## Slip Velocity and Lift

D.D. Joseph and D. Ocando

Department of Aerospace Engineering and Mechanics

and the Minnesota Supercomputer Institute,

University of Minnesota, Minneapolis, MN 55455, USA

October 2000 (revised December 2001)

**Abstract**

The lift force on a circular particle in plane Poiseuille
flow perpendicular to gravity is studied by direct numerical simulation. The
angular slip velocity , where
is the angular velocity of the
fluid at a point where the shear rate is
and
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
so that *is nearly zero
*because of* . * No
matter where the particle is placed it drifts to an equilibrium position with a
unique, slightly positive equilibrium angular slip velocity. The slip angular
velocity discrepancy defined as the difference between the slip angular
velocity of a migrating particle and the slip angular 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 centerline are discussed.

The long particle model of Choi and Joseph [2001] that gives
rise to an explicit formula for the particle velocity and the velocity profile
across the channel through the center line 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 *U*_{p} 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 center when the
particle is absent is always larger than the particle velocity; the slip
velocity is positive at steady flow.

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*Last Modified: Friday, 07-Dec-2001 14:39:27 CST*