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Mass transfer from a particle suspended in fluid with a steady linear ambient velocity distribution

Published online by Cambridge University Press:  19 April 2006

G. K. Batchelor
G. K. Batchelor
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
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Abstract

This paper is concerned with the rate of transfer of heat or mass from a force-free couple-free particle immersed in fluid whose velocity far from the particle is steady and varies linearly with position. Asymptotic results for both small and large Péclet numbers are considered. There is at least a four-parameter family of different linear ambient velocity distributions, but nevertheless a comprehensive set of results for the transfer rate may be compiled by combining previously published work with some new developments. Some of these are exact results for particular linear ambient flow fields and some are approximate results for classes of linear flow fields.

For small Péclet number (P), the non-dimensional additional transfer rate due to convection is equal to αN20P½, where N0 is the Nusselt number for P = 0 and the proportionality constant α is a parameter of the concentration distribution due to a steady point source in the given linear ambient flow field. A general method of determining α is developed, and numerical values are found for some particular linear ambient flow fields. It is estimated that the value of α for any linear ambient flow field in which the vorticity does not dominate the straining motion lies within 10% of 0·34 when P is defined in terms of a particular invariant of the ambient rate-of-strain tensor E.

At large Péclet number the transfer rate N depends on the velocity distribution near the particle, and attention is restricted to the case of a sphere in low-Reynolds-number flow. For a rigid sphere $N = \beta P^{\frac{1}{3}}$ for any ambient pure straining motion, and the Levich concentration-boundary-layer method may be used to show that β = 0·90 for both axisymmetric and two-dimensional ambient pure straining, and probably for any other pure straining motion, when P is suitably defined. When the ambient vorticity ω is non-zero, the sphere rotates, and the Levich method cannot be used. However, it is shown that the part of the velocity distribution that varies sinusoidally with the azimuthal angle around the rotation axis does not affect the transfer rate and that N is asymptotically the same as for an ambient axisymmetric pure straining motion with rate of extension in the direction of the axis of symmetry equal to Eω(= ω. E. ω/ω2). In the exceptional case Eω = 0, N approaches a constant as P → ∞.

It is possible to interpolate between the asymptotic relations for small and large Péclet number with comparatively little uncertainty for any ambient pure straining motion and for any linear ambient flow field in which ω and Eω are non-zero.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 95 , Issue 2 , 28 November 1979 , pp. 369 - 400
Copyright
© 1979 Cambridge University Press

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