What is it about?

This paper introduces a new way of thinking about pressure in fluids. Instead of treating pressure as just a number from the equations of motion, we show that it can be understood as a direct consequence of the shape (curvature) of the surface that the fluid flows around. Using a variational, geometry-based approach, we recover classical results for flow around cylinders and airfoils but now with a unified, first-principles explanation. This provides a rigorous foundation that could make it easier to extend traditional methods to more complex or realistic flows in the future.

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Why is it important?

Understanding pressure is central to predicting how fluids interact with objects from airplane wings to pipelines and medical devices. Traditionally, pressure has been explained through separate rules and numerical tricks. Our work shows that pressure actually arises from geometry itself: the way a surface curves against the fluid. This unifying perspective not only clarifies the foundations of classical methods like panel techniques, but also opens the door to more robust models of real-world flows, including cases with viscosity, turbulence, or complex geometries. By linking pressure directly to geometry, engineers gain a cleaner, more flexible framework for both analysis and design.

Perspectives

This geometric view of pressure provides a new foundation for classical fluid mechanics. By unifying interior flow equations and boundary conditions in one principle, it bridges theory and practice in a way that traditional methods never fully explained. Looking ahead, this approach could extend to more realistic cases, such as viscous and turbulent flows, where curvature-driven effects are key. It may also inspire new computational methods that are simpler, more accurate, and grounded in first principles. In the long term, linking pressure directly to geometry could reshape how we model, control, and design flows across aeronautics, energy systems, and beyond.

Marcial Sanchis
KTH

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This page is a summary of: Pressure as boundary curvature: A variational approach to potential flows, Physics of Fluids, August 2025, American Institute of Physics,
DOI: 10.1063/5.0286411.
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