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Axial drop motion in rotating fluids

Published online by Cambridge University Press:  26 April 2006

J. W. M. Bush ,
H. A. Stone  and
J. Bloxham
J. W. M. Bush
Affiliation:
Department of Earth and Planetary Sciences, Harvard University, Cambridge, MA 02138, USA Present address: Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge. CB4 5HT, UK.
H. A. Stone
Affiliation:
Division of Applied Sciences, Harvard University, Cambridge, MA 02138, USA
J. Bloxham
Affiliation:
Department of Earth and Planetary Sciences, Harvard University, Cambridge, MA 02138, USA
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Abstract

A theoretical and experimental investigation of drop motion in rotating fluids is presented. The theory describing the vertical on-axis translation of an axisymmetric rigid body through a rapidly rotating low-viscosity fluid is extended to the case of a buoyant deformable fluid drop of arbitrary viscosity. In the case that inertial and viscous effects are negligible within the bulk external flow, motions are constrained to be two-dimensional in compliance with the Taylor–Proudman theorem, and the rising drop is circumscribed by a Taylor column. Calculations for the drop shape and rise speed decouple, so that theoretical predictions for both are obtained analytically. Drop shapes are set by a balance between centrifugal and interfacial tension forces, and correspond to the family of prolate ellipsoids which would arise in the absence of drop translation. In the case of a drop rising through an unbounded fluid, the Taylor column is dissipated at a distance determined by the outer fluid viscosity, and the rise speed corresponds to that of an identically shaped rigid body. In the case of a drop rising through a sufficiently shallow plane layer of fluid, the Taylor column extends to the boundaries. In such bounded systems, the rise speed depends further on the fluid and drop viscosities, which together prescribe the efficiency of the Ekman transport over the drop and container surfaces.

A set of complementary experiments is also presented, which illustrate the effects of drop viscosity on steady drop motion in bounded rotating systems. The experimental results provide qualitative agreement with the theoretical predictions; in particular, the poloidal circulation observed inside low-viscosity drops is consistent with the presence of a double Ekman layer at the interface, and is opposite to that expected to arise in non-rotating systems. The steady rise speeds observed are larger than those predicted theoretically owing to the persistence of finite inertial effects.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 282 , 10 January 1995 , pp. 247 - 278
Copyright
© 1995 Cambridge University Press

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References

Backus, G. E. 1986 Poloidal and toroidal fields in geomagnetic field modeling. Rev. Geophys. 24, 75–109.Google Scholar
Baker, J. 1966 A technique for the precise measurement of fluid flow in the range 0–5 cm/s. J. Fluid Mech. 26, 573–575.Google Scholar
Baker, G. R. & Israeli, M. 1981 Spinup from rest of immiscible fluids. Stud. Appl. Maths 65, 249–268.Google Scholar
Berman, A. S., Bradford, J. & Lundgren, T. S. 1978 Two-fluid spin-up in a centrifuge. J. Fluid Mech. 84, 411–431.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.
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
Grace, S. F. 1926 On the motion of a sphere in a rotating liquid. Proc. R. Soc. Lond. A 113, 46–77.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University 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
Levich, V. G. 1962 Physicochemical Hydrodynamics. Prentice-Hall.
Long, R. R. 1953 Steady motion around a symmetrical obstacle moving along the axis of a rotating fluid. J. Met. 10, 197–203.Google Scholar
Maxworthy, T. 1968 The observed motion of a sphere through a short, rotating cylinder of fluid. J. Fluid Mech. 40, 453–480.Google Scholar
Maxworthy, T. 1970 The flow created by a sphere moving along the axis of a rotating, slightly viscous fluid. J. Fluid Mech. 31, 643–656.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. Phil. Trans. R. Soc. Lond. A 264, 597–634.Google Scholar
Morrison, J. W. & Morgan, G. W. 1956 The slow motion of a disc along the axis of a viscous, rotating liquid. Rep. 56207/8. Div. of Appl. Maths, Brown University.
Pritchard, W. G. 1969 The motion generated by a body moving along the axis of a uniformly rotating fluid. J. Fluid Mech. 39, 443–464.Google Scholar
Proudman, J. 1916 On the motion of solids in a liquid possessing vorticity. Proc. R. Soc. Lond. A 92, 408–424.Google Scholar
Rosenthal, D. K. 1962 The shape and stability of a bubble at the axis of a rotating liquid. J. Fluid Mech. 12, 358–366.Google Scholar
Ross, D. K. 1968a The shape and energy of a revolving liquid mass held together by surface tension. Austral. J. Phys. 21, 823–835.Google Scholar
Ross, D. K. 1968b The stability of a revolving liquid mass held together by surface tension. Austral. J. Phys. 21, 837–844.Google Scholar
Rumscheidt, F. D. & Mason, S. G. 1961 Particle motions in sheared suspensions XII. Deformation and burst of fluid drops in shear and hyperbolic flow. J. Colloid Sci. 16, 238–261.Google Scholar
Stewartson, K. 1952 On the slow motion of a sphere along the axis of a rotating fluid. Proc. Camb. Phil. Soc. 48, 168–177.Google Scholar
Stewartson, K. 1953 On the slow motion of an ellipsoid in a rotating fluid. Quart. J. Mech. Appl. Math. 6, 141–162.Google Scholar
Stewartson, K. 1958 On the motion of a sphere along the axis of a rotating fluid. Quart. J. Mech. Appl. Math. 11, 39–51.Google Scholar
Taylor, G. I. 1917 Motion of solids in fluids when the flow is not irrotational. Proc. R. Soc. Lond. A 93, 99–113.Google Scholar
Taylor, G. I. 1921 Experiments with rotating fluids. Proc. R. Soc. Lond. A 100, 114–121.Google Scholar
Taylor, G. I. 1922 The motion of a sphere in a rotating liquid. Proc. R. Soc. Lond. A 102, 180–189.Google Scholar
Taylor, G. I. 1923 Experiments on the motion of solid bodies in rotating fluids. Proc. R. Soc. Lond. A 104, 213–219.Google Scholar
Vonnegut, B. 1942 Rotating bubble method for the determination of surface and interfacial tensions. Rev. Sci. Instrum. 13, 6–9.Google Scholar