Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/117

Rh of plotting the field of flow for any finite combination of known systems. It is the geometrical equivalent of the analytical machinery employed in the mathematical solution of a vast number of cases, and as such it is due to Clerk Maxwell. Many compound systems of flow involve an infinite number of elementary components, such as some prescribed distribution of sources and sinks over certain lines and surfaces; the graphic method in such cases is not generally applicable, and the field requires to be plotted from the mathematical solution.

§ 75. ψ, φ Lines for Source and Sink System.—Let us take the case of a source and sink A and B (Fig. 40), of equal flux, in two dimensions; then the lines ψ constant for the individual fields will consist of equal-spaced radial lines extending indefinitely on all sides, as shown. If now we draw the resultant field we find that the fluid emitted by the source is absorbed by the sink, and from geometrical considerations it is obvious that the paths of flow consist everywhere of arcs of circles passing through the points A and B. Since the functions ψ and φ are interchangeable, we can in a similar manner find the resultant system of velocity potential, and we obtain the system of circles shown; if we take the latter as the lines of flow, and the arcs joining A and B as the equipotentials, we have the case of a vortex pair, that is to say, two vortex filaments with equal and opposite rotation. A.F.