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Some shear-layer and inertial modifications to the geostrophic drag on a slowly rising particle or drop in a rotating fluid

Published online by Cambridge University Press:  26 April 2006

M. Ungarish
M. Ungarish
Affiliation:
Computer Science Department, Technion - Israel Institute of Technology, Haifa 32000, Israel
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Abstract

We consider the slow axial motion of a symmetric particle or drop in a bounded rotating fluid for small Rossby and Ekman numbers, Ro and E. Previous investigations pointed out that the available linear-theory results, based on the assumption of a dominant geostrophic core and infinitesimally thin viscous layers, yield a drag force larger than the available relevant experimental results, and are unable to explain some of the observed flow-field properties, for both solid and deformable particles.

Here we attempt to improve the drag calculation model and the interpretation of the flow field by incorporating shear effects in the core (outside the E1/2 Ekman and E1/3 Stewartson layers), first in the linear (Ro = 0) formulation, then with keeping some influential nonlinear inertial terms for small but finite Ro. The major equation for the angular velocity in the core, ω(r), was usually solved by a finite-differences method, because in the practical parameter range the available analytical results are sufficiently accurate only for a disk particle or for a bubble. Results for various $\epsilon = (\frac{1}{2}H)^{1/2}$, no-slip parameter of particle surface κ, and half container height H are presented for both spherical and disk particles. The drag is below the geostrophic value, typically, by 25% for ε ≈ 0.1 and by 50% for ε ≈ 0.5. The inclusion of the inertial terms causes the lateral (‘vertical’) shear regions to contract and expand on the upstream and downstream sides, respectively, and an inertial sublayer appears in the latter when Ro ≈ O(E3/4), but the net contribution to the drag is smaller than expected. Compared with more accurate solutions and experiments the present results underestimate the drag (the reasons are discussed) but are qualitatively consistent in many respects, which indicates that many of the observed flow-field features that have been traditionally attributed to inertial effects (not sufficiently small Ro) are, rather, by-products of the lateral shear (finite value of ε).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 319 , 25 July 1996 , pp. 219 - 249
Copyright
© 1996 Cambridge University Press

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References

Barcilon, V. 1970 Some inertial modifications of the linear viscous theory of steady rotating fluid flows. Phys. Fluids 13, 537–544.Google Scholar
Barnard, B. J. S. & Pritchard, W. G. 1975 The motion generated by a body moving through a stratified fluid at large Richardson numbers. J. Fluid Mech. 71, 43–64.Google Scholar
Bennets, D. A. & Hocking, L. M. 1973 On nonlinear Ekman and Stewartson layers in a rotating fluid. Proc. R. Soc. Lond. A 333, 469–489.Google Scholar
Bennets, D. A. & Jackson, W. D. N. 1974 Source-sink flow in a rotating annulus: a combined laboratory and numerical study. J. Fluid Mech. 66, 689–705.Google Scholar
Bush, J. W. M. 1993 Drop motion in rotating fluids: a model of compositional convection in the earth's core. PhD thesis, Harvard University, Cambridge, MA, USA.
Bush, J. W. M., Stone, H. A. & Bloxham, J. 1992 The motion of an inviscid drop in a bounded rotating fluid. Phys. Fluids A 4, 1142–1147.Google Scholar
Bush, J. W. M., Stone, H. A. & Bloxham, J. 1995 Axial drop motion in rotating fluids. J. Fluid. Mech. 282, 247–278.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge Uniersity Press.
Hide, R. 1968 On source-sink flows in a rotating fluid. J. Fluid Mech. 32, 737–764.Google Scholar
Hocking, L. M., Moore, D. W. & Walton, I. C. 1979 The drag on a sphere moving axially in a long rotating container. J. Fluid Mech. 90, 781–793.Google Scholar
Maxworthy, T. 1968 The observed motion of a sphere through a short, rotating cylinder of fluid. J. Fluid Mech. 31, 643–655.Google Scholar
Moore, D. W. & Saffman, P. G. 1968 The rise of a body through a rotating fluid in a container of finite length. J. Fluid Mech. 31, 635–642.Google Scholar
Moore, D. W. & Saffman, P. G. 1969 The structure of free vertical shear layers in a rotating fluid and the motion produced by a slowly rising body. Trans. R. Soc. Lond. A 264, 597–634.Google Scholar
Stewartson, K. 1966 On almost rigid rotations. J. Fluid Mech. 26, 131–144.Google Scholar
Tanzosh & Stone. 1994 Motion of a rigid particle in a rotating viscous flow: An integral equation approach. J. Fluid. Mech. 275, 225–256.Google Scholar
Ungarish, M. 1996 Some spin-up effects on the geostrophic and quasi-geostrophic drag on a slowly rising particle in a rotating fluid, submitted.
Ungarish, M. & Vedensky, D. 1995 The motion of a rising disk in a rotating axially bounded fluid for large Taylor number. J. Fluid Mech. 291, 1–32.Google Scholar
Wedemeyer, E. H. 1964 The unsteady flow within a spinning cylinder. J. Fluid Mech. 20, 383–399.Google Scholar