Slip velocity and lift

The lift force on a circular particle in plane Poiseuille flow perpendicular to gravity is studied by direct numerical simulation. The angular slip velocity Ωs=Ωp+½γ˙, where −½γ˙ is the angular velocity of the fluid at a point where the shear rate is γ˙ and Ωp is the angular velocity of the particle, is always positive at an equilibrium position at which the hydrodynamic lift balances the buoyant weight. The particle migrates to its equilibrium position and adjusts Ωp so that Ωs > 0 is nearly zero because Ωp ≈ −1/2γ˙ No matter where the particle is placed, it drifts to an equilibrium position with a unique, slightly positive equilibrium angular slip velocity. The angular slip velocity discrepancy defined as the difference between the angular slip velocity of a migrating particle and the angular slip velocity at its equilibrium position is positive below the position of equilibrium and negative above it. This discrepancy is the quantity that changes sign above and below the equilibrium position for neutrally buoyant particles, and also above and below the lower equilibrium position for heavy particles. The existence and properties of unstable positions of equilibrium due to newly identified turning-point transitions and those near the centreline are discussed. The long particle model of Choi & Joseph (2001) that gives rise to an explicit formula for the particle velocity and the velocity profile across the channel through the centreline of the particle is modified to include the effect of the rotation of the particle. In view of the simplicity of the model, the explicit formula for Up and the velocity profile are in surprisingly good agreement with simulation values. The value of the Poiseuille flow velocity at the point at the particle's centre when the particle is absent is always larger than the particle velocity; the slip velocity is positive at steady flow.