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On the Motion of Small Particles of Elongated Form. Suspended in a Viscous Liquid

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Book cover Selected Papers of J. M. Burgers

Abstract

Introductory remarks. As stated in the preceding Chapter, p. 97, a discussion of the influence of suspended particles upon the viscosity of a liquid requires the consideration of the mathematical theory of the motion of such particles. The subject has received much attention in the existing literature, and apart from the classical work by Einstein and by Jeffery and from various subsidiary papers, mention must be made of the reviews which recently have been given by Guth and Mark in the “Ergebnisse der exakten Naturwissenschaften” 1), and by Guth and others in a series of papers which have appeared in the ‘‘KolloicUZeitschrift’’ 2). Nevertheless the Committee has considered it desirable to insert an account of some of the mathematical calculations into the present Report, as it seemed possible to arrange the matter in a form which brings out certain points involved in the theory in a more direct way. At the same time a few details which occurred in Guth’s paper no. 5, could be treated somewhat differently, and numerical evaluations of various formulae have been added. Further a discussion has been given of the information that can be derived from the results of some experimental investigations which have been published in recent years, in particular of Signer’s work.

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References

  1. E. Guth und H. Mark, Die Viskosität von Lösungen, besonders von Lösungen hochmolekularer Stoffe, Ergebnisse der exakten Naturwissenschaften 12, p. 115, 1933.

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  33. Compare: R. Signer, Helv. Chim. Acta 17, p. 70, 1934;

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  35. See: H. Staudinger, Die hochpolymeren organischen Verbindungen (Berlin 1932), which gives an extensive summary of the work up to that date. Further communications on highly polymerized compounds are continually forthcoming.

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  39. The following papers may be mentioned: (as a general summary until 1935): R. Signer, The molecular weights of polystyrenes and the shape of the molecules in solutions, Trans. Farad. Soc. 32, p. 296, 1936;

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  40. The following papers may be mentioned: (special papers): R. Signer, Zeitschr. f. physik. Chemie A 150, p. 257, 1930;

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  52. Compare: R. Signer and H. Gross, Helv. Chim. Acta 17, pp. 59, 335, 726, 1934.

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  53. See also a general exposition of the work with the ultracentrifuge by The Sved- Berg, Nature 139, p. 1051, 1937.

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  58. Compare H. Staudinger, Die hochpolymeren organischen Verbindungen pp. 65, 179. — It must be kept in mind of course that if the value of v for the lower molecular weights should be different from the value 0,92 adopted here, the results of the calculation would come out differently.

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  59. In consequence of the increase of d with molecular weight, there is no proportionality between L and M;roughly: L = 0,54 M2/3 (L in Å). As (with v =0,92) we have L d 2 = 2,90 M, this gives d = 2,3 MVg and thus d = 2,7 L 1/4 (L and d everywhere in Å). This would mean that the cross sectional area of the particle,.a d 2/4, is proportional to the square root of its length. In connection with the empirical relation L ~ M 2 / 3 we may perhaps point to a discussional remark made by H. Mark in reference to a paper by H. Staudinger, Zeitschr. f. Elektrochemie 40, p. 447, form. (3), 1934. See also Addendum II, p. 182 infra.

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  60. According to Helv. Chim. Acta 17, p. 62, 1934, the viscosity measurements for these fractions have been executed with suspensions in chloroform (η = 0,0057 Poise); estimating x= 3000, we obtain Dx= 1,0, which justifies the assumption t = 1/15 made in calculating L/d.

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  61. Compare R. Signer. Trans. Farad. Soc. 32, pp. 301–307. 1936.

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  62. R. Signer and H. Gross, Helv. Chim. Acta 17, p. 73, fig. 2, 1934.

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  64. Compare H. Lamb, Hydrodynamics (Cambridge 1932), Artt. 327, 331, pp. 576 586. The coefficient β introduced by Lamb has been replaced by η/by in the text, in order to obtain a dimensionless coefficient which is proportional to the degree of slipping. — It must be remarked that eqs. (19. 4) and (19. 5) are to be considered as rough approximations only, as they have been obtained by taking a function f(ξ)consisting of a single term; a more correct calculation will show that the numbers —0,81, +0,19 in the denominators must be replaced by slightly different ones. When the calculation is performed, not for cylindrical particles, but for ellipsoids, then (δu/ δy) y=b must be replaced by a more complicated expression; it is found that in this case it becomes much more difficult to satisfy the condition, unless it is assumed that the coefficient γ is a function of the position of the surface element considered, and gradually decreases towards the ends of the particle.

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Authors and Affiliations

  1. Delft, The Netherlands

    J. M. Burgers

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Editors and Affiliations

  1. Department of Mechanical Engineering and Maritime Technology, Delft University of Technology, The Netherlands

    F. T. M. Nieuwstadt

  2. Department of Aero and Space Engineering, Delft University of Technology, The Netherlands

    J. A. Steketee

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Burgers, J.M. (1995). On the Motion of Small Particles of Elongated Form. Suspended in a Viscous Liquid. In: Nieuwstadt, F.T.M., Steketee, J.A. (eds) Selected Papers of J. M. Burgers. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0195-0_9

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  • DOI: https://doi.org/10.1007/978-94-011-0195-0_9

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