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Flow between finite, steadily rotating eccentric cylinders

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Abstract

While concentric cylinder flows exhibit Ekman pumping at small values of the Reynolds number, once the container symmetry is broken, fluid particles migrate toward the horizontal midplane within the diverging part of the flow domain, and they move away from the midplane within the converging part of the domain. Under this condition, Ekman pumping is evident only at the widest and at the narrowest sections of the gap. When gradually increasing the Reynolds number, the flow acquires a cellular structure in a manner reminiscent of Benjamin's symmetric flow data. The cells spread from the stationary end-plates and link up at the center. The cells persist round the cylinder with varying strength and thus are truly toroidal. We show evidence of solution multiplicity at higher values of the Reynolds number by calculating two qualitatively different flows. One flow contains four cells, the other contains six, and both are supported by identical steady-state conditions.

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References

  1. Benjamin, T.B. (1978). Bifurcation phenomena in steady flows of a viscous liquid, I—theory. Proc. Roy. Soc. London Ser. A, Vol. 359, pp. 1–26.

    Google Scholar 

  2. Benjamin, T.B. (1978). Bifurcation phenomena in steady flows of a viscous liquid, II—experiments. Proc. Roy. Soc. London Ser. A, Vol. 359, pp. 27–43.

    Google Scholar 

  3. Fenstermacher, P.R., Swinney, H.L., and Gollub, J.P. (1979). Dynamic instabilities and the transition to chaotic Taylor vortex flow. J. Fluid Mech., Vol. 94, pp. 103–128.

    Google Scholar 

  4. Hide, R. (1983). On the dynamics of rotating fluids and planetary atmospheres: a summary of recent work. Pure Appl. Geophys., Vol. 121(3), pp. 365–374.

    Google Scholar 

  5. Szeri, A.Z. (1987). Some extensions of the lubrication theory of Osborne-Reynolds. ASME J. Tribology, Vol. 109, pp. 21–37.

    Google Scholar 

  6. Lin, C.C. (1967). The Theory of Hydrodynamic Stability. Cambridge University Press, Cambridge.

    Google Scholar 

  7. Coles, D (1965). Transition in circular Couette flow. J. Fluid Mech., Vol. 21, pp. 385–425.

    Google Scholar 

  8. Taylor, G.I. (1923). Stability of a viscous liquid contained between two rotating cylinders. Philos. Trans. Roy. Soc. London Ser. A, Vol. 223, pp. 289–343.

    Google Scholar 

  9. Drazin, P.G., and Reid, W.H. (1984). Hydrodynamic Stability. Cambridge University Press, Cambridge.

    Google Scholar 

  10. Landau, L., and Lipshitz, E.M. (1959). Fluid Mechanics. Academic Press, New York.

    Google Scholar 

  11. Coughlin, K.T., and Marcus, P.S. (1992). Modulated waves in Taylor-Couette flow, Parts 1 & 2. J. Fluid Mech., Vol. 234, pp. 1–46.

    Google Scholar 

  12. Benjamin, T.B., and Mullin, T. (1981). Anomalous modes in the Taylor experiment. Proc. Roy. Soc. London Ser. A, Vol. 377, pp. 221–249.

    Google Scholar 

  13. Benjamin, T.B., and Mullin, T. (1982). Notes on the multiplicity of flows in the Taylor experiment. J. Fluid Mech., Vol. 121, pp. 219–230.

    Google Scholar 

  14. Mullin, T. (1982). Mutations of steady cellular flows. J. Fluid Mech., Vol. 121, pp. 207–218.

    Google Scholar 

  15. Chandrasekhar, S. (1961). Hydrodynamic and Hydromagnetic Stability. Oxford University Press, Oxford.

    Google Scholar 

  16. Kogelman, S., and DiPrima, R.C. (1970). Stability of spatially periodic supercritical flows in hydrodynamics. Phys. Fluids, Vol. 13, pp. 1–11.

    Google Scholar 

  17. Nakaya, C. (1974). Domain of stable periodic vortex flows in a viscous fluid between concentric circular cylinders. J. Phys. Soc. Japan, Vol. 26, pp. 1146–1173.

    Google Scholar 

  18. Burkhalter, J.E., and Koschmieder, E.L. (1974). Steady supercritical Taylor vortices after sudden starts. Phys. Fluid, Vol. 17, pp. 1929–1935.

    Google Scholar 

  19. Dai, R.X., and Szeri, A.Z. (1990). A numerical study of finite Taylor flows. Internat. J. Non-Linear Mech., Vol. 25, pp. 45–60.

    Google Scholar 

  20. Wannier, G. (1950). A contribution to the hydrodynamics of lubrication. Quart. Appl. Math., Vol. 8, pp. 1–32.

    Google Scholar 

  21. Reynolds, O. (1986). On the theory of lubrication and its application to Mr. Beachamp Tower's experiments. Philos. Trans. Roy. Soc., Vol. 177, pp. 157–234.

