🌀 A Minimization Principle for Incompressible Fluid Mechanics (PMPG) 

This project introduces the Principle of Minimum Pressure Gradient (PMPG) — a new foundational principle in fluid mechanics we derived from Gauss’ Principle of Least Constraint. Unlike the classical Navier–Stokes approach, which solves for pressure as a coupling variable through Poisson's equation, PMPG reveals that nature minimizes the magnitude of the pressure gradient to satisfy incompressibility.

This reformulation turns fluid simulation into a pure minimization problem, offering a physically grounded alternative to traditional PDE-based solvers.

 🧠 Core Concept 

• Pressure acts as a constraint force, enforcing divergence-free velocity fields — much like the normal force keeps a particle on a surface.

• PMPG reframes incompressible flow as an optimization problem 

• The Navier–Stokes equations emerge as first-order optimality conditions from this principle.

• The framework is generalizable to non-Newtonian fluids, electromagnetically forced flows, and active systems.

 🎯 Why It Matters 

• PMPG makes use of the interpretation of pressure in fluid dynamics: it’s the force that enforces incompressibility.
  
  

• It eliminates the need for pressure–velocity coupling iterations in CFD.
  
  

• It defines a natural loss function for machine learning models (e.g., Physics-Informed Neural Networks).
  
  

• It opens new paths for modeling in biofluid dynamics, non-Newtonian flows, and data-driven optimization.

🧪 Applications & Results

Using PMPG, classical flow problems were solved without  using the Navier–Stokes equations, demonstrating both accuracy and insight:

• A novel lift theory applicable to any generic shape lacking a sharp trailing edge, without invoking the Kutta condition which is invalid for shapes without sharp edges, unsteady fluid or at high angles of attack.

• Predicting the separation angle on a stationary cylinder without explicitly incorporating boundary layer effects!

• Analyzing flow over a rotating cylinder within an inviscid framework! a task that previously deemed impossible.

🔗Resources 

• 🎥 Watch: A new theory of lift

• 📄 A minimization principle for incompressible fluid mechanics
  
  

• 🧮 Magnus force estimation using gauss’s principle of least constraint
  
  

• On the Separating Flow Behind a Cylinder: Insights from the Principle of Minimum Pressure Gradient