Pressure as boundary curvature: A variational approach to potential flows

We present a geometric reinterpretation of pressure in inviscid potential flows, deriving it from a unified variational principle posed over both the fluid volume and the fluid–solid interface. The bulk action enforces incompressibility via a Lagrange multiplier identified with pressure, while the curvature-dependent boundary action enforces impermeability and yields a surface stress–energy tensor. Its normal–normal projection reproduces the classical wall-pressure distribution. The surface variation also yields a Laplace–Beltrami equation for the boundary potential, whose Green's representation recovers the classical panel-method kernel from first principles. We demonstrate that this framework reproduces canonical solutions for flow around cylinders (with and without circulation) and symmetric airfoils via Joukowski mapping. By unifying the interior and boundary constraints through a single geometric action, this formulation provides a rigorous and extensible foundation for pressure as an emergent curvature-driven quantity.