Fluid-bubble interaction force

Linear stability analyses were also carried out for bubble columns by, e.g., Shnip et al (1992), Minev et al (1999), Leon-Becerril and Line (2001), Sankaranarayanan and Sundaresan (2002), Lucas et al (2005, 2006), and Monahan and Fox (2007). The reader is also referred to the review paper by Joshi (2001). The purpose of these analyses was to learn more about the stabihty of homogeneous (or uniform) bubble flow and the transition to heterogeneous flow regimes. In most cases, the results of the analyses were again strongly dependent on the exact formulation of the fluid—bubble interaction force, particularly of its lateral component. Several of these authors pinpoint the crucial effects of virtual mass and of (the sign of) the lift force. 

This conclusion provided an emphatic justification for the phenomenon of bubbles in gas-fluidized beds. As these beds represented the most widespread and important industrial applications of fluidization technology, it is not difficult to appreciate the considerable interest generated in the mid-1960s by the diverse analyses (by different authors, employing different fluid-particle interaction force expressions) that arrived at essentially the same conclusion. The inconvenient fact that liquid-fluidized beds appeared to manifest stable, homogeneous behaviour did little to detract from its almost universal acceptance. 

One therefore has to decide here which components of the phase interaction force (drag, virtual mass, Saffman lift, Magnus, history, stress gradients) are relevant and should be incorporated in the two sets of NS equations. The reader is referred to more specific literature, such as Oey et al. (2003), for reports on the effects of ignoring certain components of the interaction force in the two-fluid approach. The question how to model in the two-fluid formulation (lateral) dispersion of bubbles, drops, and particles in swarms is relevant... 

The primary physical parameters, such as the fluid/fluid and fluid/solid interaction parameters, need apriori evaluation through model calibration using numerical experiments. The fluid/fluid interaction gives rise to the surface tension force and the fluid/ solid interaction manifests in the wall adhesion force. The fluid/fluid and fluid/solid interaction parameters are evaluated by designing two numerical experiments, bubble test in the absence of solid phase... 

Ultrasound affects bubble interaction in two ways - vibration causes an attraction between two bubbles (Bjerkens force) and the Bernoulli force arises from the flow of the fluid parallel or normal to a bubble. A perpendicular flow also causes interbubble attraction. Additionally, if bubbles increase in size, due to the ultrasound, they rise much more quickly, proportional to the (radius). As mentioned above (Section 10.5.1.2), ultrasound enhances cavitation, which is also beneficial. 

X is the viscous stress tensor g is gravitational acceleration /b is the total volumetric body force acting on the liquid phase excluding the gravitational force, that is, the volumetric particle-fluid interaction force ( f) plus the volumetric bubble-fluid interaction force (fb[). Based on Newton s third law of motion, the force acting on a particle from the liquid phase, Ffy, yields a reaction force on the liquid. Therefore the momentum transfer from particles to the liquid-gas phase is taken into account in Eq. (51) by adding the volumetric particle-fluid interaction force, fp[, given below to the body force term. 

The bubble-fluid interaction force, /bf, is obtained by using a continuum surface force (CSF) model (Brackbill et al., 1992)... 

In addition to two fluid interfaces interacting with each other, the interaction of fluid interfaces with solid-liquid interfaces is important [686, 695]. One example is the interaction of particles with bubbles or drops (Figure 7.1c). The interaction of particles with bubbles in aqueous liquid is the key process in flotation [591]. The interaction with drops is essential in oil recovery. Direct particle bubble force measurements have been carried out with the AFM [739—741, 1216] (reviewed in Ref. [742]). Also, the force between individual particles and oil drops in water has been measured by atomic force microscopy [743, 744]. 

Relevant forces are external volume forces like gravity, particle-particle interactions such as contact forces or electrostatic interactions, and the force exerted by the fluid on a particle. Some of these forces (that do not act on the center of mass) have a torque associated to it. The torque associated with the fluid—particle interaction follows from the antisymmetric part of the particle stress contributions. A detailed discussion of dominant forces in the case of gas-solid, liquid-bubble, and gas-droplet interactions will be provided in the topical sections (Sections 4—6). 

The fluid particle breakage controls the maximum bubble size and can be greatly influenced by the continuous phase hydrodynamics and interfacial interactions. Therefore, a generalized breakage mechanism can be expressed as a balance between external stresses (dominating component), o, that attempts to disrupt the bubble and the surface stress, ai/d, that resists the particle deformation. Thus, at the point of breakage, these forces must balance, o This balance leads to the prediction of a critical Weber number, above which the fluid particle is no longer stable. It is defined by [36] ... 

There are three scenarios for the occurrence of a two-particle collision in a dispersion depending on the type of particle-particle interactions. (1) If the repulsive forces are predominant, the two colliding particles will rebound and the colloidal dispersion will be stable. (2) When at a given separation the attractive and repulsive forces counterbalance each other (the film formed upon particle collision is stable), aggregates or floes of attached particles can appear. (3) When the particles are fluid and the attractive interaction across the film is predominant, the film is unstable and ruptures this leads to coalescence of the drops in emulsions or of the bubbles in foams. 

Maxey and Riley [47] derived an equation of motion for a small rigid sphere of radius R in a nonuniform flow. If one considers small bubbles moving in a polar liquid, this equation might be appropriate because surfactants would tend to immobilize the surface of a bubble and make it behave like a rigid sphere. Maxey and Riley assumed that the Reynolds number based on the difference between the sphere velocity and the undisturbed fluid velocity was small compared to unity. In addition, they assumed that the spatial nonuni-formity of the undisturbed flow was sufficiently small that the modified drag due to particle rotation and the Saffman [48] lift force could be neglected. Finally, they ignored interactions between particles. 

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