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There are many physical processes for which the solution of the Laplace equation on the exterior of a body of general shape is central to the theoretical description. Previous papers [80] discussed the exterior Dirichlet problem for the Laplace equation and the calculation of the capacity C, which is the shape functional associated with this problem. The present paper discusses other functionals of shape associated with the solution of the Laplace and the Navier-Stokes equations on the exterior of objects having general shape. These functionals [26] include the electric and magnetic polarizabilities, the hydrodynamic virtual mass and the intrinsic viscosity.
New numerical and analytical results for these shape functionals, along with values from a large literature, were obtained to check a proposed relation between shape functionals associated with the Laplace equation, namely the electric polarizability and a shape functional associated with the Navier- Stokes equation, the intrinsic viscosity. Our new approximate relation is a natural generalization of a result of Hubbard and Douglas [27] relating the translational friction coefficient (Navier-Stokes equation) approximately to the electrostatic capacity (Laplace equation). These relations between hydrodynamic theory and electromagnetic theory complement the classical relation between the 'effective mass' M of 'perfect' fluids and the magnetic polarizability αm observed by Keller et al. and Kelvin [57,58].
Exact and numerical results confirm that the intrinsic viscosity [η] is proportional to the intrinsic conductivity [σ]∞ of conducting particles of arbitrary shape to within about a 5% approximation,
in three dimensions. In two dimensions we find [η] exactly equals [σ]∞ for ellipses. Data for other shapes, allowing for a usual underestimation of [η] by 2-5% in our d=2 finite element method, are in general agreement with this conjectured approximation. On the basis of this evidence, we conjecture that [η] = [σ]∞ for all shapes in d=2. Further exact calculations of [η] for some of the shapes discussed would be useful in developing a proof (or disproof) of this conjecture. All of our findings agree well with the predictions of our angular averaging approximation,
Although our primary goal in this paper was to test relations (62)-(64), the tabulated values of [σ]∞, [σ], and [η] for numerous shapes and the discussion of the general shape parameters that affect these quantities should find wide application. Inevitably, this information becomes important when we attempt to resum the virial expansions , such as (1), to provide a useful description of suspensions of arbitrarily-shaped particles in a matrix at high concentrations.