substitution. We may look upon this figure as representing in section a theoretical wing-form, or aerofoil, appropriate to an inviscid fluid with its accompanying lines of flow; as such it is merely one of an infinite number of possible forms, its only virtue being that of representing the simplest possible case of peripteroid motion.
§ 122. Peripteroid Motion.—An infinite cylinder, of any sectional form whatever, divides infinite space into a doubly connected region, and in such a region cyclic motion becomes possible. From the hydrodynamic standpoint irregularity of contour is no detriment, as obstructing neither the cyclic motion nor that of translation. The consequence is that peripteroid motion is theoretically possible in the case of a cylinder of infinite extent, no matter what its cross-section. This conclusion applies naturally only in the case of the inviscid fluid; in a real fluid we are threatened with discontinuity. The position is analogous in every way to that of simple translation. In the inviscid fluid all bodies are of stream-line form, in real fluids only those that in their motion do not set up a discontinuity. Again, just as in the simple translation only certain simple cases are capable of solution by known analytical methods, so in peripteroid motion the cases capable of solution are very limited in number.
In order that a case of peripteroid motion should be solvable, the boundary conditions (both internal and external) must, generally speaking,[1] be such that their lines of flow for both translation and cyclic motion are separately known. The author has succeeded in plotting the stream lines in the following cases:—
Fig. 70, a filament of infinite lateral extent in an infinite expanse of fluid.
- ↑ A case, such as Fig. 70, is an exception. Here neither system is known separately for a cylinder the form of the shaded section. In a case of this description, where a body is substituted for a self-contained system of flow, we have an exception to the fact stated.
163
