Force balance bubble volume

The final volume VF can be evaluated by finding individually the value of the force-balance bubble volume vfb and the volume Qtc entering the bubble during detachment, and then adding the two. Thus,... 

The model for inviscid liquids is equally well applicable to viscous liquids also, provided that the resistance due to viscous drag is included in the analysis. As a first approximation, the viscous drag may be evaluated by a Stokes resistance term, since the bubble is not followed by a wake. Thus we proceed as before, first evaluating the force-balance bubble volume Kr and then the total bubble volume by reference to the detachment stage. 

Knowing the force balance bubble volume Vfb and the other system charctaeristics, the final bubble volume VF can be directly calculated from Eq. (61). 

Considering the case of vanishingly small flows (Q a 0), both the first and the second terms on the right-hand side of Eq. (55) which contain Q, become zero, and the force balance bubble volume is obtained directly by equating the surface tension force with the buoyancy force. Even in the detachment stage,... 

As before, the force balance bubble volume is first found, and its volume is then used to determine the final bubble volume. 

Equation (89) represents an implicit relationship between the force balance bubble volume and the other variables. 

The force balance bubble volume equation, on the introduction of the drag term given by Eq. (99) becomes... 

The force balance bubble volume at the end of the expansion stage is given by... 

Tate s law makes use of bubble formation in the measurement of the surface tension of a liquid, and hence the fact that surface tension influences the bubble volume is obvious. At flow rates tending to zero, the bubble volume is such that the upward force due to buoyancy is balanced by the downward force of surface tension. So, an increase in surface tension should... 

At the end of the expansion stage, when the bubble volume is Vf and its radius rfb, the force balance becomes ... 

The bubble volume achieved in the first stage is predicted from a force balance ... 

With such an understanding on system complexity in mind, the DBS model is composed of two simple force balance equations, respectively, for small or large bubble classes, and one mass conservation equation as well as the stability condition serving as a variational criterion and a closure for conservative equations. For a given operating condition of the global system, six structure parameters for small and large bubble classes (their respective diameters dg, dL, volume fraction... 

In a polymeric system, it would be reasonable to examine the possibility that the free volume concept described earlier, which explains so well phenomena like molecular diffusion and viscosity, might perhaps also explain nucleation phenomena. The critical radius re of a stable bubble can be obtained from a simple mechanical-force balance, yielding the Laplace equation ... 

The flow of the continuous phase is considered to be initiated by a balance between the interfacial particle-fluid coupling - and wall friction forces, whereas the fluid phase turbulence damps the macroscale dynamics of the bubble swarms smoothing the velocity - and volume fraction fields. Temporal instabilities induced by the fluid inertia terms create non-homogeneities in the force balances. Unfortunately, proper modeling of turbulence is still one of the main open questions in gas-liquid bubbly flows, and this flow property may significantly affect both the stresses and the bubble dispersion [141]. 

Bubble Volume at Incipient Detachment Using Energy and Force Balance... 

The buoyancy force acting on bubbles in the control volume is balanced by the drag force acting on the control volume. 

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