Journal of Fluid Mechanics

Chaotic streamlines inside drops immersed in steady Stokes flows

H. A.  Stone a1, Ali  Nadim a2 and Steven H.  Strogatz a2
a1 Division of Applied Sciences, Harvard University, Cambridge, MA 02138, USA
a2 Department of Mathematics, MIT, Cambridge, MA 02139, USA

Article author query
stone ha   [Google Scholar] 
nadim a   [Google Scholar] 
strogatz sh   [Google Scholar] 


Motivated by the recent work of Bajer & Moffatt (1990), we investigate the kinematics of bounded steady Stokes flows. Specifically, we consider the streamlines inside a neutrally buoyant spherical drop immersed in a general linear flow. The Eulerian velocity field internal to the drop, known analytically, is a cubic function of position. For a wide range of parameters the internal streamlines, hence the fluid particle paths, may wander chaotically. Typical Poincaré sections show both ordered and chaotic regions. The extent and existence of chaotic wandering is related to (i) the orientation of the vorticity vector relative to the principal axes of strain of the undisturbed flow and (ii) the magnitude of the vorticity relative to the magnitude of the rate-of-strain tensor. In the limit of small vorticity, we use the method of averaging to predict the size of the dominant island region. This yields the critical orientation of the vorticity vector at which this dominant island disappears so that particle paths fill almost the entire Poincaré section. The problem studied here appears to be one of the simplest, physically realizable, bounded steady Stokes flows which produces chaotic streamlines.

(Published Online April 26 2006)
(Received December 12 1990)