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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Extra info for Treatise on analytical dynamics of particles and rigid bodies
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.