Abstract
The classical approaches to the problem of flow past a sphere are presented, along with the more modern technique of matched inner and outer expansions. For the huge literature on flow past nonspherical obstaces, a list of references is provided. Stokeslets are introduced and related to Lighthill’s compelling work on the propulsion of microorganisms.
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- 1.
The component \(\upsilon _{r}\) vanishes identically on the θ = π∕2 plane, where ζ r remains finite. Hence it might be objected that the inertia term in Eq. (3.102) cannot be neglected in the neighborhood of this plane, no matter how small the Reynolds number . However the vanishing of \(\upsilon _{r}\) is merely an accident resulting from the selection of coordinate system. It would not, in general, be observed in a coordinate system with origin, say, off the axis of symmetry. The Navier-Stokes equation is a vector equation and it is the relative vector magnitudes of ζ i and \(\upsilon _{i}\) that matter.
- 2.
- 3.
The exact solution to the problem was given by Goldstein (1929). Investigations of this sort are motivated by the idea that Oseen’s equation and the Navier-Stokes equation are qualitatively similar, so that solutions of the former might be expected to yield qualitative information about solutions of the latter for all Reynolds numbers.
- 4.
This is most easily seen by observing that it vanishes in a Cartesian system.
- 5.
In fact it does worse than neglect the inertia in this region: it misrepresents it. However the misrepresentation is also negligible.
- 6.
Lighthill was an accomplished open-water swimmer, believed to be the first to complete the 14 km swim around Sark. He first did it at age 49, and repeated it five times subsequently. He had nearly completed his seventh attempt, at age 74, when he incurred a fatal rupture of his mitral valve.
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Langlois, W.E., Deville, M.O. (2014). Flow Past a Sphere. In: Slow Viscous Flow. Springer, Cham. https://doi.org/10.1007/978-3-319-03835-3_6
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