Bubbles spherical shape

David W. Taylor Model Basin, Washington, September 1953 Jackson, loc. cit. Valentin, op. cit.. Chap. 2 Soo, op. cit.. Chap. 3 Calderbank, loc. cit., p. CE220 and Levich, op. cit.. Chap. 8). A comprehensive and apparently accurate predictive method has been publisned [Jami-alahamadi et al., Trans ICE, 72, part A, 119-122 (1994)]. Small bubbles (below 0.2 mm in diameter) are essentially rigid spheres and rise at terminal velocities that place them clearly in the laminar-flow region hence their rising velocity may be calculated from Stokes law. As bubble size increases to about 2 mm, the spherical shape is retained, and the Reynolds number is still sufficiently small (<10) that Stokes law should be nearly obeyed. 

Observations of bubbles emerging through the bed surface show that bubble shape is markedly dependent on liquid velocity. This indicates the existence of a relationship between bed viscosity and liquid velocity. A bed near incipient fluidization is characterized by a high viscosity, and an emerging bubble is of nearly spherical shape, whereas a fluidized bed of high porosity is characterized by a viscosity not very much higher than that of water, so that an emerging bubble is of spherical cap shape. 

Equation (17) indicates that the entire distribution may be determined if one parameter, av, is known as a function of the physical properties of the system and the operating variables. It is constant for a particular system under constant operating conditions. This equation has been checked in a batch system of hydrosols coagulating in Brownian motion, where a changes with time due to coalescence and breakup of particles, and in a liquid-liquid dispersion, in which av is not a function of time (B4, G5). The agreement in both cases is good. The deviation in Fig. 2 probably results from the distortion of the bubbles from spherical shape and a departure from random collisions, coalescence, and breakup of bubbles. 

Imagine a stream of bubbles rapidly rising to the surface of glass of soda. What shape are these bubbles Without any prompting, all of us would picture spherically shaped bubbles. What chemical principles preclude oval-shaped or cubic-shaped bubbles ... 

A bubble of air in a liquid is, as we know, spherical, and it is obvious that this spherical shape can only be maintained if the pressure on the inside is greater than that outside. Let P be the excess of pressure inside per unit surface, and a the radius of the sphere the pressure tending to force the two hemispheres apart is then evidently P x area of largest circle, i.e., P naa. This pressure is balanced by the pull arising from surface tension, which acts round the circumference of the same circle, and is, accordingly, 2ira[Pg.17]

The deviation of the experimental values from the theoretical ones has been explained as being due partly to the upward current induced by the bubbles in the liquid surrounding the orifice and partly to the distortion in the spherical shape of the forming bubbles. 

All attempts to introduce fluorosilicone oils, in which the fluoro-alkyl group was covalently bound to the Si-O-Si backbone, were without success [31,32], A new approach to create a heavier than water silicone oil was the mixture of partially fluorinated compounds and ultra-purified silicone oil. The first product on the market was Oxane Hd (Bausch Lomb Inc., Rochester, NY, USA), a mixture of 1-perfluorooctyl-5-methylhex-2-en and silicone oil 5000 mPas. The specific density of this clear mixture is 1.02 g/ml. This creates the possibility to treat the lower quadrant of the retina. An intra-ocular bubble of Oxane Hd has an ideal spherical shape as demonstrated in Fig. 4. 

All the work discussed in the preceding sections is subject to the assumptions that the fluid particles remain perfectly spherical and that surfactants play a negligible role. Deformation from a spherical shape tends to increase the drag on a bubble or drop (see Chapter 7). Likewise, any retardation at the interface leads to an increase in drag as discussed in Chapter 3. Hence the theories presented above provide lower limits for the drag and upper limits for the internal circulation of fluid particles at intermediate and high Re, just as the Hadamard-Rybzcynski solution does at low Re. 

Transfer from large bubbles and drops may be estimated by assuming that the front surface is a segment of a sphere with the surrounding fluid in potential flow. Although bubbles are oblate ellipsoidal for Re < 40, less error should result from assumption of a spherical shape than from the assumption of potential flow. 

