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Published online by Cambridge University Press: 24 October 2008
It is recognized that the main hope for advance in a rational theory of aeroplane propulsion lies in extensive use of the general principle of conservation of resultant momentum, even more than on considerations of energy; for energy can be dissipated by friction, while the relation of momentum to force is absolute.
* The sign of (L, M, N) has been reversed.
† The factor ½ in this definition identifies vorticity with differential rotation. The formulae imply that the velocity falls off towards infinity owing to all the vortical sources being at finite distance.
* Applied to a solid of density ρ0 moving in water of density ρ, the integrations would naturally be taken over all space, at the constant density ρ, leaving over a separate momentum of the solid at reduced (Archimedean) density ρ0 − ρ.
† [or carries with it in renewed circulation, as infra at end.]
* More generally, familiar Observation appears to indicate that laminar motion, in a layer that would be unstable by the Rayleigh rule, would be stabilized by inserting in it a row of vortices: cf. a theoretical note by Kelvin, Scientific Papers, iv, p. 186.Google Scholar
† Scientific Papers, vol. iv, Hydrodynamics; a playful epistolary controversy between Stokes and Kelvin on the possibility of slip of the Helmholtz type is there abstracted. The force of the explanation by cavitation is substantially reduced for the case of gases such as air. It is also asserted from observation that vortices break right off from the back of the edge. Incipient cavitations, continually collapsing against the wing itself, might very well maintain an irrotational circulation round it. The loads actually sustained are astonishing. In any case the momentum criterion involves that a free wing has to adjust its tilt so that the double vortex sheet left behind it is at its source of small breadth and so of slight account. Eddy resistance, in relation to fineness of lines in the rear, was considered by Stokes as early as 1843: cf. Collected Papers, vol. i, pp. 53, 90, 311.
* Recent experiments by Bénard, reported in Comptet Rendus, seem to show that regular spacing on the two sheets, which simplifies the Karman mathematical theory by making the motion steady, is not essential for stability of the vortex trail.
* A vortex absolutely steady relative to the wing would by the present formulation have constant momentum in the relative motion, and therefore would exert no force, which is in confirmation of this statement.
* See Bairstow, L., Applied Aerodynamics (1920), p. 360: also L.Prandtl, Die Naturwissenschaften, Feb. 1925.Google Scholar
† The discrepancy with the formula for a ring (p. 623) is adjusted by considering the ring made up of two conjugate straight vortices connected by vortical arcs at the ends: the latter make a contribution equal to that of the straight parts.