    Google Scholar 

  22. Wood, W. (1957). The asymptotic expansions at large Reynolds numbers for steady motion between non-coaxial rotating cylinders. J. Fluid Mech., Vol. 3, pp. 159–175.

    Google Scholar 

  23. Kamal, M.M. (1966). Separation in the flow between eccentric rotating cylinders. ASME J. Basic Engrg., Vol. 88, pp. 717–724.

    Google Scholar 

  24. Ashino, I. (1975). Slow motion between eccentric rotating cylinders. Bull. JSME, Vol. 18, pp. 280–285.

    Google Scholar 

  25. Ballal, B., and Rivlin, R.S. (1976). Flow of a Newtonian fluid between eccentric rotating cylinders. Arch. Rational Mech. Anal., Vol. 62, pp. 237–274.

    Google Scholar 

  26. Kulinski, E., and Ostrach, S. (1967). Journal bearing velocity profiles for small eccentricity and moderate modified Reynolds number. ASME J. Appl. Mech., Vol. 89, pp. 16–22.

    Google Scholar 

  27. Yamada, Y. (1968). On the flow between eccentric cylinders when the outer cylinder rotates. Japan Soc. Mech. Engrs., Vol. 45, pp. 455–462.

    Google Scholar 

  28. Sood, D.R., and Elrod, H.G. (1972). Numerical solution of the incompressible Navier-Stokes equations in doubly-connected regions. AIAA J., Vol. 12, pp. 636–641.

    Google Scholar 

  29. DiPrima, R.C., and Stuart, J.T. (1972). Non-local effects in the stability of flow between eccentric rotating cylinders. J. Fluid Mech., Vol. 54, pp. 393–415.

    Google Scholar 

  30. San Andres, A., and Szeri, A.Z. (1984). Flow between eccentric rotating cylinders. ASME J. Appl. Mech., Vol. 51, pp. 869–878.

    Google Scholar 

  31. Dai, R.X., Dong, Q.M., and Szeri, A.Z. (1992). Flow of variable viscosity fluid between eccentric rotating cylinders. Internat. J. Non-Linear Mech., Vol. 27, pp. 367–389.

    Google Scholar 

  32. Christie, I., Rajagopal, K.R., and Szeri, A.Z. (1987). Flow of a non-Newtonian fluid between eccentric rotating cylinders. Internat. J. Engrg. Sci., Vol. 25, pp. 1029–1047.

    Google Scholar 

  33. DiPrima, R.C., and Stuart, J.T. (1972). Flow between eccentric rotating cylinders. ASME J. Lubrication Tech., Vol. 94, pp. 266–274.

    Google Scholar 

  34. Dai, R.X., Dong, Q.M., and Szeri, A.Z. (1992). Approximations in hydrodynamic lubrication. ASME J. Tribology, Vol. 146, pp. 15–25.

    Google Scholar 

  35. Wilcock, D.F. (1950). Turbulence in high speed journal bearings. Trans. ASME, Vol. 72, pp. 825–834.

    Google Scholar 

  36. Smith, M.I., and Fuller, D.D. (1956). Journal bearing operations at superlaminar speeds. ASME J. Lubrication Tech., Vol. 78, pp. 467–474.

    Google Scholar 

  37. DiPrima, R.C. (1963). A note on the stability of flow in loaded journal bearings. Trans. ASLE, Vol. 6, pp. 249–253.

    Google Scholar 

  38. Vohr, J.H. (1968). An experimental study of Taylor vortices and turbulence in flow between eccentric rotating cylinders. ASME J. Lubrication Tech., Vol. 90, pp. 285–296.

    Google Scholar 

  39. DiPrima, R.C., and Stuart, J.T. (1975). The non-linear calculation of Taylor vortex flow between eccentric rotating cylinders. J. Fluid Mech., Vol. 67, pp. 85–111.

    Google Scholar 

  40. Castle, P., and Mobbs, F.R. (1968). Hydrodynamic stability of the flow between eccentric rotating cylinders: visual observations an torque measurements. Proc. Inst. Mech. Engrs., Vol. 182, pp. 41–52.

    Google Scholar 

  41. Mobbs, F.R., and Younes, M.A. (1974). The Taylor vortex regime in the flow between eccentric rotating cylinders. ASME J. Lubrication Tech., Vol. 96, pp. 127–134.

    Google Scholar 

  42. Versteegen, P.L., and Jankowski, D.F. (1969). Experiments in the stability of viscous flow between eccentric rotating cylinders. Phys. Fluids, Vol. 12, pp. 1138–1143.

    Google Scholar 

  43. Frene, J., and Godet, M. (1974). Flow transition criteria in a journal bearing. ASME J. Lubrication Tech., Vol. 96, pp. 135–140.

    Google Scholar 

  44. Koschmieder, E.L. (1976). Taylor vortices between eccentric cylinders. Phys. Fluids, Vol. 79, pp. 1–4.

    Google Scholar 

  45. Weinstein, M. (1977). Wavy vortices in teh flow between long eccentric cylinders, I—linear theory. Proc. Roy. Soc. London Ser. A, Vol. 354, pp. 441–457.

