On the solution of Stokes’ equations between confocal ellipses

• Published in: Physics of fluids (1994) Vol. 6; no. 12; pp. 3833 - 3846
• Main Authors: Saatdjian, Estéban, Midoux, Noël, André, Jean Claude
• Format: Journal Article
• Language: English
• Published: 01.12.1994
• Subjects: NAVIER−STOKES EQUATIONS; BOUNDARY−VALUE PROBLEMS; STRUCTURAL STABILITY; CHAOTIC SYSTEMS; ELLIPTICAL CONFIGURATION; VISCOUS FLOW; STREAMLINES; POINCARE MAPPING; LAMINAR FLOW; ANNULAR SPACE; INCOMPRESSIBLE FLOW; ANALYTICAL SOLUTION
• ISSN: 1070-6631; 1089-7666
• DOI: 10.1063/1.868375

The analytical solution of Stokes’ equations between two concentric, confocal ellipses is derived here. This bounded flow, similar in certain respects to the journal bearing flow, was imagined in order to investigate two‐dimensional mixing and Lagrangian chaos in a bounded flow with two symmetry axis. The derived streamfunction is in the form of a Fourier cosine series and, when the eccentricity ratio of the inner ellipse is not very low, the solution converges very rapidly. When the ellipses turn in opposite directions, there are cases where two saddle points are connected by two different streamlines, a necessary and sufficient condition for structural instability according to Peixoto’s theorem. This flow geometry could be particularly effective for mixing of viscous fluids since the number of low period hyperbolic and elliptical points during time periodic boundary motion is greater than for the eccentric rotating cylinder system. The Poincaré sections obtained with a discontinuous velocity protocol suggest that the size of regions of poor mixing can be reduced by increasing the inner ellipse motion per period. For this geometry, the Poincaré sections indicate that counter‐rotation yields a more chaotic long term behavior than co‐rotation.