Orbital instability and chaos in the Stokes flow between two eccentric cylinders

Abstract

The motion of fluid particles due to the slow flow between two eccentric cylinders rotating alternately is examined experimentally and numerically. In ‘return experiments’ composed of alternate rotations of the cylinders by N periods and their time reversal, the dye starting from one region almost returns to its initial position even for large N, whereas the deviation of the dye starting from the other region from its initial position is large and rapidly increases with N. These two regions correspond to the regular and chaotic regions in the numerically computed Poincaré plot for the alternate rotations of the cylinders. These results suggest the significance of orbital instability in the chaotic region in the experiments with unavoidable inperfections. A part of the experimental results can be explained qualitatively using a loccl Lyapunov exponent (L.L.E.) for finite evolution time. The importance of the stagnation point of the flow due to the rotation of a cylinder in the orbital instability is also suggested using the L.L.E.