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Motion of a Sphere in a Viscous Fluid

Published online by Cambridge University Press:  24 October 2008

T. J. I'A. Bromwich
T. J. I'A. Bromwich
Affiliation:
St John's College
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Extract

This problem was originally attacked by Stokes, and the formula obtained by Stokes for the case of steady motion (given at the end of § 1 below) is now generally used in experimental work. However, this formula throws no light on the way in which the velocity of the sphere approaches its steady value: and the only known discussion of this problem was given by Boggio (his results will be found in § 2 below). Unfortunately, when the specific gravity of the sphere exceeds ⅝ (or the constant σ, defined later, is less than 4) the solution so obtained cannot be converted into numerical results, because the constants α, β become complex numbers; and even when α, β are real, the formula is not very convenient for large values of t (as the velocity approaches the steady value).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 25 , Issue 4 , October 1929 , pp. 369 - 383
Copyright
Copyright © Cambridge Philosophical Society 1929

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References

* The function is of the order 1/2x when x> 5; but to evaluate it, from the tables of the Erf function, is practically impossible, since , so that 1−Erfx must be less than 10−11.

† For a convenient summary, reference may be made to the Tract by Jeffreys, H. (Operational Methods in Mathematical Physics, Cambridge, 1927)Google Scholar. The first application of complex integrals to the operational solutions of the equation of diffusion (which is the fundamental equation of the present problem) will be found in my paper, Proc. Camb. Phil. Soc. vol. 20, 1921, p. 411.Google Scholar

‡ The summation requires from 30 to 40 terms of alternate signs; the maximum terms are of order 107 and the final value of the sum is of order 10−8.

§ Math, and Phys. Papers, vol. 3, p. 1.

* A general treatment will be found in a paper of mine on Electromagnetic Waves (Phil. Mag. vol. 38, 1919, p. 143)Google Scholar. To adapt the work given there, to operational forms, it is necessary to replace ik by q, simply: the particular solution needed for our present problem arises from the solution given for n= 1.

* The necessary modifications, when initial motions exist, are found from the rules given in my paper, Proc. Land. Math. Soc. (2), vol. 15, 1916, p. 401Google Scholar. See, in particular, § 8; and H. Jeffreys, Tract, §§ 4, 5.

† It is at this point that the sign to be attached to the real part of q is assumed to be positive; otherwise we cannot distinguish between the above solution and the one which contains e+qr.

‡ Compare the formulae for X, Y in § 4 of my 1919 Phil. Mag. paper.

* This formula has been repeatedly used in modern experiments on the drift of clouds of small particles through a viscous medium.

* See the foot of p. 33 in the Collected Papers, vol. 3.

† See p. 34 (loc. cit.): and Lamb's, Hydrodynamics (3rd ed.), Art. 339 (26).Google Scholar

‡ For a heavy sphere, falling under gravity, F=(M−M′)g.

* Proc. Camb. Phil. Soc. vol. 20, 1921, p. 411 (see § 4);Google Scholar or H. Jeffreys, Tract, pp. 23–26. The result is due to Heaviside.

* For an account of the various notations, adopted by different writers, see Jeffreys, H., Tract, pp. 94–95Google Scholar. In my book on Infinite Series the faetor (2/ √π) has been omitted.

† Rendiconti R. Soc. dei Lincei, vol. 16, 1907, pp. 613, 730: compare Basset, , Quart. Journal of Maths. vol. 41, 1910, p. 369.Google Scholar Boggio reduces the problem to an integral equation, which can be solved by means of Abel's solution.

‡ For then M>M′ and σ<3, so that α, β are complex. For instance, M>4M′ gives σ = 1, and .

§ See, for instance, Infinite Series (2nd ed.), pp. 332–4.

* See my 1921 paper, quoted above. We can easily establish the asymptotic property by first replacing our operator in (27) with a complex integral: and then proceeding on the lines adopted in that paper (§ 4).

* See Millikan, E. A., The Electron (Chicago, 1924), Chs. IV, V.Google Scholar

* These numbers, although simplified to shorten the calculations, are suggested by those observed in an actual experiment tabulated by Millikan (loc. cit. p. 109).

† It seems likely that the experimenters have tested this point by direct observation, although no reference has come to my notice. But no theoretical investigation has been given previously.

‡ Electrician, vol. 53 (1904), p. 995Google Scholar: and Heaviside, , Electromagnetic Theory, vol. 3, pp. 194–211.Google Scholar

§ See also Heaviside, , Electromagnetic Theory, vol. 2, pp. 38–42.Google Scholar

* For instance, Figs. 7, 8 (loc. cit. pp. 955, 956) give graphs of the potential W for values corresponding to σ=20, 10, 4, 2, 1; σ is called z in this paper. In Fig. 7, τ/σ = r 2S/R =.00031846 sec. In Fig. 8, στ = s 2R/S =.0050954 sec.

* The value for t/τ = 160 was derived from formula (36) only.