On the solution of Stokes' equations between confocal ellipses
Identifieur interne : 000F66 ( PascalFrancis/Corpus ); précédent : 000F65; suivant : 000F67On the solution of Stokes' equations between confocal ellipses
Auteurs : Estéban Saatdjian ; Noël Midoux ; Jean Claude AndréSource :
- Physics of Fluids [ 1070-6631 ] ; 1994-12.
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- Pascal (Inist)
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Abstract
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. © 1994 American Institute of Physics.
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NO : | PASCAL 94-0689029 AIP |
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ET : | On the solution of Stokes' equations between confocal ellipses |
AU : | SAATDJIAN (Estéban); MIDOUX (Noël); ANDRÉ (Jean Claude) |
AF : | ENSIC-LSGC, 1 rue Grandville, BP 451, 54001 Nancy Cédex, France (1 aut., 2 aut.); GRAPP-CNRS, 1 rue Grandville, BP 451, 54001 Nancy Cédex, France (3 aut.) |
DT : | Publication en série; Niveau analytique |
SO : | Physics of Fluids; ISSN 1070-6631; Coden PHFLE6; Etats-Unis; Da. 1994-12; Vol. 6; No. 12; Pp. 3833-3846 |
LA : | Anglais |
EA : | 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. © 1994 American Institute of Physics. |
CC : | 001B40G15G; 001B40G52 |
FD : | Etude théorique; 4715G; 4752; Ecoulement visqueux; Ecoulement incompressible; Equation Navier Stokes; Configuration elliptique; Problème valeur limite; Solution analytique; Ligne courant; Ecoulement laminaire; Stabilité construction; Section Poincaré; Système chaotique; Espace annulaire |
ED : | Theoretical study; Viscous flow; Incompressible flow; Navier-Stokes equations; Elliptical configuration; Boundary-value problems; Analytical solution; Streamlines; Laminar flow; Structural stability; Poincaré mapping; Chaotic systems; Annular space |
LO : | INIST-8651A |
ID : | 94-0689029 |
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Pascal:94-0689029Le document en format XML
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<ET>On the solution of Stokes' equations between confocal ellipses</ET>
<AU>SAATDJIAN (Estéban); MIDOUX (Noël); ANDRÉ (Jean Claude)</AU>
<AF>ENSIC-LSGC, 1 rue Grandville, BP 451, 54001 Nancy Cédex, France (1 aut., 2 aut.); GRAPP-CNRS, 1 rue Grandville, BP 451, 54001 Nancy Cédex, France (3 aut.)</AF>
<DT>Publication en série; Niveau analytique</DT>
<SO>Physics of Fluids; ISSN 1070-6631; Coden PHFLE6; Etats-Unis; Da. 1994-12; Vol. 6; No. 12; Pp. 3833-3846</SO>
<LA>Anglais</LA>
<EA>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. © 1994 American Institute of Physics.</EA>
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