On the Mathematical Analysis and Physical Implications of the Principle of Minimum Pressure Gradient

Abstract

In this paper, we establish a two-way equivalence between the incompressible Navier- Stokes equation (INSE) and the principle of minimum pressure gradient (PMPG). We prove that a candidate smooth flow field is a solution of the INSE if and only if its instantaneous evolution minimizes, at every instant, the norm of the pressure force, required to enforce incompressibility. We show that the PMPG is precisely the minimization formulation of the Leray-Helmholtz projection. Any admissible instantaneous evolution (e.g., onset of separation) resulting from the INSE necessarily minimizes the PMPG cost. Conversely, any other kinematically admissible evolution, requiring a strictly larger pressure force to ensure the same constraints, does not satisfy the INSE. Thus, the PMPG offers a variational perspective through which intricate incompressible flow behaviors may be interpreted. In a finite-dimensional setting with divergence-free modes, we show that the PMPG yields the same dynamics as classical Galerkin projection. Moreover, the PMPG provides a natural generalization of classical Galerkin projection beyond linear modal expansions, accommodating nonlinear and non-modal representations. We then examine the relation between instantaneous dynamical minimization and steady variational selection, including its connection to the variational theory of lift. Motivated by these observations, we formulate conjectures concerning necessary conditions for stability and the convergence of Navier-Stokes solutions to Euler's in the vanishing-viscosity limit.

Figures (9)

• Figure 1: Schematic illustration of the instantaneous plane of admissible motion, defined by (\ref{['eq:Constraints_Linear']}), at a particular configuration. Infinitely many instantaneous motions lie this plane (dotted vectors), each satisfying the linear constraint (\ref{['eq:Constraints_Linear']}) at the expense of a constraint force $\bm{R}$. Gauss's principle asserts that, among all kinematically admissible motions, Nature selects the one that requires the least constraint force. This minimization is simply equivalent to projecting the impressed force $\bm{F}$ onto the plane of admissible motions.
• Figure 2: A Schematic for a double-pendulum oscillating in a horizontal plane.
• Figure 3: A Schematic for the plane of admissible accelerations at a given point ($\bm{q},\dot{\bm{q}})$ in the tangent bundle, along with the contours of the Gaussian cost $Z$ in Eq. (\ref{['eq:Gaussian_Double_Pendulum_Quadratic']}) with the accelerations $\ddot\theta$, $\ddot\phi$ at the point $\bm{q}=(0^\circ,5^\circ)$, $\dot{\bm{q}}=(1,1)$. The minimizing accelerations $(\ddot\theta^*,\ddot\phi^*)$ satisfy Newton's equations of motion. However, the ones that minimize the 4-norm $\sum_i m_i \bm{a}_i^4$ do not necessarily coincide with the Newtonian evolution.
• Figure 4: A Schematic diagram illustrating the Helmholtz-Leray projection and geometry of incompressible flows.
• Figure 5: Schematic illustrating the equivalence between Newton's equations and Gauss's principle in particle mechanics, and the corresponding equivalence between the Navier-Stokes equation and the Principle of Minimum Pressure Gradient (PMPG) in incompressible continuum mechanics.