# Variational Principle for Velocity-Pressure Formulation of Navier-Stokes Equations

## TL;DR

This paper develops a variational principle for the velocity-pressure formulation of Navier-Stokes equations, linking the original and adjoint systems to produce equivalent Euler-Lagrange equations under certain conditions.

## Contribution

It introduces a modified variational principle that yields the same solutions as Navier-Stokes equations, extending the theoretical framework for fluid dynamics analysis.

## Key findings

- The variational principle can be adapted to Navier-Stokes equations with a unique adjoint solution.
- The approach is valid for steady-state flows with finite Reynolds number.
- The method maintains the same order as the original Navier-Stokes equations.

## Abstract

The work described here shows that the known variational principle for the Navier-Stokes equations and the adjoint system can be modified to produce a set of Euler-Lagrange variational equations which have the same order and same solution as the Navier-Stokes equations provided the adjoint system has a unique solution, and provided in the steady state case, that the Reynolds number remains finite.