# Bulletin of the American Physical Society

# 75th Annual Meeting of the Division of Fluid Dynamics

## Sunday–Tuesday, November 20–22, 2022; Indiana Convention Center, Indianapolis, Indiana.

### Session Q26: General Fluid Dynamics: Theory

1:25 PM–3:22 PM,
Monday, November 21, 2022

Room: 234

Chair: Carlo Casciola, Univ of Rome La Sapienza

### Abstract: Q26.00004 : What Does Nature Minimize In Every Incompressible Flow?

2:04 PM–2:17 PM

#### Presenter:

Haithem E Taha

(UC Irvine)

#### Authors:

Haithem E Taha

(UC Irvine)

Cody Gonzalez

(University of California, Irvine)

*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.

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