Small inertial effects on a spherical bubble, drop or particle moving near a wall in a time-dependent linear flow

The problem of a spherical drop of arbitrary density and viscosity moving near a wall under the effect of a body force is analysed theoretically in the limit where the wall lies in the inner region of the flow disturbance, the distance between the drop and the wall being large compared to the drop radius. The drop may move in an arbitrary direction with respect to the wall, and the undisturbed flow field is assumed to comprise a steady uniform shear or solid-body rotation and a time-dependent uniform stream, the variations of which take place over time scales large compared to the viscous diffusion time. An exact force balance with no limitation on the magnitude of inertial effects is obtained by using the reciprocal theorem. Explicit expressions for the contributions of temporal acceleration, slip and shear or rotation to the total hydrodynamic force are derived in the limit of small-but-finite inertial effects. The connection between these near-wall results and inertial lift and drag corrections in an unbounded flow is discussed. Situations of particular interest in which the lift force results from a combination of contributions due to unsteadiness and advection, like the case of a particle moving near the bottom wall of a centrifuge, are also examined.