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An analytical study of transport in Stokes flows exhibiting large-scale chaos in the eccentric journal bearing

Published online by Cambridge University Press:  26 April 2006

Tasso J. Kaper  and
S. Wiggins
Tasso J. Kaper
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA Present address: Department of Mathematics, Boston University, Boston, MA 02215, USA.
S. Wiggins
Affiliation:
Applied Mechanics, California Institute of Technology, Pasadena, CA 91125, USA
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Abstract

In the present work, we apply new tools from the field of adiabatic dynamical systems theory to make quantitative predictions of various important mixing quantities in quasi-steady Stokes flows which possess slowly varying saddle stagnation points. Many of these quantities can be obtained before experiments or numerical simulations are performed using only knowledge of the streamlines in steady-state flows and the externally determined flow parameters. The location and size of the main region in which mixing occurs is determined to leading order by the slowly sweeping instantaneous stagnation streamlines. Tracer patches get stretched by an amount O(1/ε) during each period, and the average striation thickness of the patch decreases by a factor of ε in the same time. Also, the rate of stretching of material interfaces is bounded from below with an analytically obtained exponentially growing lower bound. Finally, the highly regular appearance of islands in quasi-steady Stokes’ flows is explained using an extension of the KAM theory. As an example to illustrate these results, we study the transport of passive scalars in a low Reynolds number flow in the two-dimensional eccentric journal bearing when the angular velocities of the cylinders are slowly modulated, continuously and periodically in time, with frequency ε. In contrast to the flows usually studied with dynamical systems, these slowly varying systems are singular perturbation (apparently far from integrable) problems exhibiting large-scale chaos, in which the non-integrability is due to the slow, continuous O(1) modulation of the position of the saddle stagnation point and the two streamlines stagnating on it.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 253 , August 1993 , pp. 211 - 243
Copyright
© 1993 Cambridge University Press

