Bernoulli Principle in Ferroelectrics.

Ferroelectric materials, characterized by spontaneous electric polarization, exhibit remarkable parallels with fluid dynamics, where polarization flux behaves similarly to fluid flow. Understanding polarization distribution in confined geometries at the nanoscale is crucial for both fundamental physics and technological applications. Here, we show that the classical Bernoulli principle, which describes the conservation of the energy flux along velocity streamlines in a moving fluid, can be extended to the conservation of polarization flux in ferroelectric nanorods with varying cross-sectional areas. Geometric constrictions lead to an increase in polarization, resembling fluid acceleration in a narrowing pipe, while expansions cause a decrease. Beyond a critical expansion, phase separation occurs, giving rise to topological polarization structures such as polarization bubbles, curls and Hopfions. This effect extends to soft ferroelectrics, including ferroelectric nematic liquid crystals, where polarization flux conservation governs the formation of complex mesoscale states.