Saffman lift

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... 

Saltation of solids occurs in the turbulent boundary layer where the wall effects on the particle motion must be accounted for. Such effects include the lift due to the imposed mean shear (Saffman lift, see 3.2.3) and particle rotation (Magnus effect, see 3.2.4), as well as an increase in drag force (Faxen effect). In pneumatic conveying, the motion of a particle in the boundary layer is primarily affected by the shear-induced lift. In addition, the added mass effect and Basset force can be neglected for most cases where the particle... 

In particular applications alternative relations for the slip velocity (3.428) can be derived introducing suitable simplifying assumptions about the dispersed phase momentum equations comparing the relative importance of the pressure gradient, the drag force, the added mass force, the Basset force, the Magnus force and the Saffman lift force [125, 119, 58]. For gas-liquid flows it is frequently assumed that the last four effects are negligible [201, 19[. 

Rigid spheres sometimes experience a lift force perpendicular to the direction of the flow or motion. For many years it was believed that only two mechanisms could cause such a lift. The first one described is the so-called Magnus force which is caused by forced rotation of a sphere in a uniform flow field. This force may also be caused by forced rotation of a sphere in a quiescent fluid. The second mechanism is the Saffman lift. This causes a particle in a shear flow to move across the flow field. This force is not caused by forced rotation of the particle, as particles that are not forced to rotate also experience this lift (i.e., these particles may also rotate, but then by an angular velocity induced by the flow field itself). 

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. 

Particles moving in a fluid with mean shear experience a lift force perpendicular to the direction of fluid flow. The shear lift originates from inertia effects in the viscous flow around the particle and depends on the mean vorticity of the fluid phase evaluated at the particle location x = X (r). For a spherical particle, the particle acceleration due to the lift force (also known as the Saffman lift force) is equal to (Auton, 1987 Drew Lahey, 1993 Drew Passman, 1999 Saffman, 1965)... 

When the conditions required by the inequality (274) are met, the slip-shear Saffman lift force is larger, by an order of magnitude, than the slip-spin Rubinow-Keller lift force for from Eqs. (275) and (276) we find... 

With the above assumptions, the equation of motion for small bubbles in a shear flow is obtained from Eq. (14) by neglecting the virtual mass, gravitational, and Bassett history terms as well as the Faxen correction term in the Stokes drag and including the Saffman lift term given in Eq. (21). 

The Saffman lift force in Eq. (22) does not include a number of effects that are discussed below. 

The Saffman lift force is valid provided that the following conditions are satisfied ... 

Maxey and Riley pointed out that the Magnus force is of order and, for that reason, it is less important than the Saffman lift force. 

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