Lift of a Spherical Particle Subject to Vorticity and/or Spin

Abstract

A LTHOUGH the drag and gravitational forces are typically the most important to particle motion, the lift force can often be significant and can even approach the magnitude of the other two forces in some circumstances for surface spin velocities on the order of the translational particle velocity. As such, there are a number of multiphase systems in which consideration of lift is vital. The lift has been noted to be important for lateral migration in tubes (Saffman [1]), effectiveness of microcentrifuges (Heron et al. [2]), and particle deposition in boundary layers (Young and Leeming [3]). Another flowfield for which particle lift effects are important is that of the rotating bioreactor, which allows animal tissue growth under idealized conditions, similar to that obtained in microgravity conditions [4]. In many of these conditions, the particle can be considered as spherical and solid and the surrounding fluid as incompressible and Newtonian, as will be the focus of this review. However, there have beenmany excellent reviews on particle lift that discuss other conditions. In particular, the review by Leal [5] discussed dynamics and theoretical results for nonspherical particles, deformable particles, and particles in non-Newtonian fluids. On the subject of clean bubbles, Drew [6], Magnaudet and Eames [7], and Tomiyama et al. [8] discussed theory and models for lift in shear, rotational, and straining flows. For noncontinuum conditions, Wang [9] investigated the effect of particle spin in free-molecular flow and obtained a lift force that acted in a direction opposite to that of drag in continuum creeping-flow conditions. Recently, Volkov [10] investigated the transition between the continuum and the freemolecular regimes for spinning particles to obtain the critical Knudsen number at which particle spin did not yield lift. To discuss solid particle lift for incompressible continuum Newtonian flow, it is helpful to first define the dimensionless parameters that influence lift, as well as the various forms of the lift coefficient. In general, the particle velocity v is defined as the translational velocity of the particle center of mass xp. The continuous-fluid velocity is generally defined in all areas of the domain unoccupied by particles. However, a hypothetical continuous-phase velocity can be extrapolated to the particle centroid and will be denoted as u and termed the “unhindered velocity.”The relative velocity of the particlesw is then based on the unhindered velocity (i.e., along a particle trajectory):