Sy et al (S8, S9) and Morrison and Stewart (M12) analyzed the initial motion of fluid spheres with creeping flow in both phases. For bubbles (y = 0, k = 0), the condition that internal and external Reynolds numbers remain small is sufficient to ensure a spherical shape. However, for other k and y, the Weber number must also be small to prevent significant distortion (S9). For k = 0, the equation governing the particle velocity may be transformed to an ordinary differential equation (Kl), to give a result corresponding to Eq. (11-16), i.e.,... 

Consider the vacuum forming of a polymer sheet into a conical mold as shown in Figure 7.84. We want to derive an expression for the thickness distribution of the final, conical-shaped product. The sheet has an initial uniform thickness of ho and is isothermal. It is assumed that the polymer is incompressible, and it deforms as an elastic solid (rather than a viscous liquid as in previous analyses) the free bubble is uniform in thickness and has a spherical shape the free bubble remains isothermal, but the sheet solidifies upon confacf wifh fhe mold wall fhere is no slip on fhe walls, and fhe bubble fhickness is very small compared fo ifs size. The presenf analysis holds for fhermoforming processes when fhe free bubble is less than hemispherical, since beyond this point the thickness cannot be assumed as constant. 

In Figure 7.84, we note that after a certain time the free bubble contacts the mold at height z and has a spherical shape of radius R. The radius is determined by the mold geometry and bubble position and is given by... 

If there are sufficiently strong repulsive interactions, such as from Ihe electric double-layer lorce. then the gas bubbles at the lop of u froth collect together without bursting. Furthermore, their interfaces approach as closely as these repulsive forces allow typically on the order of 100 nm. Thus bubbles on top of a froth can pack together very closely and still allow most uf the liquid to escape downward under the influence of gravity while maintaining their spherical shape. Given sufficient liquid, such a foam can resemble the random close-packed structure formed by hard spheres. 

It is well known that short-range forces of attraction exist between molecules (see page 215), and are responsible for the existence of the liquid state. The phenomena of surface and interfacial tension are readily explained in terms of these forces. The molecules which are located within the bulk of a liquid are, on average, subjected to equal forces of attraction in all directions, whereas those located at, for example, a liquid-air interface experience unbalanced attractive forces resulting in a net inward pull (Figure 4.1). As many molecules as possible will leave the liquid surface for the interior of the liquid the surface will therefore tend to contract spontaneously. For this reason, droplets of liquid and bubbles of gas tend to attain a spherical shape. 

Consider the molecules in a liquid. As shown in Figure 3.1, for a liquid exposed to a gas the attractive van der Waals forces between molecules are felt equally by all molecules except those in the interfacial region. This imbalance pulls the latter molecules towards the interior of the liquid. The contracting force at the surface is known as the surface tension. Since the surface has a tendency to contract spontaneously in order to minimize the surface area, droplets of liquid and bubbles of gas tend to adopt a spherical shape this reduces the total surface free energy. For two immiscible liquids a similar situation applies, except that it may not be so immediately obvious how the interface will tend to curve. There will still be an imbalance of intermolecular forces resulting in an interfacial tension and the interface will adopt a configuration that minimizes the interfacial free energy. 

When detergent molecules find themselves in water, they tend to associate with each other. A good example of this association is seen in the bubbles that slosh around in the washing machine while you re doing laundry. The bubbles consist of very thin layers of detergent (plus some water) in which the molecules are packed side by side. The spherical shape of the bubbles is due to a physical force called surface tension, which acts to reduce the area of the bubble to the smallest area able to accommodate the detergent. If you take the molecules from a cell membrane, purify them away from all the other components of a cell, and dissolve them in water, they will often pack together into a spherical, enclosed shape. 

Bubbles with diameter smaller than 0.01 cm (Re < 0.5) rise as solid particles and obey Stokes law [13,14], This is due to the fact that when bubbles are small even negligible amounts of surface active agents are sufficient to achieve complete immobility of the surfaces. When Re > 0.5, a deviation from Stokes law is observed but the spherical shape of bubbles is retained up to Re values close to 1500. When Re > 200-300 the velocity of bubble rise in the absence of surfactant satisfies the following equation [14]... 

Bubble of size of the order of tens of micrometers floating on the surfactant surface only little deviates from the spherical shape. This fact has been used in the method of diminishing bubble [128,129] which allows to measure the contact angle of the black film, the linear tension of the contact line film/meniscus and the coefficient of the gas permeability through the film. Fig. 2.24 presents the scheme of this device. 

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