    Google Scholar 

  46. Weinstein, M. (1977). Wavy vortices in the flow between long eccentric cylinders, II—non-linear theory. Proc. Roy. Soc. London Ser. A, Vol. 354, pp. 459–489.

    Google Scholar 

  47. Frank, G., and Meyer-Spasche, R. (1981). Computation of transitions in Taylor vortex flows. Z. Angew. Math. Phys., Vol. 32, pp. 710–720.

    Google Scholar 

  48. Marcus, P.S. (1984). Simulation of Taylor-Couette flow, I—numerical methods and comparison with experiment. J. Fluid Mech., Vol. 146, pp. 45–64.

    Google Scholar 

  49. Marcus, P.S. (1984) Simulation of Taylor-Couette flow, II—numerical results for wavy vortex flow with one travelling wave. J. Fluid Mech., Vol. 146, pp. 65–113.

    Google Scholar 

  50. Cliff, K.A., and Mullin, T. (1985). A numerical and experimental study of anomalous modes in the Taylor experiment. J. Fluid Mech., Vol. 153, pp. 243–258.

    Google Scholar 

  51. Cliff, K.A., and Spence, A. (1986). Numerical calculations of bifurcations in the finite Taylor problem. In Numerical Methods for Fluid Dynamics (K.W. Morton and M.J. Baines, eds.), pp. 155–176. Clarendon Press, Oxford.

    Google Scholar 

  52. Meyer-Spasche, R., and Keller, H.B. (1978). Numerical study of Taylor-vortex flows between rotating cylinders. California Institute of Technology.

  53. Meyer-Spasche, R., and Keller, H.B. (1980). Computations of the axisymmetric flow between rotating cylinders. J. Comput. Phys., Vol. 35, pp. 100–109.

    Google Scholar 

  54. Oikawa, M., Karasndani, T., and Funakoshni, M. (1989). Stability of flow between eccentric rotating cylinders. J. Phys. Soc. Japan, Vol. 58, pp. 2355–2364.

    Google Scholar 

  55. Oikawa, M., Karasndani, T., and Funakoshni, M. (1989). Stability of flow between eccentric rotating cylinders with wide gap. J. Phys. Soc. Japan, Vol. 58., pp. 2209–2210.

    Google Scholar 

  56. Dai, R.X., Dong, Q.M., and Szeri, A.Z. (1992). Flow between eccentric rotating cylinders: bifurcation and stability. Internat. J. Engrg. Sci., Vol. 30, pp. 1323–1340.

    Google Scholar 

  57. Zienkiewicz, O.C., and Woo, J. (1991). Incompressibility without tears: how to avoid restrictions of mixed formulation. Internat. J. Numer. Methods Engrg., Vol. 32, pp. 1189–1203.

    Google Scholar 

  58. Ritchie, G.S. (1968). On the stability of viscous flow between eccentric rotating cylinders. J. Fluid Mech., Vol. 32, pp. 131–144.

    Google Scholar 

  59. Hughes, T.J.R., Franca, L.P., and Balestra, M. (1986). A new finite element formulation for computation fluid dynamics. Comput. Methods Appl. Math. Engrg., Vol. 59, pp. 85–99.

    Google Scholar 

  60. de Sampaio, P.A.B. (1991). Galerkin formulation for the incompressible Navier-Stokes equations using equal order interpolation for velocity and pressure. Internat. J. Numer. Methods Engrg., Vol. 31, pp. 1135–1149.

    Google Scholar 

  61. Fletcher, C.A.J. (1991). Computation Techniques for Fluid Dynamics. Springer-Verlag, New York.

    Google Scholar 

  62. deBoor, C. (1978). A Practical Guide to Splines. Springer-Verlag, New York.

    Google Scholar 

  63. Ortega, J.M., and Rheinboldt, W.C. (1970). Iterative Solution of Non-Linear Equations in Several Variables. Academic Press, New York.

    Google Scholar 

  64. Joseph, D.D. (1976). Stability of Fluid Motions. Springer-Verlag, New York.

    Google Scholar 

  65. Keller, H.B. (1977). Numerical solutions of bifurcation and non-linear eigenvalue problems. In Applications of Bifurcation Theory (P. Rabinowitz, ed.). Academic Press, New York.

    Google Scholar 

  66. Patankar, S.V. (1980). Numerical Heat Transference and Fluid Flow. Hemisphere, New York.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Pittsburgh, 15261, Pittsburgh, PA, USA

    A. Z. Szeri & A. Al-Sharif

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  1. A. Z. Szeri
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  2. A. Al-Sharif
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Communicated by M.Y. Hussaini

This material is based upon work supported by the National Science Foundation under Grant No. CTS-8920956.

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Szeri, A.Z., Al-Sharif, A. Flow between finite, steadily rotating eccentric cylinders. Theoret. Comput. Fluid Dynamics 7, 1–28 (1995). https://doi.org/10.1007/BF00312397

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