Force on a sphere in a general potential flow

The force on a spherical particle moving arbitrarily in a fairly general class of unsteady potential flows is calculated. It is found that the force, which can be expressed as the gradient of a potential, takes the form of an infinite series of progressively higher order in the ratio of the sphere’s radius to the scales of the surrounding flow. The first few terms conform with the known form usually referred to as ‘added mass force’ but, when the remaining terms are non-negligible, this concept becomes less relevant. Thus, unlike Faxèn’s extension of Stokes’s drag law, it proves impossible to generalise the idea of added mass to particles that are not much smaller than the flow scales. The kinetic energy of the fluid in excess of that which it would have in the absence of the sphere is also calculated when the flows considered are unbounded.