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Full text of "Scientific Papers - Vi"

1911] HYDRODYNAMICAL NOTES 15 mere height, which is a source of inconvenience and even danger to small craft. The above is nothing more than a suggestion. I do not know of any detailed account of the special character of these waves, on which perhaps a better opinion might be founded. Rotational Fluid Motion in a Corner. The motion of incompressible inviscid fluid is here supposed to take place in two dimensions and to be bounded by two fixed planes meeting at an angle a. If there is no rotation, the stream-function ty, satisfying V2^ = 0, may be expressed by a series of terms rir/a sm TJ-Q^ r27r/a gm 27r#/a, . . . rnw/a sin mrO/a, where n is an integer, making -^ = 0 when 6 = 0 or 6 = a. In the immediate vicinity of the origin the first term predominates. For example, if the angle be a right angle, ^ =rzmi26=2xy, ........................... (1) if we introduce rectangular coordinates. The possibility of irrotational motion depends upon the fixed boundary not being closed. If a < TT, the motion near the origin is finite ; but if a > TT, the velocities deduced from ty become infinite. If there be rotation, motion may take place even though the boundary be closed. For example, the circuit may be completed by the arc of the circle r = 1. In the case which it is proposed to consider the rotation M is uniform, and the motion may be regarded as steady. The stream- function then satisfies the general equation V^ = d-^ldai~+d^/df- = ^a}, ..................... (2) or in polar coordinates d*± 1 d^r I d*ty W>* r ~dr+r* ~d&~^ ...................... W When the angle is a right angle, it might perhaps be expected that there should be a simple expression for ty in powers of x and y, analogous to (1 ) and applicable to the immediate vicinity of the origin ; but we may easily satisfy ourselves that no such expression exists*. In order to express the motion we must find solutions of (3) subject to the conditions that ^=0 when 6 = 0 and when 6 — ot. For this purpose we assume, as we may do, that •^ = "2,Rn sin mrd/a, -s ........ . ....... , ........... (4) * In strictness the satisfaction of (2) at the origin is inconsistent with the evanescence of if/ on the rectangular axes. ...t memoir could be very simply derived from the expression for the