The inertial migration of a small rigid spherical particle, suspended in a fluid flowing between two plane boundaries, is investigated theoretically to find the effect on the lateral motion. The channel Reynolds number is of order unity and thus both boundary-induced and Oseen-like inertial migration effects are important. The particle Reynolds number is small but non-zero, and singular perturbation techniques are used to calculate the component of the migration velocity which is directed perpendicular to the boundaries of the channel. The particle is non-neutrally buoyant and thus its buoyancy-induced motion may be either parallel or perpendicular to the channel boundaries, depending on the channel alignment. When the buoyancy results in motion perpendicular to the channel boundaries, the inertial migration is a first-order correction to the magnitude of this lateral motion, which significantly increases near to the boundaries. When the buoyancy produces motion parallel with the channel boundaries, the inertial migration gives the zeroth-order lateral motion either towards or away from the boundaries. It is found that those particles which have a velocity exceeding the undisturbed shear flow will migrate towards the boundaries, whereas those with velocities less than the undisturbed flow migrate towards the channel centreline. This calculation is of practical importance for various chemical engineering devices in which particles must be filtered or separated. It is useful to calculate the forces on a particle moving near to a boundary, through a shear flow. This study may also explain certain migration effects of bubbles and crystals suspended in molten rock flow flowing through volcanic conduits.
An extended nonlinear Schrödinger equation for water waves with linear shear flow, wind, and dissipation
