Cylinder of arbitrary cross section in a weakly rotational biparametric incompressible inviscid flow (Q922077)

In \textit{P. N. Shankar}, Z. Angew. Math. Phys. 36, 172-173 (1985) it was shown how to construct the steady-state inviscid flow around an infinite cylinder when the free stream has two velocity components: one perpendicular to the generator and the other parallel to it. Generally speaking, both components are nonconstant. The solution obtained is expressed in terms of the stream function for plane flow in the absence of the second (longitudinal) component. As the author himself points out, the result obtained is, unfortunately, of limited significance since the exact expression for the stream function is known only in cases when the free-stream velocity is either constant or a linear, harmonic or exponential function of the transverse coordinate. In this article it is shown that this limitation can be overcome if it is assumed that at points remote from the cylinder the flow differs little from uniform flow with velocity component perpendicular and parallel to the generator. In this case the problem can be reduced to two problems independent in the first approximation: the determination of the longitudinal velocity component and the stream function of the disturbed flow. In this approximation the streamlines corresponding to uniform flow over the cylinder serve as longitudinal velocity isotachs. As for the stream function and pressure, they coincide with the corresponding values for plane flow over the same cylinder in the absence of a longitudinal velocity. Using the idea put forward by \textit{P. N. Shankar} and \textit{U. N. Sinha} [J. Méc. 19, 125-148 (1980; Zbl 0423.76019)], we investigate the flow over a circular cylinder when the cross flow has an arbitrary periodic velocity profile far upstream. By assumption, the vorticity is weak. Apart from the requirement that it differs little from a constant value and is differentiable with respect to the transverse coordinate, no restrictions are placed on the longitudinal velocity component, which at points remote from the cylinder depends only on the transverse coordinate.