Distributions of sources and normal dipoles over a quadrilateral panel
The potential due to a distribution of sources or normal dipoles on a flat quadrilateral panel is evaluated for the cases where the density of the singularities is constant, linear, bilinear, or of arbitrary polynomial form. The results in the first two cases are consistent with those derived previously, but the present derivation is considered to be simplified. In particular, the constant dipole distribution is derived from a geometric argument which avoids direct integration; this derivation applies more generally on a curvilinear panel bounded by straight edges.
Also presented are multipole expansions for the same potentials, suitable for use when the distance to the field point is substantially larger than the panel dimensions. Algorithms are derived to evaluate the coefficients in these expansions to an arbitrary order.
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