Virtual-mass force importance

The second important term is the virtual mass coefficient (Cv). When the dispersed phase accelerates (or decelerates) with respect to the continuous phase, the surrounding continuous phase has to be accelerated (or decelerated). For such a motion, additional force is needed, which is called added or virtual mass force. This force was given by the second term in Eq. (8). The constant Cy is called the virtual or added mass coefficient. It is difficult to estimate the value of Cv with the present status of knowledge. Therefore, many recommendations are available in the published literature. In an extreme case of potential flow, the value of Cy is 0.5. 

Apart from the drag force, there are three other important forces acting on a dispersed phase particle, namely lift force, virtual mass force and Basset history force. When the dispersed phase particle is rising through the non-uniform flow field of the continuous phase, it will experience a lift force due to vorticity or shear in the continuous phase flow field. Auton (1983) showed that the lift force is proportional to the vector product of the slip velocity and the curl of the liquid velocity. This suggests that lift force acts in a direction perpendicular to both, the direction of slip velocity... 

The change of momentum for a particle in the disperse phase is typically due to body forces and fluid-particle interaction forces. Among body forces, gravity is probably the most important. However, because body forces act on each phase individually, they do not result in momentum transfer between phases. In contrast, fluid-particle forces result in momentum transfer between the continuous phase and the disperse phase. The most important of these are the buoyancy and drag forces, which, for reasons that will become clearer below, must be defined in a consistent manner. However, as detailed in the work of Maxey Riley (1983), additional forces affect the motion of a particle in the disperse phase, such as the added-mass or virtual-mass force (Auton et al., 1988), the Saffman lift force (Saffman, 1965), the Basset history term, and the Brownian and thermophoretic forces. All these forces will be discussed in the following sections, and the equations for their quantification will be presented and discussed. 

Summarizing the forces introduced above, tests carried out in different multiphase systems have shown that the order of importance of the different forces involved typically ranks buoyancy and drag in the first positions and then lift and virtual-mass forces for fluid-solid systems and virtual-mass and lift forces for fluid-fluid systems (see, for example, the studies on non-drag forces by Diaz et al (2008) and Barton (1995)), whereas the most common values for the corresponding constants are Cl = 0.25 and Cv = 0.5 both for fluid-fluid and for fluid-solid systems. Naturally, since it is straightforward to implement all the forces in a computational code (Vikas et al, 201 lb), it is best to include them all for the sake of generality. 

The virtual mass coefficient Cgp of an isolated spherical bubble is 0.5. The BP gradient is then added to the right-hand side of the gas momentum (Eq. (5.4)) and acts as a driving force for bubbles to move from areas of higher to areas of lower d and facilitates stabilization of the bubbly flow regime. However, Sankaranarayanan and Sundaresan (2002) indicate that as increases, the collisional and hydrodynamic contributions become important. 

We have so far described drag and lift forces acting on a suspended particle. There are, however, additional hydrodynamic forces, such as Basset history, Faxen correction, and virtual mass effects that act on the particles. Some of these forces could become important especially for the particles suspended in a liquid. The general equation of motion of a small spherical particle suspended in fluid as obtained by Maxey and Riley is given as... 

The small attraction referred to as dispersion forces is the source of electrostatic attraction between temporary dipoles. The strength of dispersion forces depends on how easily an electron cloud can be polarized. Electrons in smaller atoms and molecules are held closer to their nuclei and, therefore, are not easily polarized. The strength of dispersion forces tends to increase with increasing molecular mass and size. Intermolecular attractive forces between Clj molecules and between Br2 molecules are estimated to be 2.9 kJ (0.7 kcal)/mol and 4.2 kJ (1.0 kcal)/mol, respectively. Dispersion forces are inversely proportional to the sixth power of the distance between interacting atoms or molecules. For them to be important, the interacting atoms or molecules must be in virtual contact with one another. 

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