In single-fluid turbulence, the energy cascade — the process by which kinetic energy transfers from large eddies to small ones — obeys exact scaling laws. Kolmogorov's 4/5 law relates the third-order structure function to the energy dissipation rate. The 2/15 law extends the relationship to anisotropic components. These laws are among the few exact results in the theory of turbulence: derived from the Navier-Stokes equations without approximation.
When two immiscible fluids mix turbulently — oil and water, for instance — there is an interface. The interface has surface tension. Surface tension stores energy and resists deformation. The standard assumption has been that the interface is passively advected by the turbulent flow — carried along, deformed, broken up, but not participating in the energy cascade itself. The energy cascade happens in the fluid; the interface just goes along for the ride.
Crialesi-Esposito, Chibbaro, and Boffetta (arXiv:2603.12143, March 2026) derive exact scaling laws for isotropic binary fluid turbulence. The new laws are analogs of the 4/5 and 2/15 laws, but they include explicit contributions from the interfacial terms — surface tension forces that appear directly in the scaling relations.
The interface participates. It doesn't just absorb and release energy at the boundaries of the cascade; it contributes to the cascade itself, appearing as a source term in the exact scaling law. The surface tension between the two fluids modifies the rate at which energy transfers across scales, entering the equation on equal footing with the inertial and dissipative terms.
The structural lesson: a boundary between two systems can participate in the dynamics it appears merely to separate. The interface between oil and water looks like a surface — a dividing line between two domains. In the scaling law, it is a force — a contributor to energy transfer at every scale. The distinction between “the system” and “the boundary of the system” breaks down when the boundary carries its own dynamics.