
As early as 1906, Lorentz reported the Green's function for flow near a flat surface. Seventy years later, Blake [62] recognized that Lorentz's result could be reinterpreted by analogy to electrostatics. He suggested that the flow due to a stokeslet could be canceled on a bounding surface by conceptually placing its hydrodynamic image on the opposite side. Solutions to the Stokes equations being unique, the resulting flow must be Lorentz's Green's function for bounded flow.
The electrostatic image needed to cancel a charge distribution's field on a surface is just an appropriately scaled mirror image of the initial source. In hydrodynamics, the image of a stokeslet is not simply another stokeslet, but rather a more complicated construction including sources which Blake dubbed a stokeslet doublet (D) and a source doublet (SD). This combination is depicted schematically in Fig. 6. The flow due to the entire image system is described by the Green's function [62]
and  (37)  
(38) 
Applying Faxén's first law and identifying leads to
Figure 6 shows typical data obtained with optical tweezers and digital video microscopy for a silica sphere's diffusion above a wall. The particular sphere for this data set was one of the pair studied in the previous section. The sphere's height above the wall was repeatedly reset by the optical tweezer at intervals. During this period of free motion, it could diffuse out of plane only , on average. Advancing the microscope's focus in steps allowed us to sample the dynamics' dependence on . The measured heightdependent diffusivity agrees well with both Eqs. (35) and (39).