Chaotic streamlines in steady bounded three-dimensional Stokes flows

The streamline structure inside a spherical liquid drop immersed in a steady Stokes flow is studied for the case that the external flow fields are characterized by (a) the translational velocity and vorticity vectors and (b) the rate-of-strain tensor and the vorticity vector. The velocity fields internal to the drop, known analytically, are, respectively, quadratic and cubic functions of position in cases (a) and (b). Recently, the cubic flow field in problem (b) was shown to exhibit chaotic streamlines [H.A. Stone, A. Nadim, S.H. Strogatz, J. Fluid Mech. 232 (1991) 629]. Here it is demonstrated that the quadratic flow field in (a) may also produce chaotically wandering streamlines. In each flow the axisymmetric case is considered first and it is shown that the equations of motion may be expressed in canonical Hamiltonian form and so are integrated analytically. In the nearly axisymmetric case, the onset of streamline chaos in each of the situations (a) and (b) is investigated analytically by deriving the area-preserving maps giving the trajectories of fluid particles close to the separatrices of the unperturbed motion. Estimates for the width of the chaotic layers are obtained as a function of the components of the vorticity vector and the analytical predictions are verified by direct numerical simulations.