Number Morton

Flow Past Deformable Bodies. The flow of fluids past deformable surfaces is often important, eg, contact of Hquids with gas bubbles or with drops of another Hquid. Proper description of the flow must allow for both the deformation of these bodies from their shapes in the absence of flow and for the internal circulations that may be set up within the drops or bubbles in response to the external flow. DeformabiUty is related to the interfacial tension and density difference between the phases internal circulation is related to the drop viscosity. A proper description of the flow involves not only the Reynolds number, dFp/p., but also other dimensionless groups, eg, the viscosity ratio, 1 /p En tvos number (En ), Api5 /o and the Morton number (Mo),giJ.iAp/plG (6). 

Morton number, 11 775 Morveau, Louis Bernard Guyton de,... 

The value of ki depends on the Eotvos number, Ed, and the Morton number, Gj, as shown in Figure 7.12. 

For bubbles and drops rising or falling freely in infinite media it is possible to prepare a generalized graphical correlation in terms of the Eotvds number, Eo Morton number, M and Reynolds number. Re (Gl, G2) ... 

The conditions under which fluid particles adopt an ellipsoidal shape are outlined in Chapter 2 (see Fig. 2.5). In most systems, bubbles and drops in the intermediate size range d typically between 1 and 15 mm) lie in this regime. However, bubbles and drops in systems of high Morton number are never ellipsoidal. Ellipsoidal fluid particles can often be approximated as oblate spheroids with vertical axes of symmetry, but this approximation is not always reliable. Bubbles and drops in this regime often lack fore-and-aft symmetry, and show shape oscillations. 

There is considerable evidence (D3, G7, PI, P4, SI) that bubbles in liquid metals show the behavior expected from studies in more conventional liquids. Because of the large surface tension forces for liquid metals, Morton numbers tend to be low (typically of order 10 ) and these systems are prone to contamination by surface-active impurities. Figure 8.10a shows a two-dimensional nitrogen bubble in liquid mercury. For experimental convenience, the bubbles studied have generally been rather large, so that there are few data available for spherical or slightly deformed ellipsoidal bubbles in liquid metals. Data... 

Morton number, =gii Ap/p a acceleration modulus, = d/U ) dU /dt) displacement modulus, =xjd Mach number, = characteristie velocity/c... 

Mach number Morton number Molecular weight Power law exponent Blend time number Best number Power number Pumping number Pressure... 

Fig. 12. Typical results reported by Tomiyama ei al. (1993) on the effect of the Morton number M (atEotvSs number Eo = 10) on the shape and dynamics of a single bubble rising in (a) a Newtonian liquid, and (b) graphical correlation due to Grace (1973) and Grace et al. (1976). [Part (a) reprinted from Nuclear Engineering and Design, Volume 141, Tomiyama, A., Zun, I., Sou, A., and Sakaguchi, T., Numerical analysis of bubble motion with the VOF method, pp. 69-82, Copyright 1993, with permission from Elsevier Science. Part (b) reprinted from Grace, R., Clift, R., and Weber, M.E., Bubbles, Drops, and Particles. Academic Press, Orlando, 1976. Reprinted by permission of Academic Press.)...

For bubble (b) the Eotvos number is 6.24 and the Morton number is 6.31 X 10 . The bubbles have totally different motion depending on whether the bubble deforms or not. Bubble (a) does not deform significantly, and rotates with the flow as it rises, and eventually experiences a lift to the right. Bubble (b) deforms due to the shear and the upward motion. The bubble thus takes the form of an airfoil, and experiences lift to the left. The circulation of bubble (b) changes as the bubble deforms, and settles in the opposite direction of the circulation of bubble (a). 

Ap is the difference in material density between the liquid and gas phases. This situation is typically handled by describing the bubbles with a single internal coordinate (i.e. the equivalent-sphere diameter) and by introducing an aspect ratio, defined as the ratio between the minor and the major axes of the bubble. This aspect ratio E can be calculated by using the empirical equation proposed by Moore (1965) as a function of the Morton number E = 1/(1 + 0.043RCp Mo ). An alternative to this is the use of the correlation proposed by Wellek et al. (1966) for liquid-liquid droplets E = 1/(1 + 0.1613Eo° ), which is valid for Eo < 40 and Mo < 10 , whereas for Eo > 40 and RCp > 1.2 fluid particles are typically of spherical shape. Once the characteristic E value is known, the ratio of the real area of the bubble Ap and the area Aeq of a sphere with an equivalent volume can be calculated as follows ... 

Mach number for continuous phase Mach number for disperse phase Morton number... 

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