What Does Nature Minimize In Every Incompressible Flow?

Abstract

In this paper, we discover the fundamental quantity that Nature minimizes in almost all flows encountered in everyday life: river, rain, flow in a pipe, blood flow, airflow over an airplane, etc. We show that the norm of the pressure gradient over the field is minimum at every instant of time! We call it the principle of minimum pressure gradient (PMPG). The principle is deeply rooted in classical mechanics via Gauss' principle of least constraint. Therefore, while we prove mathematically that Navier-Stokes' equation represents the necessary condition for minimization of the pressure gradient, the PMPG stands on its own philosophy independent of Navier-Stokes'. It turns any fluid mechanics problem into a minimization one. We demonstrate this intriguing property by solving three of the classical problems in fluid mechanics using the PMPG without resorting to Navier-Stokes' equation. 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. Moreover, the result challenges the accepted wisdom about lift generation on an airfoil, which has prevailed over a century. The PMPG is expected to be transformative for theoretical modeling of fluid mechanics as it encodes a complicated nonlinear partial differential equation into a simple minimization problem. The principle even transcends Navier-Stokes' equations in its applicability to non-Newtonian fluids with arbitrary constitutive relations and fluids subject to arbitrary forcing (e.g. electric or magnetic).