Bubble spherically symmetric

For swarms of spherical bubbles, the field may be expected to be approximately spherically symmetric when the origin of coordinates is fixed on the center of mass of a typical particle. Therefore, by using spherical coordinates and the initial condition ... 

The apparatus used in the present study is based on the axi-symmetric bubble shape analysis, i.e., a firmly established technique for the measurement of static and dynamic surface tension as well as of the geometrical properties of the bubble (Loglio et al. 1996, 2001, Kovalchuk et al. 2001, Miller et al. 2000, Rusanov and Prokhorov 1996, Neumann and Spelt 1996, Cheu et al. 1998). In essence, the shape of a bubble (or of a drop) is determined by a combination of surface tension and gravity effects. Surface forces tend to make drops and bubbles spherical whereas gravity tends to elongate them. 

Problem 12-6. The Linear Stability of a Spherically Symmetric Fluid Interface to Radial Accelerations. The classical Rayleigh Taylor analysis that is described in Section B examines the stability of a plane interface between two fluids of different density to accelerations normal to the interface and shows that the interface is unstable or stable, depending on whether the acceleration is directed from the heavier fluid to the lighter fluid, or vice versa. In this problem, we consider the related problem of a spherically symmetric interface that is subjected to radial accelerations. This is a generalization of the problem of an expanding or contracting gas bubble that was considered in Chap. 4. 

In chemical technology, one often meets the problem about a spherically symmetric deformation (contraction or extension) of a gas bubble in an infinite viscous fluid. In the homobaric approximation (the pressure is homogeneous inside the bubble) [306, 312], only the motion of the outer fluid is of interest. The Navier-Stokes equations describing this motion in the spherical coordinates have the form... 

Consider the growth of a small vapor or gas bubble, far from any other body, initially in equilibrium with a large volume of stationary liquid, as a result of a small displacement from equilibrium. The bubble maintains a spherical shape because of surface tension, and the liquid motion is purely radial. Note that this does not imply that the temperature or concentration field is likewise spherically symmetric. The governing equations and boundary... 

In this form the parametric solutions, Eqs. (73) and (74), for the spherically symmetric problem can be applied immediately. Physically this implies that a bubble growing in an initially arbitrary temperature field grows at precisely the same rate as if the initial temperature were averaged in each thin spherical shell surrounding the bubble center. Two special cases are considered in detail (7) the linear thermal boundary layer, of thickness /, next to the heating surface, outside of which the temperature is uniform and (2) the exponential boundary layer, where the temperature is assumed to decay exponentially with distance from the wall. The latter distribution is of the form... 

Consider now the diffusion growth of an isolated motionless bubble in a binary solution. Denote by pn the mass concentration of the dissolved component in the liquid. Suppose the bubble consists only of component 1, the process is spherically symmetrical, and the distribution p i is described by the equation of convective diffusion (22.1), in which should be should take Ug = 0 and Ur = (R/r) R under the condition p Q p. ... 

For the turbulent flow regime, the flux of bubbles of volume co toward the test bubble of volume V can be considered as a diffusion flux with the effective coefficient of diffusion Dr. Consider a bubble of volume V, placed in a turbulent flow of liquid containing bubbles of volume co with the number concentration n. Assuming that the process is stationary and spherically symmetric, we have the following equation describing the distribution n(r) ... 

Fast motions of a bubble surface produce sound waves. Small (but non-zero) compressibility of the liquid is responsible for a finite velocity of acoustic signals propagation and leads to appearance of additional kind of the energy losses, called acoustic dissipation. When the bubble oscillates in a sound field, the acoustic losses entail an additional phase shift between the pressure in the incident wave and the interface motion. Since the bubbles are much more compressible than the surrounding liquid, the monopole sound scattering makes a major contribution to acoustic dissipation. The action of an incident wave on a bubble may be considered as spherically-symmetric for sound wavelengths in the liquid lr >Ro-When the spherical bubble with radius is at rest in the liquid at ambient pressure, pg), the internal pressure, p, differs from p by the value of capillary pressure, that is... 

For a quantitative discussion of the model, the bubble is regarded as a spherically symmetric potential well in the interior of which the Ps atom moves in a potential-free space. The parameters of the bubble may be determined by quantum-mechanical methods on the basis of the experimentally measured lifetime and angular distribution data [Le 73b]. 

For polymers interfacial and surface tensions are more practically obtainable from analysing the shapes of pendant or sessile drops or bubbles, all of which are examples of axisymmetrical drops. Bubbles may be used to obtain surface tensions at liquid/vapour interfaces over a range of temperatures and for vapours other than air. Drops can also be used to obtain vapour/liquid surface tensions but they are particularly suited to determination of liquid/liquid interfacial tensions, for example for polymer/polymer interfaces. All the methods are based on the application of equation (2.2.1). The principles are illustrated in figure 2.4, in which a sessile drop is used as the specific example. Just like for the capillary meniscus, the drop has two principal radii of curvature, R in the plane of the axis of symmetry and / 2 normal to the plane of the paper. At the apex, O, the drop is spherically symmetrical and R = Rz = b and equation (2.2.12) becomes... 

In what follows we present the equations and boundary conditions for the problems which are addressed here, namely axisymmetric jets (two-phase generalizations are straightforward) and spherically symmetric drops or bubbles. Before presentation of the mathematical models, we also give a brief description of the applications and what we hope too understand by the theoretical studies. For the sake of brevity, the elements of tensor algebra which are needed in deriving interfacial conditions will be stated and described as needed. For general descriptions the reader is referred to the texts by Aris [1], Edwards et al. [18] and McConnell... 

Bubble radius gives the extensional strain and strain rate. If we can ignore diffusion and the influence of the capillary and assume that the bubble collapses symmetrically in an incompressible fluid, then the continuity equation in spherical coordinates (Table 1.7.1) reduces to... 

Bubbles in polymer solutions also collapse at constant rate as indicated in Figure 7.6.3. Note that for these lower viscosity materials, bubble collapse can be very rapid. When the bubble gets too small, necking occurs near the capillary and the collapse is no longer spherically symmetric (Figure 7.6.3b). 

Differential Equations 2.35 and 2.36 are easily solved for the nonstationary axial-symmetric nontwisted turbulent flow of a continuous incompressible Newtonian two-phase medium without ta g interfacial heat and mass exchange into consideration. Therefore, the source of f in Equation 2.35 is the force of interfacial interaction caused by tension. It has been assumed that all the dispersion inclusions (droplets, bubbles, and so on) are spherical. 

Photograph A shows the instantaneous flow field where a marked increase in the heat transfer coefficient results as the bubble approaches the heat transfer surface. It is also observed that the instantaneous heat transfer coefficient starts increasing well before the bubble approaches the lower edge of the probe, although the photograph for this case is not shown here. This increase in heat transfer coefficient is attributed to the local turbulence caused by the approaching bubble. The bubble is a spherical cap, and the wake structure appears to be symmetrical about the vertical axis of bubble movement. 

The pressure field parameters, impulse and time characteristics of waves and gas bubble pulsation occurring at an underwater gas explosion were measured in [45,46]. The underwater gas detonations were triggered in a half-closed vessel and a spherical shell. The pressure field of the last case is symmetrical and all-directional. The experimental data obtained for C3H8 + 5O2 and 2H2 + O2 mixtures were in good agreement with the theoretical results. 

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