Transformation of elliptic partial differential equations for solving two‐dimensional boundary value problems in fluid flow

Abstract

Abstract It is possible to transform elliptic partial differential equations to exchange the dependent with one of the independent variables. The Laplace equation for a stream function ‘Ψ’ over the X and Y co‐ordinate system, for example, can be transformed into a relationship expressing the Y position of streamlines in terms of strem function Ψ and X . Although the resulting new partial differential equation is much more complex it is much more convenient to use in a computer. An irregular Y boundary becomes, with the new relationship, merely the boundary values assigned to the outer streamlines and the computer always need only deal with a rectangular array. The resulting answer is in the form of the position of streamlines which is the information directly required for plotting flow maps.