Lagrangian Chaos in the Stokes Flow Between Two Eccentric Rotating Cylinders

Abstract

Chaotic motion of the fluid particles in the Stokes flow between two eccentric cylinders rotating alternately is investigated numerically, analytically and experimentally. We examine the dependence of the motion of the fluid particles on the eccentricity ε, focusing on an equilibrium point of the Poincaré plot. When the bifurcation of the equilibrium point from the elliptic to the hyperbolic type occurs at ε = εb, the area of the chaotic region takes a maximum around εb. The results from the perturbation analysis show good agreement with the numerical results. The orbital instability of the motion of the fluid particles is also investigated experimentally. The orbital instability is visualized by injected dye in the "return experiment", in which the two cylinders are rotated alternately by N periods in the first half, and then rotated in its time reversal way for N periods in the second half. The dye starting from the regular region of the numerically computed Poincaré plot of particle positions after every period returns well to its initial position even for large N. However, the deviation of the dye starting from the chaotic region of the Poincaré plot from its initial position is large and rapidly increases with N.