Cody Gonzalez | Searcharxiv

arXiv2022

What Does Nature Minimize In Every Incompressible Flow?

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).

physics.flu-dyn↗

arXiv2021

A Variational Theory of Lift

In this paper, we revive a special, less-common, variational principle in analytical mechanics (Hertz' principle of least curvature) to develop a novel variational analogue of Euler's equations for the dynamics of an ideal fluid. The new variational formulation is fundamentally different from those formulations based on Hamilton's principle of least action. Using this new variational formulation, we generalize the century-old problem of the flow over a two-dimensional body, to find that lift is a direct consequence of curvature. The developed variational principle reduces to the classical Kutta-Zhukovsky condition in the special case of a sharp-edged airfoil, which challenges the accepted wisdom about the Kutta condition being a manifestation of viscous effects. Rather, we found that it represents conservation of momentum. Moreover, the developed variational principle provides, for the first time, a theoretical model for lift over smooth shapes without sharp edges where the Kutta condition is not applicable. We discuss how this fundamental divergence from current theory can explain discrepancies in computational studies and experiments with superfluids.

physics.flu-dyn↗