundary conditions appropriate to the noslip condition on physical rods, and that, while expressions such as Eq ͑12͒ involve singularities, none of these lie in the flow domain itself
The desire to simulate the trajectories of fluid particles in chaotic Stokes flows necessitates a accurate solution for the velocity field
Summary
In this paper we investigate a two-dimensional model for a highly viscous Newtonian fluid, stirred in a vat by the motion of one or more stirring rods: a “batch stirring device” ͑or BSD. As one might expect from everyday experience, a fluid can readily be stirred effectively in such a device, even in the Stokes flow regime, as examined here. A attractive feature of the BSD, as noted by Boyland, Aref, and Stremler, is that it allows one to use a multiplicity of stirring rods. Reliable simulation of the motion of fluid particles in the BSD necessitates a knowledge of the corresponding velocity field to high accuracy. This poses a particular numerical challenge, even though the governing equation is thelinearbiharmonic equation ٌ4 = 0 for the stream function , because the flow domain geometry is complicated and time dependent.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
