Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-19T09:27:35.710Z Has data issue: false hasContentIssue false
  • English
  • Français

Heat-transfer enhancement due to slender recirculation and chaotic transport between counter-rotating eccentric cylinders

Published online by Cambridge University Press:  26 April 2006

S. Ghosh ,
H.-C. Chang  and
M. Sen
S. Ghosh
Affiliation:
Department of Chemical Engineering, University of Notre Dame, Notre Dame. IN 46556, USA
H.-C. Chang
Affiliation:
Department of Chemical Engineering, University of Notre Dame, Notre Dame. IN 46556, USA
M. Sen
Affiliation:
Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Using Stokes flow between eccentric, counter-rotating cylinders as a prototype for bounded, nearly parallel lubrication flow, we investigate the effect of a slender recirculation region within the flow field on cross-stream heat or mass transport in the important limit of high Péclet number Pe where the enhancement over pure conduction heat transfer without recirculation is most pronounced. The steady enhancement is estimated with a matched asymptotic expansion to resolve the diffusive boundary layers at the separatrices which bound the recirculation region. The enhancement over pure conduction is shown to vary as ε½ at infinite Pe, where ε½ is the characteristic width of the recirculation region. The enhancement decays from this asymptote as Pe−½. If one perturbs the steady flow by a time-periodic forcing, fast relative to the convective and diffusive times, the separatrices undergo a homoclinic entanglement which allows fluid elements to cross the separatrices. We establish the existence of this homoclinic entanglement and show that the resulting chaotic particle transport further enhances the cross-stream flux. We estimate the penetration of the fluid elements across the separatrices and their effective diffusivity due to this chaotic transport by a Melnikov analysis for small-amplitude forcing. These and the steady results then provide quantitative estimates of the timeaveraged transport enhancement and allow optimization with respect to system parameters. An optimum forcing frequency which induces maximum heat-transfer enhancement is predicted and numerically verified. The predicted optimum frequency remains valid at strong forcing and large Pe where chaotic transport is as important as the recirculation mechanism. Since most heat and mass transport devices operate at high Pe, our analysis suggests that chaotic enhancement can improve their performance and that a small amplitude theory can be used to optimize its application.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 238 , May 1992 , pp. 119 - 154
Copyright
© 1992 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Acharya, N., Sen, M. & Chang, H.-C. 1992 Heat transfer enhancement in coiled tubes by chaotic mixing. Intl J. Heat Mass Transfer (to appear).Google Scholar
Aref, H. 1983 Integrable, chaotic and turbulent vortex motion in two-dimensional flows. Ann. Rev. Fluid Mech. 15, 345–389.Google Scholar
Aref, H. 1984 Stirring by chaotic advection. J. Fluid Mech. 143, 1–21.Google Scholar
Aref, H. & Balachandar, S. 1986 Chaotic Advection in a Stokes flow. Phys. Fluids 29, 3515–3521.Google Scholar
Ballal, B. Y. & Rivlin, R. S. 1976 Flow of a Newtonian fluid between eccentric cylinders: inertial effects. Arch. Rat. Mech. Anal. 62, 237–294.Google Scholar
Batchelor, G. K. 1956 On steady laminar flow with closed streamlines at large Reynolds number. J. Fluid Mech. 1, 177–189.Google Scholar
Broomhead, D. S. & Ryrie, S. C. 1988 Particle paths in wavy vortices. Nonlinearity 1, 409–434.Google Scholar
Chaiken, J., Chevray, R., Tabor, M. & Tam, Q. M. 1986 Experimental study of Lagrangian turbulence in Stokes flow. Proc. R. Soc. Lond. A 408, 165–174.Google Scholar
