On the two-dimensional steady flow of a viscous fluid behind a solid body.-II

One reason for carrying out the calculations of the previous paper was to provide material for an experimental study of the transition to turbulence in the wake behind a plate parallel to the stream. A second reason was to compare the results with certain results due to Filon, who has calculated both the List and second approximations to the velocity at a considerable distance from a fixed cylindrical obstacle in an unlimited stream whose velocity at infinity is constant.* He also uses the notions of the Oseen approximation; that is to say, he assumes that the departures from the undisturbed velocity are small, and neglects terms quadratic in these departures for the first approximations, etc .; but he does not assume that v is small and does not use the Prandtl equations. Thus the formulæ of paper 1, paragraph 2, should be limiting forms, for small v, of Filon's formulæ for a symmetrical wake. This is verified in paragraph 2 below; and the calculations in paper 1, paragraph 2, other than the attempt at a third approximation, may be regarded as a simplified form of Filon's calculations. The direct simplification of Filon's results gives the formulæ 2 (31) (p. 569), for the velocity at a sufficient distance downstream in any symmetrical wake provided that the motion is steady, whether v is small or not. these formulæ differ only in the last terms from the formulæ 2 (27) on p. 553 of paper 1, obtained from the Prandtl equations, and these terms are negligible, compared with the others, when v is small, (For the meaning of the symbols, see paragraph 1.3 of paper 1.) Thus the first asymptotic approximation is exactly the same here as in the previous paper ; in the second approximation the more accurate results of this paper contain extra terms, which it is shown on p. 567 arise entirely from the previous neglect of the pressure gradient in the direction of the stream.