What Does Nature Minimize In Every Incompressible Flow?

Here, we revive Gauss' principle of least constraint, which is not commonly found in textbooks of classical physics. It asserts that the motion of a constrained mechanical system is such that the the magnitude of constraint forces must be minimum at every instant. It is an actual minimal principle in contrast to the stationary principle of least action. For incompressible fluids, the pressure gradient is a constraint force; its main role is ensure the continuity constraint. Hence, according to Gauss' principle, the integral of the magnitude of the pressure gradient is minimum at every instant. We prove that Navier-Stokes equations represent the necessary condition for minimization of the pressure gradient. We call it The Principle of Minimum Pressure Gradient (PMPG). It turns any fluid mechanics problem into a minimization problem. We demonstrate this intriguing property by solving classical fluid mechanics problems (e.g., channel flow, Stokes' 2nd problem) without resorting to Navier-Stokes' equations, rather by minimizing the pressure gradient with respect to the free parameters. The PMPG is applicable to non-Newtonian fluids with arbitrary forcing (e.g., electromagnetic). In this case, Navier-Stokes' equations must be modified, while the PMPG is applicable verbatim. Moreover, its inviscid version, which provides a minimization formulation of Euler's euqations, may provide closure for two-dimensional problems where Euler's equations do not have a unique solution. For example, the century-old problem of the ideal flow over an airfoil with an arbitrary shape (not necessary with a sharp trailing edge) is not solvable using classical techniques. In contrast, the PMPG provides a unique solution (i.e., a unique circulation that minimizes the pressure gradient). Interestingly, this unique circulation reduces to Kutta's in the special case of a sharp-edged airfoil, which challenges the accepted wisdom about the viscous nature of the Kutta condition.