The physics behind Heron's fountain

Heron's fountain looks like a simple trick: water spurts upward as if the device could run forever. In reality, it is a showcase of several physical phenomena at work—gravity, air compression, viscosity, turbulence, and energy dissipation all intertwined. Surprisingly, this classic demonstration has rarely been treated beyond a qualitative level. In this work, we build a consistent theoretical framework, extending Bernoulli's equation to include viscous effects and clarifying the limits of its applicability within the system. We complement the model with experiments that allow a direct comparison between predictions and data. The result is a new perspective on a two-thousand-year-old invention, in which a familiar classroom curiosity becomes a precise tool to explore fundamental fluid mechanics. From an educational standpoint, the system also provides a flexible platform for laboratory or classroom activities, suitable for advanced undergraduate and early graduate students, and well-suited to illustrate hydrostatics, viscous flow, and the interplay between theory and experiment.