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The hydrodynamic interaction of two spheres moving in an unbounded fluid at small but finite Reynolds number. (English) Zbl 0541.76042
The Stokes equations are in general used to analyze the interaction of two or more bodies immersed in an unbounded fluid. It is clear however that the omitted inertial term may introduce qualitative changes to the obtained picture. Thus the problem of hydrodynamic interactions with finite Reynolds number is an important one.
The solution obtained by the authors is based on the method of matched asymptotic expansion and the main assumptions are: (a) the distance l between spheres is much larger than their radii; (b) \(| U_ A-U_ B|\), where \(U_ A\), \(U_ B\) are velocities of the spheres A and B, is sufficiently small to allow to consider the flow as a steady one; (c) the Reynolds number must fulfill \(R\ll 1\) and, moreover, the distance between the spheres 1 is much smaller than a/R, a being the radius of the spheres. Provided these assumptions are fulfilled, the second sphere, B, is located in the inner region of expansion of the sphere A.
The consecutive steps in solving the problem are as follows: (a) the method of H. Brenner and R. G. Cox [(*) ibid. 17, 561-595 (1963; Zbl 0116.180)] is used to write the inner and outer expansions and to formulate the matching conditions; (b) force and torque are obtained by integration over a volume bounded internally by the surfaces of the spheres and externally by a spherical surface \(S_ L\) the radius of which is later sent to infinity; (c) all but one of the resulting integrals are calculated using the Brenner and Cox technique (*). The calculation of the remaining integral is most complicated and is finally performed using the algebraic manipulation language REDUCE-2 [A. C. Hern, REDUCE-2 user’s manual, UPC-19 Univ. of Utah (1973)]. Results of this integration are checked numerically.
It is shown that the obtained results are in agreement with those of S. I. Rubinow and J. B. Keller [ibid. 11, 447-459 (1961; Zbl 0103.195)] for one sphere and with P. Vasseure and R. G. Cox [ibid. 80, 561-591 (1977; Zbl 0351.76121)] for \(\ell \gg a/R\) (in a particular case allowing this comparison).
It is shown also that for two spheres of equal size sedimenting vertically the leading sphere experiences larger drag than the trailing one (i.e. they attract each other).
The authors claim that their method is applicable for the most interesting case when \(a/\ell\) is of order one and they promise to show it in their next paper.
Reviewer: R.Herczyński

MSC:
76D07 Stokes and related (Oseen, etc.) flows
76-04 Software, source code, etc. for problems pertaining to fluid mechanics
Software:
REDUCE
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References:
[1] DOI: 10.1017/S0022112077001840 · Zbl 0351.76121 · doi:10.1017/S0022112077001840
[2] DOI: 10.1017/S0022112061000640 · Zbl 0103.19503 · doi:10.1017/S0022112061000640
[3] DOI: 10.1017/S002211206300152X · Zbl 0116.18003 · doi:10.1017/S002211206300152X
[4] DOI: 10.1017/S0022112072002927 · Zbl 0247.76088 · doi:10.1017/S0022112072002927
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