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References

Abraham, R. H. & Shaw, C. D. 1985 Dynamics– The Geometry of Behavior. Santa Cruz: Ariel.
Aref, H. & Balachandar, S. 1986 Chaotic advection in a Stokes flow. Phys. Fluids 29, 3515–3521.Google Scholar
Aref, H. & Jones, S. 1989 Enhanced separation of diffusing particles by chaotic advection. Phys. Fluids A 1, 470–474.Google Scholar
Arnol'd, V. I., Neishtadt, A. I. & Kozlov, X. (eds.) 1988 Dynamical Systems, vol. III, Encyclopaedia of Mathematical Science. Springer.
Ballal, B. Y. & Rivlin, R. S. 1976 Flow of a Newtonian fluid between eccentric rotating cylinders: inertial effects. Arch. Rat. Mech. Anal. 62, 237–294.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bruhwiler, D. L. & Cary, J. R. 1989 Diffusion of particles in a slowly-modulated wave. Physica D 40, 265–282.Google Scholar
Cary, J. R., Escande, D. F. & Tennyson, J. 1986 Adiabatic invariant change due to separatrix crossing. Phys. Rev. A 34, 4256–4275.Google Scholar
Cary, J. R. & Skodje, R. T. 1989 Phase change between separatrix crossings. Physica D 39, 287.Google Scholar
Chaiken, J., Chevray, R., Tabor, M. & Tan, Q. M. 1986 Experimental study of lagrangian turbulence in a Stokes flow. Proc. R. Soc. Lond. A 408, 165–174.Google Scholar
Chaiken, J., Chu, C. K., Tabor, M. & Tan, Q. M. 1987 Lagrangian turbulence and spatial complexity in a Stokes flow. Phys. Fluids 30, 687–699.Google Scholar
Chandrasekhar, S. 1943 Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15, 1–89.Google Scholar
Chien, W.-L., Rising, H. & Ottino, J. M. 1986 Laminar mixing and chaotic mixing in several cavity flows. J. Fluid Mech. 170, 355–377.Google Scholar
Coppola, V. T. & Rand, R. H. 1991 Chaos in a system with a periodically disappearing separatrix. Nonlinear Dyn. 1, 401–420.Google Scholar
Dutta, P. & Chevray, R. 1991 Effect of diffusion on chaotic advection in Stokes flow. Phys. Fluids A 3, 1440.Google Scholar
Elskens, Y. & Escande, D. 1991 Slowly pulsating separatrices sweep homoclinic tangles where islands must be small: an extension of classical adiabatic theory. Nonlinearity 4, 615.Google Scholar
Franjione, J. G., Leong, C. W. & Ottino, J. M. 1989 Symmetries within chaos: a route to effective mixing. Phys. Fluids A 1, 1772–1783.Google Scholar
Franjione, J. G. & Ottino, J. M. 1987 Feasibility of numerical tracking of material lines in chaotic flows. Phys. Fluids 30, 3641–3643.Google Scholar
Ghosh, S., Chang, H.-C. & Sen, M. 1991 Heat transfer enhancement due to slender recirculation and chaotic transport between counter-rotating eccentric cylinders, preprint.
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer.
Hoffman, N. R. A. & Mckenzie, D. P. 1985 The destruction of geochemical heterogeneities by differential fluid motions during mantle convection. Geophys. J. R. Astron. Soc. 82, 163–206.Google Scholar
Kaper, T. J. 1991 Part I: On the structure in separatrix-swept regions of slowly-modulated Hamiltonian systems; Part II: On the quantification of mixing in chaotic Stokes flows: the eccentric journal bearing. Ph.D. thesis, California Institute of Technology.
Kaper, T. J., & Kovacic, G. & Wiggins, S. 1990 Melnikov functions, action, and lobe area in Hamiltonian systems. Tech. Rep. LAUR 90-2455, Los Alamos National Lab.
Kaper, T. J. & Wiggins, S. 1989 Transport, mixing, and stretching in a chaotic Stokes flow: the two-roll mill. Los Alamos Tech. Rep. LA-UR 90-2638; also in Proc. Third Annual Joint ASCE/ASME Mechanics Conf., La Jolla, July, 1989.
Kaper, T. J. & Wiggins, S. 1991a Lobe area in adiabatic Hamiltonian systems. Physica D 51, 205–212.Google Scholar
Kaper, T. J. & Wiggins, S. 1991b On the structure of separatrix-swept regions in singularly-perturbed Hamiltonian systems, Diffl Integral Equat. 5, 1363–1381.Google Scholar
Kaper, T. J. & Wiggins, S. 1991c A commentary ‘On the periodic solutions of a forced second-order equation' by S. P. Hastings and J. B. McLeod. J. Nonlin. Sci. 1, 247–253Google Scholar
Kruskal, M. 1962 Asymptotic theory of Hamiltonian and other systems with all solutions nearly periodic. J. Math. Phys. 3, 806–828.Google Scholar
LEONG, C. W. & OTTINO, J. M. 1989 Experiments on mixing due to chaotic advection in a cavity. J. Fluid Mech. 209, 463–499.Google Scholar
Muzzio, F. J., Ottino, J. M. & Swanson, P. D. 1991 The statistics of stretching and stirring in chaotic flows. Phys. Fluids A 3, 822–834.Google Scholar
Neishtadt, A. I. 1975 Passage through a separatrix in a resonance problem with a slowly-varying parameter. Prikl. Mat. Mecl 39, 594–605.Google Scholar
Neishtadt, A. I., Chaikovskii, D. K., Chernikov, A. A. 1991 Adiabatic chaos and particle diffusion. Sov. Phys. JETP 72, 423–430.Google Scholar
Ng, R. C.-Y. 1989 Semi-dilute polymer solutions in strong flows. Part I: Birefringence and flow modification in extensional flows; Part II Chaotic mixing in time-periodic flows. Ph.D. thesis, California Institute of Technology.
Ng, R. C.-Y., James, D. F. & Leal, L. G. 1990 Chaotic mixing and transport in a two-dimensional time-periodic Stokes flow – the blinking two-roll mill, in preparation.
Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos and Transport. Cambridge University Press.
Palmer, K. 1986 Transversal heteroclinic points and Cherry's example of a nonintegrable Hamiltonian system. J. Diffl Equat. 65, 321–360.Google Scholar
Robinson, C. 1983 Sustained resonance for a nonlinear system with slowly-varying coefficients. SIAM J. Math. Anal. 14, 847–860.Google Scholar
Rom-Kedar, V., Leonard, A. & Wiggins, S. 1990 An analytical study of the transport, mixing and chaos in an unsteady vortical flow. J. Fluid Mech. 214, 347–394.Google Scholar
Rom-Kedar, V. & Wiggins, S. 1990 Transport in two-dimensional maps. Arch. Rat. Mech. Anal. 109, 239–298.Google Scholar
Swanson, P. D. 1991 Regular and chaotic mixing of viscous fluids in eccentric rotating cylinders. Ph.D. thesis, University of Massachusetts, Amherst.
Swanson, P. D. & Ottino, J. M. 1990 A comparative computational and experimental study of chaotic mixing in viscous fluids. J. Fluid Mech. 213, 227–249.Google Scholar
Wiggins, S. 1988a Global Bifurcations and Chaos: Analytical Methods, 1st edn. Springer.
Wiggins, S. 1988b On the detection and dynamical consequences of orbits homoclinic to hyperbolic periodic orbits and normally hyperbolic invariant tori in a class of ordinary differential equations. SIAM J. Appl. Maths 48, 262–285.Google Scholar
Wiggins, S. 1988c Adiabatic chaos. Phys. Lett. A 128, 339–342.Google Scholar
Wiggins, S. 1992 Chaotic Transport in Dynamical Systems, 1st edn. Springer.