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Advances in Applied Mechanics, Vol. 39 by Erik van der Giessen, Hassan Aref

By Erik van der Giessen, Hassan Aref

The most important advancements within the box of fluid and reliable mechanics are scattered all through an array of clinical journals, making it frequently tough to discover what the true advances are, particularly for a researcher new to the sphere. The Advances in utilized Mechanics booklet sequence attracts jointly the hot major advances in a variety of subject matters in utilized mechanics. released due to the fact 1948, Advances in utilized Mechanics goals to supply authoritative evaluation articles on issues within the mechanical sciences, basically of curiosity to scientists and engineers operating within the quite a few branches of mechanics, but additionally of curiosity to the numerous who use the result of research in mechanics and diverse program components. Advances in utilized Mechanics remains to be a book of excessive effect. evaluation articles are supplied via major scientists within the box on a call for participation in basic terms foundation. the various articles released became classics inside their fields. quantity 39 within the Mechanics sequence includes articles on vortex dynamics, the numerical simulation of two-phase flows, environmental difficulties in China, and piezoelectrics.

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Kimura and Okamoto, 1987). Thus, if the sum of the strengths, Ga ; is zero, the 54 H. Aref et al. point vortices carry all the vorticity in the flow. If not, one has to remember the existence of a largely passive, uniform vorticity in the background. ) Using nb ¼ xb =R in the numerator, and lxa 2 xb l2 ¼ 2ðR2 2 xa ·xb Þ in the denominator, Eq. 1) may be recast as N xa £ xb dxa 21 X 0 ¼ G : 4pR b ¼ 1 b R2 2 xa ·xb dt ð10:2Þ Another interesting reformulation arises by projecting the surface of the sphere stereographically onto the plane tangent to the sphere at its north pole.

3 as n ! 1. The existence and stability of these ‘double-rings’ was discussed by Havelock (1931) in an important paper, where particular attention was paid to the staggered case for g ¼ 2 1, a circular counterpart of the Ka´rma´n vortex street. With the notation introduced above, the equation determining the ratio of ring radii in this case is ðn 2 1Þcosh½ðn þ 2ÞrŠ ¼ ðn þ 1Þcosh½ðn 2 2ÞrŠ: ð6:11Þ For large n, the ratio of radii, r, will tend to 1 as 1 þ n 21. 32, somewhat larger than von Ka´rma´n’s ratio for the least unstable point vortex street.

5) now gives ! n ! 2 : ð9:6Þ Formally, the quantities on both sides of Eq. 6) are ‘variances’ of the complex coordinates of the vortices in the two populations. 6) suffices to show that Eqs. 2) have no solution for n ¼ 2. This is somewhat surprising, since one might have thought that two pairs, placed at a great distance from one another, would translate independently with minimal mutual influence and so would approximate a translating state for n ¼ 2. One might further have assumed that slight adjustments to this state would make it exactly satisfy Eqs.

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