Magnus Force Estimation Using Gauss’s Principle of Least Constraint

Abstract

The flow around a rotating cylinder is one of the fundamental problems that have piqued the interests of many venerable fluid mechanicians since the time of Rayleigh. The force caused by the rotation of the cylinder has always been considered as an immediate consequence of viscosity, since the potential flow model failed entirely to predict the value of the circulation due to the lack of a Kutta-like condition. On the other hand, Glauert modeled the flow outside the boundary layer of a rotating cylinder as a potential flow with an unknown circulation. He then obtained an approximate solution of Prandtl’s boundary-layer equations and applied the no-slip condition to estimate the circulation in the outer flow. Interestingly, for rapidly rotating cylinders ([Formula: see text]), up to fourth-order in the small parameter [Formula: see text], the obtained circulation is independent of viscosity. In this work, we use Glauert’s model of the outer flow (i.e., a potential flow with an unknown circulation). However, instead of the tedious boundary-layer calculations, we rely on Gauss’s principle of least constraint to obtain the unknown circulation. A perfect match with Glauert’s solution is found. Moreover, our solution, in contrast to Glauert’s, points to the existence of different physics at small rotational speeds. The obtained results, given their perfect matching with Glauert’s solution (relying on the no-slip condition), point to a potential equivalence between the no-slip condition and fluid body forces.