ON THE FLOW OF COMPRESSIBLE FLUID PAST AN OBSTACLE. [Philosophical Magazine, Vol. xxxn. pp. 1—6, 1916.]
IT is well known that according to classical Hydrodynamics a steady stream of frictionless incompressible fluid exercises no resultant force upon an obstacle, such as a rigid sphere, immersed in it. The development of a " resistance " is usually attributed to viscosity, or when there is a sharp edge to the negative pressure which may accompany it (Helmholtz). In either case it would seem that resistance involves something of the nature of a wake, extending behind the obstacle to an infinite distance. When the system of disturbed velocities, although it may mathematically extend to infinity, remains as it were attached to the obstacle, there can be no resistance.
The absence of resistance is asserted for an incompressible fluid ; but it can hardly be supposed that a small degree of compressibility, as in water, would affect the conclusion. On the other hand, high relative velocities, exceeding that of sound in the fluid, must entirely alter the conditions. It seems worth while to examine this question more closely, especially as the first effects of compressibility are amenable to mathematical treatment.
The equation of continuity for a compressible fluid in steady motion is in the usual notation
4. . 4. . 4. - _ o H 1
dx dy dz " \dx dy dz) ' ...............
or, if there be a velocity-potential <£,
= o (2)
............. ^ '
dx dso dy dy dz dz
In most cases we may regard the pressure p as a given function of the density p, dependent upon the nature of the fluid. The simplest is that of Boyle's law where p = a2p, a being the velocity of sound. The general equation
f-O-W...............................(3)um of Rotating Masses. On all these subjects the reader will find expositions which could hardly be improved, together with references to original writings of the author and others where further developments may be followed.