Saffman lift force

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[. 

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... 

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. 

---