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Flow in a differentially rotated cylindrical drop at moderate Reynolds number

Published online by Cambridge University Press:  20 April 2006

George M. Harriott  and
Robert A. Brown
George M. Harriott
Affiliation:
Department of Chemical Engineering and Material Processing Center, Massachusetts Institute of Technology, Cambridge, MA 02139
Robert A. Brown
Affiliation:
Department of Chemical Engineering and Material Processing Center, Massachusetts Institute of Technology, Cambridge, MA 02139
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Abstract

Galerkin finite-element approximations are combined with computer-implemented perturbation methods for tracking families of solutions to calculate the steady axisymmetric flows in a differentially rotated cylindrical drop as a function of Reynolds number Re, drop aspect ratio and the rotation ratio between the two end disks. The flows for Reynolds numbers below 100 are primarily viscous and reasonably described by an asymptotic analysis. When the disks are exactly counter-rotated, multiple steady flows are calculated that bifurcate to higher values of Re from the expected solution with two identical secondary cells stacked symmetrically about the axial midplane. The new flows have two cells of different size and are stable beyond the critical value Rec. The slope of the locus of Rec for drops with aspect ratio up to 3 disagrees with the result for two disks of infinite radius computed assuming the similarity form of the velocity field. Changing the rotation ratio from exact counter-rotation ruptures the junction of the multiple flow fields into two separated flow families.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 144 , July 1984 , pp. 403 - 418
Copyright
© 1984 Cambridge University Press

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References

Brady, J. F. 1981 On the flow between two finite rotating disks. Bull. Am. Phys. Soc. 26, 1269.Google Scholar
Brady, J. F. 1983 On the flow between stationary and rotating finite disks. Submitted to J. Fluid Mech.Google Scholar
Brady, J. F. & Acrivos, A. 1982 Closed-cavity laminar flows at moderate Reynolds number. J. Fluid Mech. 115, 427–442.Google Scholar
Brown, R. A., Scriven, L. E. & Silliman, W. J. 1980 Computer-aided analysis of nonlinear problems in transport phenomena. In New Approaches to Non-linear Problems in Dynamics (ed. P. J. Holmes). SIAM
Carruthers, J. R. & Grasso, M. 1972 Studies of floating zones in simulated zero gravity. J. Appl. Phys. 43, 436–443.Google Scholar
Fowle, A. A., Soto, L., Strong, P. F. & Wang, C. A. 1980 Experimental and flow characteristics of floating liquid columns confined between rotating disks. Final Rep. to NASA, Washington, D.C., Contract NASW-3186 by Arthur D. Little.
Greenspan, H. P. 1969 The Theory of Rotating Fluids. Cambridge University Press.
Griffiths, D. E., Jones, D. T. & Walters, K. 1969 A flow reversal due to edge effects. J. Fluid Mech. 36, 161–175.Google Scholar
Harriott, G. M. & Brown, R. A. 1983 Flow in a differentially rotated cylindrical drop at low Reynolds number. J. Fluid Mech. 126, 269–285.Google Scholar
Hill, C. T. 1972 Nearly viscometric flow of viscoelastic fluids in the disk and cylinder system. II: Experimental. Trans. Soc. Rheol. 16, 213–245.Google Scholar
Holodniok, M., Kubičcek, M. & Hlavaček, V. 1981 Computation of the flow between two rotating coaxial disks: multiplicity of steady-state solutions. J. Fluid Mech. 108, 227–240.Google Scholar
Huyakorn, P., Taylor, C., Lee, R. & Gresho, P. 1978 A comparison of various mixed-interpolation finite elements in the velocity-pressure formulation of the Navier-Stokes equations. Comp. Fluids 6, 25–35.Google Scholar
Iooss, G. & Joseph, D. D. 1980 Elementary Stability and Bifurcation Theory. Springer.
Kármán, T. von 1921 Über laminare und turbulente Reibung. Z. angew. Math. Mech. 1, 233–251.Google Scholar
Keener, J. P. & Keller, H. B. 1973 Perturbed bifurcation theory. Arch Rat. Mech. Anal. 50, 159–175.Google Scholar
Keller, H. B. & Szeto, R. K.-H. 1980 Calculation of flows between rotating disks. In Computing Methods in Applied Sciences and Engineering (ed. R. Glowinski & J. L. Lions). North-Holland.
Kobayaski, N. & Wilcox, W. R. 1982 Computational studies of convection due to rotation in a cylindrical floating zone. J. Crystal Growth 59, 616–624.Google Scholar
Kramer, J. M. & Johnson, M. W. 1972 Nearly viscometric flow in a disk and cylinder system. I: Theoretical. Trans. Soc. Rheol. 16, 197–212.Google Scholar
Kubiček, M., Holodniok, M. & Hlavaček, V. 1976 Calculation of flow between two rotating coaxial disks by differentiation with respect to an actual parameter. Comp. Fluids 4, 59–64.Google Scholar
Matkowsky, B. J. & Reiss, E. L. 1977 Singular perturbation of bifurcations. SIAM J. Appl. Maths 33, 230–255.Google Scholar
Nguyen, N. D., Ribault, J. P. & Florent, P. 1975 Multiple solutions for flow between coaxial disks. J. Fluid Mech. 68, 369–388.Google Scholar
Pao, H.-P. 1970 A numerical computation of a confined rotating flow. Trans. ASME E: J. Appl. Mech. 37, 480–487.Google Scholar
Pearson, C. E. 1965 Numerical solutions for the time-dependent viscous flow between two rotating coaxial disks. J. Fluid Mech. 21, 623–633.Google Scholar
Roberts, S. M. & Shipman, J. S. 1976 Computation of the flow between a rotating and a stationary disk. J. Fluid Mech. 73, 53–63.Google Scholar
Szeto, R. K.-H. 1978 The flow between rotating coaxial disks. Ph.D. thesis, California Institute of Technology.
Thomasset, F. 1981 Implementation of Finite Element Methods for Navier-Stokes Equations. Springer.
Thompson, J. M. T. & Hunt, G. W. 1973 A General Theory of Elastic Stability. Wiley.
Ungar, L. H. & Brown, R. A. 1982 The dependence of the shape and stability of captive rotating drops on multiple parameters. Phil. Trans. R. Soc. Lond. A 306, 347–370.Google Scholar
Yamaguchi, Y., Chang, C. J. & Brown, R. A. 1984 Multiple buoyancy-driven flows in a vertical cylinder heated from below. Phil. Trans. R. Soc. Lond. (in press).Google Scholar