Estimates for the sedimentation speed

The classical problem for a cloud of identical spheres settling steadily in an un-bounded incompressible fluid is revisited and a rigorous upper bound on the sed-imentation speed is derived. Within the frame of the Rayleigh analogy between equations of linear (incompressible) elasticity and Stokesian hydrodynamics a more general Robin’s type problem in elastostatics is first discussed, in which an array of rigid spheres, cemented into an elastic matrix, are displaced due to a given force. In-stead of a variational principle a Bounjakowsky-Schwarz type inequality is applied in order to avoid the quest for trial fields that satisfy the boundary conditions. The energy of the system is thus explicitly estimated which generates a lower bound on the effective constant α ∗ that interconnects the force with the mean displacement of the spheres. In the incompressible case, invoking the Rayleigh analogy, a rigor-ous bound on the familiar hindered settling speed of the dispersion immediately follows. The bound turns out to coincide with the ‘far-field ’ approximation for the sedimentation speed, proposed by Brady & Durlofsky