Green's function approach to the theory of hydrodynamic interactions in suspensions

Abstract

A new approach to the theory of many-sphere hydrodynamic interactions in suspensions is developed starting from the linearized Navier-Stokes equations reformulated in terms of the induced force formalism. This approach is based on the idea of the direct series expansion of the Green's function involved which leads in a very straightforward and controlled manner to the presentation of the velocity moments by means of the irreducible multipoles of the induced forces. To utilize this idea the calculation of the Green's function surface and volume moments of arbitrary order is performed using the method of generating functions. In principle, this offers the possibility to evaluate the translational and rotational mobility tensors in a power series expansions in two natural (for the system under consideration) parameters, namely, a/R and αa where a is a typical radius of the spheres, R a typical distance between spheres and α is expressed through the fluid density π and the viscosity of the fluid η as α=−iωρ/η. The correspondence between our and earlier theories of hydrodynamic interactions is discussed up to the seventh order in parameter (a/R) in the quasistatic limit and up to the third order in combination of both parameters, a/R and αa, in the case of finite frequencies.