Hydrodynamic Interactions Between Widely Separated Particles - The Method of Reflections

Hydrodynamic Interactions Between Widely Separated Particles - The Method of Reflections 

The best available analytic technique is known as the method of reflections. It is an iterative scheme that applies when the separation distance between the particles or between the particle and wall is large relative to the characteristic dimension of the particle(s).27 The basic idea is to approximate the solution as a series of terms that satisfy the creeping-flow 

Similarly, the leading-order approximation in the vicinity of B is the Stokes velocity field that satisfies the boundary condition 

neither uf nor uf is exact, because neither satisfies the no-shp condition on the surface of the other particle. For example, the motion of particle A produces a disturbance velocity in the vicinity of B of order Ify a/d and the motion of B similarly produces a disturbance velocity of the fluid in the vicinity of A. Thus, to improve the approximation of the velocity field near particle A, we add to uf a correction uf that is a solution of the creeping-flow equations that vanishes at infinity and satisfies the condition 

Near particle B, we add a correction uf that satisfies the creeping-flow equations, vanishes at infinity, and satisfies the condition 

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