Conformal transformation with the aid of an electrical tank

Abstract

1—When a steady electric current flows in a flat sheet of uniforms conductivity, the electric potential V can be represented by a plane-harmonic function, since at every point ∇ 2 V ≡ ∂ 2 V /∂ x 2 + ∂ 2 V /∂ y 2 = 0 (1) when the ( x, y ) plane is taken parallel to the sheet. The " stream function " ψ of an inviscid fluid moving steadily in two dimensions is governed by the similar equation ∇ 2 ψ = 0, (2) and advantage has been taken of this analogy to solve the hydrodynamic problem, in cases of mathematical difficulty, by exploring the distribution of potential in an electrolyte of uniform depth, contained within an "electrical tank" (Relf 1924; Taylor and Sharman 1928). An inviscid fluid can slip on the surface of a solid body, but (when the body is stationary) it can have no velocity in a direction normal thereto: this means that Ψ in the hydrodynamical problem has a constant value at all points in the surface of the body, and so, in the electrical analogue, the corresponding boundary must be an equipotential. The condition can be satisfied by inserting a highly-conducting body, of suitable shape, within the electrolyte in the tank. 2—Plane-harmonic functions appear in may other problems, but usually the boundary condition is less easy to satisfy in the electrical experiment. For example, de Saint Venant's solution of the problem of torsion in a nonciricular shaft (Love 1927, § 216; Southwell 1936, § 336) introduces a function Ψ which is governed by (2) at all points in the cross-section; but the boundary condition is Ψ -½( x 2 + y 2 ) = const., (3) and hence, in the tank, different values of the electric potential would have to be maintained at different points in a specified boundary.