Numerical Solution of Axisymmetrical Problems, with Applications to Electrostatics and Torsion

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Abstract
Numerical methods are given for solution of axisymmetrical problems involving the partial differential equation ∂2ψ∂z2+∂2ψ∂ρ2+Kρ∂ψ∂ρ=0,where ρ is the radial coordinate and z the coordinate parallel to the axis. The various values of K which occur in physical situations are discussed, and common iteration methods for handling these problems are given. For Laplace's equation, K = 1. For the Stoke's stream function, K = − 1, but it is pointed out that for numerical work a new function, called the flow-disturbance function, having K = 3, is more tractable. Similarly a new function with K = 5, the stress-concentration function, is much easier to compute than the usual stress function (K = − 3) for the case of the torsion of a circular shaft of varying diameter. The methods are illustrated by computation of the equipotentials for an electron lens, and by a complete computation of the stresses and strains in a particular grooved circular shaft under torsion.

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