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Full text of "Scientific Papers - Vi"

1916] ON THE FLOW OF COMPRESSIBLE FLUID PAST AN OBSTACLE 403 where q is the resultant velocity, so that (4) reduces in this case to or log p = 0 ^qz, or a2 log(p//>o) = - %q\ .... if pa correspond to ij = 0. From (2) and (5) we get (5) .(6) __ __-t . _j____i_ J I r 2a2 [dsc dx dy dy dz dz When <f is small in comparison with a2, this equation may be employed to estimate the effects of compressibility. Taking a known solution for an incompressible fluid, we calculate the value of the right-hand member and by integration obtain a second approximation to the solution in the actual case. The operation may be repeated, and if the integrations can be effected, we obtain a solution in series proceeding by descending powers of a'-. It may be presumed that this series will be convergent so long as </ is less than a2. There is no difficulty in the first steps for obstacles in the form of spheres or cylinders, and I will detail especially the treatment in the latter case. If U, parallel to 6 = 0, denote the uniform velocity of the stream at a distance, the velocity-potential for the motion of incompressible fluid is known to be 0 = U(r + c2/r) cos 0, ...........................(7) the origin of polar coordinates (r, 6} being at the centre of the cylinder. At the surface of the cylinder r = c, d$ldr = 0, for all values of 6. On the right hand of (6) dq- _ and from (7) d$ df dcj> d<f = dai dx dy dy dr dr r* dO ~dQ ' drj .(8) 2/9 Also Accordingly O v "U dr I ^ = _4c; U"~a dr r5 Urde 4c2 + r COS A-tO ~ in rdv ar8 The terms on the right of (10) are all of the form rŪ cos n0, so that for the present purpose we have to solve - w r dr f^-.. (11) 262<n