By E. T. Whittaker

There might be few books on mathematical mechanics as well-known as this, a piece that varieties a entire account of the entire classical result of analytical dynamics.

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This variation in local curvature plays an important role in the surrounding 28 SINGLE BUBBLE RISE CHARACTERISTICS flow fields; the magnitude of vorticity generated at any point on the bubble surface increases proportionally with the local curvature at that point [see Eq. 11); Ryskin and Leal, 1984a]. It should be noted, however, that the resulting changes in the vorticity distributions, and thus the velocity distributions, have little influence on the overall bubble shape. 2 Bubble Shape Prediction under Small Distortion Once the bubble shape is identified, it can be quantified in terms of geometric parameters such as the aspect ratio [minor (vertical) axis/major (horizontal) axis, or h/b], the eccentricity (E, inverse of the aspect ratio), the ratio of equivalent diameter to the major axis (c?

2 where Ε ι 2/3 i (2 . o I 0 2 1 1 1 +°ni^^ «* • Λ · a. » Eq. 2 - 1 Data from Kubota et al. 77x10"* Ethanol Contaminated liquid ο 2 . 6(a) Aspect ratio of air bubbles in liquids as a function of Weber number. Ks takes into account the effect of possible separation of the flow around the bubble (refer to Chapter 3 for the existence of a wake behind a spiraling bubble). Under the assumption of ideal flow around the bubble, Κ s = 1, but K is greater than unity if flow separation occurs. Saffman (1956) suggested typical values of K to be between 1 and 2.

2 BUBBLE RISE VELOCITY 41 a generalized correlation equation for C/& is introduced which takes into consideration the existing literature data. , Peebles and Garber, 1953; Levich, 1962; Wallis, 1969): for instance, Stokes' solution (Stokes, 1851) for sufficiently small spherical bubbles (Ree < 1), and the equation of Davies and Taylor (1950) for large spherical-cap bubbles (We > 20 or Eo > 40). Predictive capability of these equations has proven satisfactory for bubbles in various liquids provided the proper regime is selected.

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