General solution of the particle momentum equation in unsteady Stokes flows

Abstract

The general solution of the particle momentum equation for unsteady Stokes flows is obtained analytically. The method used to obtain the solution consists of applying a fractional-differential operator to the first-order integro-differential equation of motion in order to transform the original equation into a second-order non-homogeneous equation, and then solving this last equation by the method of variation of parameters. The fractional differential operator consists of a three-time-scale linear operator that stretches the order of the Riemann–Liouville fractional derivative associated with the history term in the equation of motion. In order to illustrate the application of the general solution to particular background flow fields, the particle velocity is calculated for three specific flow configurations. These flow configurations correspond to the gravitationally induced motion of a particle through an otherwise quiescent fluid, the motion of a particle caused by a background velocity field that accelerates linearly in time, and the motion of a particle in a fluid that undergoes an impulsive acceleration. The analytical solutions for these three specific cases are analysed and compared to other solutions found in the literature.