Schrödinger–Navier–Stokes equation for capillary fluids

We highlight some properties of the Schrödinger–Navier–Stokes (SNS) equation [Salasnich et al., Int. J. Mod. Phys. C 35, 2450100 (2024)] of potential relevance for microfluidics and soft matter. Specifically, we show that the SNS equation with generic parameters is formally equivalent to the Navier–Stokes–Korteweg equations for capillary fluids, with the equivalence established at the level of an action functional that naturally decomposes into a Korteweg conservative and a dissipative contribution. We derive the dispersion relation for sound modes, showing that the dispersive parameter controls capillary stiffness while the dissipative parameter controls viscous damping, and that the Bogoliubov dispersion relation is recovered in the quantum limit. We also derive an effective one-dimensional SNS equation for a fluid confined in a narrow capillary tube.