Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/113

Rh have a cyclic system in which the origin represents the axis of a cylindrical body of infinite length making the space round it a doubly connected region. The velocity of the fluid is everywhere inversely as the length of its path of flow, consequently if we suppose the cylinder be made smaller the velocity at its surface will be proportionately greater, so that in the limit if we suppose the cylinder to become evanescent the velocity becomes infinite. The circulation round any such evanescent filament is indeterminate, for it is equal to $$\infty \times 0$$. The physical signification of this is that we have a system of flow that may be regarded as rotational or irrotational according as we regard the cylinder as non-existent or merely evanescent. If we regard the cylinder as non-existent and the motion as rotational, then the rotation is measured by the circulation round any of the lines of flow (for the circulation round each is the same), so that the whole rotation must be supposed concentrated at the geometric centre.

Such a motion is known as vortex motion, and the system figured constitutes a vortex filament. It will be seen that if $$r$$ represent the radius of the path of flow and $$v$$ the corresponding velocity, $$v\ r =$$ constant, and if the angular velocity $$\omega = v/r$$ we have $$\omega\ r^2 =$$ constant,—that is to say, for any circuit of flow the area $$\times$$ angular velocity is constant, which is the relation for vortex motion established generally by the theorem of Helmholtz and Kelvin. The discussion of this type of motion will be resumed later in the chapter.

'''§ 73. Irrotational Motion. Fundamental or Elementary Forms. Compounding by Superposition.'''—All known forms of irrotational motion can be regarded as being compounded from a limited number of different types. These are:—(a) Uniform motion of translation; (b) rectilinear motion to or from a point, i.e., sources and sinks; (c) cyclic motion (in multiply connected regions only).

Let us examine first the simple case of a fluid mass possessed only of a uniform motion of translation, and let us suppose that its