Chávez, M., Zhixue, W. & Sen, M. 1988 Turbulent convection in helicoidal coils. Wärme- und Stoffübertragung 22 (1–2), 55–60.Google Scholar
Chirikov, B. V. 1979 A universal instability of many-dimensional oscillator systems. Phys. Rep. 52, 263–379.Google Scholar
Cox, S. M., Drazin, P. G., Ryrie, S. C. & Slater, K. 1990 Chaotic advection of irrotational flows and of waves in fluids. J. Fluid Mech. 214, 517–524.Google Scholar
Eckhardt, B. 1990 Lagrangian chaos and transport of scalars. In Topological Fluid Mechanics (ed. H. K. Moffat & A. Tsinober), pp. 23–33. Cambridge University Press.
Escande, D. & Elskens, Y. 1992 Slowly pulsating separatrices sweep homoclinic tangles where islands must be small: an extension of classical adiabatic theory. Nonlinearity (to appear).Google Scholar
Ghaddar, N. K., Magen, M., Mikic, B. B. & Patera, A. T. 1986 Numerical investigation of incompressible flow in grooved channels, part 2. Resonance and oscillatory heat transfer enhancement. J. Fluid Mech. 168, 541–567.Google Scholar
Ghosh, S. 1991 Transport enhancement in steady and time-periodic Stokes flow between counterrotating cylinders. MS thesis, Department of Chemical Engineering, University of Notre Dame.
Goren, S. L. & Mani, P. V. S. 1968 Mass transfer through horizontal liquid films in wavy motion. AIChE J. 14, 57–61.Google Scholar
Guckenheimer, J. & Holmes, P. 1983 Non Linear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer.
Holmes, P. 1984 Some remarks on chaotic particle paths in time-periodic, three-dimensional swirling flows. Contemp. Maths 28, 393–404.Google Scholar
Karney, C. F. F. 1983 Long-time correlations in the stochastic regime. Physica 8D, 360–380.Google Scholar
Moffat, H. K. 1983 Transport effects associated with turbulence with particular attention to the influence of helicity. Rep. Prog. Phys. 46, 621–664.Google Scholar
Ohadi, M. M., Sharaf, N. & Nelson, D. A. 1991 Electrohydrodynamic enhancement of heat transfer in a shell-and-tube heat exchanger. Heat Transfer 4, 19–39.Google Scholar
Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press.
Rhines, P. B. & Young, W. R. 1983 How rapidly is a passive scalar mixed with closed streamlines. J. Fluid Mech. 133, 133–145.Google Scholar
Rom-Kedar, V. 1990 Transport rates of a class of two-dimensional maps and flows. Physica D 43, 229–268.Google Scholar
Rom-Kedar, V., Leonard, A. & Wiggins, S. 1990 An analytical study of transport, mixing and chaos in an unsteady vortical flow. J. Fluid Mech. 214, 347–394.Google Scholar
Rosenbluth, M. N., Berk, H. L., Doxas, I. & Horton, W. 1987 Effective diffusion in laminar convective flows. Phys. Fluids 30, 2636–2647.Google Scholar
Sagues, F. & Horsthemke, W. 1986 Diffusive transport in spatially periodic hydrodynamic flows. Phys. Rev. A 34, 4136–4143.Google Scholar
Saxena, A. K. & Nigam, K. D. P. 1984 Coiled configuration for flow inversion and its effect on residence time distribution. AIChE J. 30, 363–368.Google Scholar
Shraiman, B. I. 1987 Diffusive transport in a Rayleigh-Bénard convection cell. Phys. Rev. A 36 (1), 261–267.Google Scholar
Solomon, T. H. & Gollub, J. P. 1988 Chaotic particle transport in time-dependent Rayleigh-Bénard convection. Phys. Rev. A 38 (12), 6280–6285.Google Scholar
Stewart, W. E. 1977 Physicochemical Hydrodynamics. V. G. Levich Festschrift. Spalding.
Swanson, P. D. & Ottino, J. M. 1990 A comparative computational and experimental study of chaotic mixing of viscous fluids. J. Fluid Mech. 213, 227–249.Google Scholar
Weiss, J. B. & Knobloch, E. 1989 Mass transport and mixing by modulated travelling waves. Phys. Rev. A 40, 2579–2589.Google Scholar
Young, W., Pumir, A. & Pomeau, Y. 1989 Anomalous diffusion of tracer in convective rolls. Phys. Fluids A 1, 462–469.Google Scholar