High order computation of the history term in the equation of motion for a spherical particle in a fluid

The historical evolution of the equation of motion for a spherical particle in a fluid and the search for its general solution are recalled. The presence of an integral term that is nonzero under unsteady motion and viscous conditions allowed simple analytical or numerical solutions for the particle dynamics to be found only in a few particular cases. A general solution to the equation of motion seems to require the use of computational methods. Numerical schemes to handle the integral term of the equation of motion have already been developed. We present here adaptations of a first order method for the implementation at high order, which may employ either fixed or variable computation time steps. Some examples are shown to establish comparisons between diverse numerical methods.