General Basset–Boussinesq–Oseen equation: existence, uniqueness, approximation and regularity of solutions

The Basset–Boussinesq–Oseen equation (BBO equation) is the mathematical formulation of the Lagrangian acceleration of a spherical particle, moving with Lagragian velocity, in an unsteady flow as the sum of the viscous, gravitational, buoyancy, virtual mass, and Basset forces acting respectively on the particle. Despite the widespread use of the equation in applications, the basic properties of its solutions have remained unexplored. Here we fill this gap by proving global existence and uniqueness of solutions in an appropriate partially ordered Banach space. The method also gives an approximate solution to the problem with accurate result which is easy to implement. Meanwhile, regularity properties of the solutions are proved under some conditions which gives a representation of the solutions. Illustrative examples exhibit the efficiency of our method.