A minimization principle for incompressible fluid mechanics

Abstract

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).