The internal problems of two dimensional potential theory

The Magnetostatic Characteristics of Two Non‐Magnetic Elliptic Cylinders

“…When /J = 45° the values of G = -7r M / J.L/2 as d = b /2c varies are given by Table II. The moment of the couple decreases as the cylinders separate, the rate of decrease being greater at smaller values of d. 6. Further Considerations.…”

“…It can be proved from equation (28) that the coefficients An are all purely imaginary, and are zero when n is odd. When n is even, the coefficients are in general non-zero, the first few coefficients being A2 = ip.lc 2 /411', As = 5ip.lc 6 Having determined the coefficients An it is now possible to deduce the magnetic field intensities both inside and outside the conducting cylinders.…”

“…It is assumed that 71 = + 00 corresponds to z = + 00, thus the z-plane outside the line -2c ~ x ~ 2c is mapped conformally on to the semi-infinite strip in the s-plane given by 0 ~ 71 < 00, 0 ~ ~ ~ 211". Strictly the two-fold Riemann surface in the z-plane with branch points at z = ±2c is mapped conformally on to the infinite strip -00 < 71 < 00, 0 ~ ~ ~ 211", in the s-plane, [6], [7]. From equation (4), it can be shown that where An,T and "In,.…”

Exact solutions and physical analogies for unidirectional flows

“…The same problem arises in solid mechanics for beam torsion and bending [19][20][21], and myriad exact solutions have been derived. Beginning with the seminal paper of de Saint-Venant [12], many complex shapes were analyzed by conformal mapping, notably by Morris [22], Muskhelishvili [19], and Pólya and Szegö [23]. Morris also applied her general solution for beam torsion to Poiseuille flow [22], and Tao [24,25] later solved related problems in forced convection.…”

“…Beginning with the seminal paper of de Saint-Venant [12], many complex shapes were analyzed by conformal mapping, notably by Morris [22], Muskhelishvili [19], and Pólya and Szegö [23]. Morris also applied her general solution for beam torsion to Poiseuille flow [22], and Tao [24,25] later solved related problems in forced convection. In contrast to the more familiar case of Laplace's equation, however, conformal mapping cannot be as easily applied to Poisson's equation, since it is not conformally invariant [26].…”