Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/102

§ 61 $$\phi =$$ constant; then the cell cut off, $$a ,\ b ,\ c ,\ d ,$$ will be approximately square, and if we choose to subdivide for intermediate values of flux and velocity potential as indicated, the cellules so formed will approximate still more closely, and the whole field may be regarded as ultimately built up of a number of such square elements.

If in two-dimensional motion the successive increments of flux be represented by increments of a quantity $$\psi ,$$ it can be shown that the $$\phi$$ lines and the $$\psi$$ lines may be interchanged, the lines of equal flux becoming equipotentials, and vice versâ. The applied impulse will of course require to be different for the two systems.

When the motion takes place in three dimensions, the $$\psi$$ lines and $$\phi$$ surfaces still divide the fields into a number of rectangular elements, or cubes (Fig. 34), but the conjugate property no longer exists; the $$\phi ,\ \psi ,$$ functions are not interchangeable.

The foregoing principles may be illustrated by the simple case of a source and sink.

§ 62. Sources and Sinks.—A source is a hypothetical conception, and may be defined as a point at which fluid is being continuously generated, and conversely a sink is a point at which fluid is supposed to disappear. Nothing actually resembling a source or sink is known to experience, the utility of the conception resting in its application to theory. A point source gives rise to three-dimensional motion; a line source gives rise to two-dimensional motion. A line source may also be described as a point source in two dimensions.

The field of flow from a source or towards a sink in an infinite expanse of fluid can be laid down from considerations