The motion of a particle in a viscous fluid under gravity,vibration and Basset's force

The sedimentation of a heavy solid spherical particle from a state of rest in an incompressible viscous fluid in a vessel with a vibrating bottom is investigated. Taking the Basset force into account, the problem is reduced to solving a Cauchy problem for a linear integro-differential equation. An exact solution of this problem and simple asymptotic formulae are obtained and a complete analysis of the effect of the Basset force on the oscillations and sedimentation of the particles is carried out. It is shown that a consideration of the Basset force introduces a considerable correction to the classical amplitude-frequency relation, reducing its value and also considerably slowing the arrival of the amplitude at a constant value. When there are no vibrations, it follows from the solution of the problem that there is a slow establishment of the limiting velocity (inversely proportional to the square root of the time), which differs considerably from the case of the sedimentation of particles in accordance with Stokes's law (the establishment of the limiting velocity occurs exponentially).