The behaviour of particles in a sinusoidal velocity field

Abstract

A non-linear Langevin equation has been developed that represents the general behaviour of free particles in a sinusoidal velocity field. Analysis shows that a linear drag law, in which the frictional force is directly proportional to the relative particle-fluid velocity (Stokes’s law), does not result in a stable particle motion. However, a non-linear relation in which the drag is proportional to the square of the relative velocity (Newton’s law) leads to the Mathieu equation, wiien it may be shown that stable particle trajectories may occur in certain ranges of frequency and amplitude. Stability criteria based on the Mathieu equation are applied to bubble-liquid systems in the sonic and ultrasonic regions, to liquid drop systems in pulse columns and to solid-liquid systems by using drag coefficients obtained from measurements of terminal velocities. The agreement between the theoretical and observed frequencies indicates that the velocity profile around a particle in a sinusoidal field is approximated by a quasi-steady state distribution and that transient effects may be neglected for the separated flow conditions considered in the present communication.