Schr\"odinger-Navier-Stokes equation for capillary fluids

TL;DR

This paper explores the Schr"odinger-Navier-Stokes equation's properties, showing its equivalence to the Navier-Stokes-Korteweg equations for capillary fluids, and discusses its implications for microfluidics, soft matter, and quantum simulations.

Contribution

It demonstrates the formal equivalence of the SNS equation to the Navier-Stokes-Korteweg equations and derives dispersion relations relevant for capillary fluids and quantum limits.

Findings

SNS equation is equivalent to Navier-Stokes-Korteweg equations for capillary fluids.

Dispersion relation shows capillary stiffness controlled by dispersive parameter.

Quantum limit recovers Bogoliubov dispersion relation.

Abstract

We highlight some properties of the Schr\"odinger-Navier-Stokes (SNS) equation [Salasnich, Succi, and Tiribocchi (2024)] of potential relevance for microfluidics and soft matter. Specifically, we show that the SNS equationwith 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 decomposes naturally into a Korteweg conservative and a Rayleigh dissipative components, respectively. 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. Finally, it is argued that the SNS may facilitate the quantum simulation of complex states of flowing matter.