A Minimization Principle for Incompressible Fluid Mechanics

Fluid mechanics is a branch of mechanics concerned with the motion of fluid flows. Mechanics is typically classified into Newtonian and variational mechanics. The great victories in mechanics achieved by physicists in the past century were mainly enabled by variational approaches. While classical mechanics fails at small scales (e.g., atoms) and large scales (e.g., planets), variational mechanics has provided a fundamental approach in both Einstein’s general relativity and quantum mechanics. On the other hand, when we focus our attention on the mechanics of fluids, we find little success has been achieved beyond the Newtonian approach which was culminated in the Navier-Stokes equations of motion. Certainly, the distribution of research efforts between Newtonian and variational formulations in fluid mechanics is exorbitantly uneven; the overwhelming majority of research efforts are concerned with the Newtonian-mechanics formulation of Navier-Stokes' equations, which may be stalling. Indeed, there is a real need to seriously consider other branches in mehcanics and what they may offer to the motions of fluids.

Most variational principles in classical mechanics are based on the principle of least action, which is only a stationary principle; the “least action” is a misnomer. In contrast, Gauss’ prin-

ciple 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). The PMPG is expected to be of paramount importance for theoretical modeling of fluid mechanics as it encodes a complicated nonlinear partial differential equation into a minimization problem. It even transcends Navier-Stokes’ equations in its applicability to non-Newtonian fluids with arbitrary constitutive relations and fluids subject to arbitrary forcing (e.g. electromagnetic).