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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. 

Most variational principles in classical mechanics are based on the principle of least action, which is only a stationary principle. In contrast, Gauss' principle of least constraint is a true minimum principle. In this paper, we apply Gauss' principle to the mechanics of incompressible flows, thereby discovering the fundamental quantity that Nature minimizes in most flows encountered in everyday life. We show that the magnitude of the pressure gradient over the domain is minimum at every instant of time. We call it the principle of minimum pressure gradient (PMPG). It turns a fluid mechanics problem into a minimization one. We demonstrate this intriguing property by solving four classical problems in fluid mechanics using the PMPG without resorting to Navier–Stokes' equation. In some cases, the PMPG minimization approach is not any more efficient than solving Navier–Stokes'. However, in other cases, it is more insightful and efficient. In fact, the inviscid version of the PMPG allowed solving the long-standing problem of the aerohydrodynamic lift over smooth cylindrical shapes where Euler's equation fails to provide a unique answer. The PMPG transcends Navier–Stokes' equations in its applicability to non-Newtonian fluids with arbitrary constitutive relations and fluids subject to arbitrary forcing (e.g., electromagnetic).