Laminar Flow About a Rotating Body of Revolution in an Axial Airstream

We have set ourselves the problem of calculating the laminar flow on a body of revolution in an axial flow which simultaneously rotates about its axis. The problem mentioned above, the flow about a rotating disk in a flow, which we solved some time ago, represents the first step in the calculation of the flow on the rotating body of revolution in a flow insofar as, in the case of a round nose, a small region about the front stagnation point of the body of revolution may be replaced by its tangential plane. In our problem regarding the rotating body of revolution in a flow, for laminar flow, one of the limiting cases is known: that of the body which is in an axial approach flow but does not rotate. The other limiting case, namely the flow in the neighborhood of a body which rotates but is not subjected to a flow is known only for the rotating circular cylinder, aside from the rotating disk. In the case of the cylinder one deals with a distribution of the circumferential velocity according to the law v = omega R(exp 2)/r where R signifies the cylinder radius, r the distance from the center, and omega the angular velocity of the rotation. The velocity distribution as it is produced here by the friction effect is therefore the same as in the neighborhood of a potential vortex. When we treat, in what follows, the general case of the rotating body of revolution in a flow according to the calculation methods of Prandtl's boundary-layer theory, we must keep in mind that this solution cannot contain the limiting case of the body of revolution which only rotates but is not subjected to a flow. However, this is no essential limitation since this case is not of particular importance for practical purposes.