System bispherical

Many such coordinate systems exist. Besides cylindrical, spherical, and Cartesian, the most common are probably elliptical, elliptical cylindrical, bispherical, and bicylindrical. A very useful compilation of the properties of vectors and vector operators, in a large number of orthogonal coordinate systems, may be found in an appendix to the book by Happel and Brenner, op. cit. 

In the bispherical system of coordinates, the Laplace equation, Fq. (12.1), transforms to ... 

Other. The latter condition means that we cannot use a bispherical system of coordinates, Eq. (12.6), but rather we need a confluent bispherical system of coordinates /7,0) (Fig. 12.8), derived from the bispherical system ((, //, O) by the limiting transition 0 and rj 0 [93]. 

Equations for touching spheres in the confluent bispherical system of coordinates are rj = rh and rj = ti2- We also need expressions for the surface element ds, and projections of the external normal to this element on axes of Cartesian coordinates ... 

The general solution of the Laplace equation, Eq. (12.51), in the degenerate bispherical system of coordinates, limited at (J 0, takes the form ... 

However, many numerical solutions of the PB have become available now which can be exploited for estimating the validity of the approximate models. In these calculations, pioneered by Hoskin and Levine [21,44], one uses the finite-difference method and the PB equation is formulated in the bispherical coordinate system. The advantage of this orthogonal coordinate system is that the boundary conditions at the sphere surfaces can be accurately expressed. This coordinate system (with more mesh points) was subsequently used by Camie et al. [45], who performed calculations of the interaction force for two spherical particles in a 1-1 electrolyte. The authors proved that the electrostatic fields distribution within the particles exerted a negligible effect on interaction force characterized by 8 < 5 (e.g., polystyrene latex particles). 

For smaller separations, the force on a dielectric sphere in a imiform field E can only be calculated numerically using the bispherical coordinate system. The results plotted in Fig. 6 indicate that at h /a < 1, the force decreases with the distance less abruptly than for larger distances and significantly depends on the relative dielechic constant Ep/E. In the limit of Ep/E 0, the force can be interpolated by the function... 

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