Dynamics of the Bubble in a Solution

If we integrate (6.167) over the entire surface of a plate of length h and width h, we will obtain the total diffusion flux to the plate  

The last expression can be presented in dimensionless form. Introduce a dimensionless parameter, known in problems on natural convection as Grashof s number, 

Dynamics of gas bubbles in liquids presents significant interest for many reasons. First, bubble motion research provides information about properties of the elementary boundary between liquid and gaseous phases, about the laws governing phase transitions (evaporation, condensation), and about chemical reactions at the surface. Second, this process is also of interest from a purely technical viewpoint. Such branches of industry as gas, petroleum, and chemical engineering commonly utilize processes and devices whose operation is directly interrelated with the laws of bubbles motion. Applications include separation of gas from liquid barbotage of bubbles through a layer of mixture, which is thus enriched by various reagents contained in the bubbles flotation, which is employed in treatment of polluted liquids, etc. 

In the present section, the main attention will be given to the motion of the bubble s surface caused by the change of its volume, rather than the motion of bubbles relative to the liquid. This volume can change as a result of evaporation of the liquid phase or condensation of the gaseous phase. If both the liquid and the bubble are multi-component mixtures, then chemical reactions are possible at the bubble surface, which too can result in a change of the bubble s size. 

Consider a spherical gas bubble of initial radius R, placed in a quiescent liquid [16]. Assume that the bubble center does not move relative to the liquid, but the bubble volume changes with time due to the difference of pressures inside the bubble and in the ambient liquid, and also as a result of dynamic and heat- 

---

Eqs. (22.1)-(22.5) with conditions (22.6) describe the dynamics of the bubble in a solution under the condition that the velocity field of the flow that goes around the bubble is given, and that the bubble itself is not subject to deformation. 

If a bubble of radius Ro was initially placed into a supersaturated (at the given pressure and temperature) mixture, then, due to the concentration difference Ap = pi — pi between the gas dissolved in the liquid far away from the bubble and the appropriate equilibrium value at the bubble surface, there arises a directional diffusion flux of the dissolved substance toward the surface. At the interface, the transition of substance from the liquid to the gaseous state takes place. The result is the increase in the bubble volume. The growth of the bubble, in its turn, results in the increase of its lift velocity, as well as the increase of convective diffusion flux. The statement of the problem and the basic dynamic equations for a bubble in a solution were described in Section 6.8. 

Drag reduction can be achieved by direct injection of microbubbles through slots or porous skin (193-196) or the generation of hydrogen by electrolysis at the wall (197). The primary parameters, independent of gas type and Reynolds number, appear to be the actual gas flow rate referenced to injector conditions of temperature and pressure (198-200) and the location of the bubbles in the turbulent boundary layer (198,199,201-203). Merkle and Deutsch (196) have provided a comprehensive review on skin friction reduction by microbubble injection. Mahadevan and co-workers (204) postulated that microbubbles like polymer solution destroy turbulence production by selectively increasing the viscosity near the buffer region. They increase the local dynamic viscosity. Pal and co-workers (205) demonstrated that microbubble and polymer solution shear stress statistics as measured by flush moimted hot film sensors are similar at equivalent value of drag reduction. 

Here was adopted for simplicity a = A and a 1 (the latter inequality is satisfied for bubbles with Rg > >1 mkm). Phase plot of this equation is presented in Figme 7.2.2. It is seen that for k = -1 (collapsing cavity) z->Zj ast- oo rfzo>Z2. The stationary point z = Zg is unstable. The rate of the cavity collapse z = Zj in the asymptotic regime satisfies inequality Zp < Zj < 0, where Zp = -RCp is equal to the collapse rate of the cavity in a pure viscous fluid with viscosity of polymeric solution q. It means that the cavity closure in viscoelastic solution of polymer at asymptotic stage is slower than in a viscous liquid with same equilibrium viscosity. On the contrary, the expansion under the same conditions is faster at k = 1, Zp < Zj < z, where Zp = RCp and z = Re = (1 - P) RCp is the asymptotic rate of the cavity expansion in a pure solvent with the viscosity (1 - P)q. This result is ejqrlained by different behavior of the stress tensor component controlling the fluid rheology effect on the cavity dynamics, in extensional and compressional flows, respec-... 

A recent design of the maximum bubble pressure instrument for measurement of dynamic surface tension allows resolution in the millisecond time frame [119, 120]. This was accomplished by increasing the system volume relative to that of the bubble and by using electric and acoustic sensors to track the bubble formation frequency. Miller and co-workers also assessed the hydrodynamic effects arising at short bubble formation times with experiments on very viscous liquids [121]. They proposed a correction procedure to improve reliability at short times. This technique is applicable to the study of surfactant and polymer adsorption from solution [101, 120]. 

The surface elasticity force is considered as the most important factor of stability of steady-state foams [113]. In the model of Malysa [123] it is assumed that a dynamic foam is a non-equilibrium system and phenomena occurring in the solution have an influence on the formation and stability of the foam. The foam collapse takes place only at the top of the foam bubbles at thickness larger than 100 nm, where fl = 0. So, the lifetime of the bubbles at the... 

A bubble-bubble interaction model based on potential flow over bluff bodies was developed and incorporated in a code called, BuDY (for Bubble DYnamics). The model is based on an assumption that the instantaneous velocity of an individual bubble in a fluidized bed can be obtained by adding to its rise velocity in isolation, the velocity which the emulsion phase would have had at the nose of the bubble, if the bubble was absent. The details of model development, model equations and solution procedures are described in Ranade (1997a). Appropriate representation of bubble formation, coalescence and exit of bubbles from the dense bed were included in the model. With the knowledge of initial bubble positions and bubble size, subsequent bubble positions can be tracked to predict instantaneous velocities and bubble positions within the dense bed. 

---