A hydrodynamic origin of Korteweg stresses from shear-induced horizontal buoyancy

TL;DR

This paper reveals that shear-induced horizontal buoyancy in non-Boussinesq fluids can be understood as a Korteweg stress arising from self-coupled transport, linking micro-scale effects to macro-scale stresses.

Contribution

It demonstrates that a shear-induced buoyancy force is equivalent to a divergence of Korteweg stress, originating from internal flow coupling rather than molecular interactions.

Findings

The emergent stress depends on Prandtl and Grashof numbers.

A transition occurs at Pr=1/2 between internal inertia and hydrostatic effects.

Quadratic Korteweg stresses may be universal in gradient-driven flows.

Abstract

Recent study \cite{rajamanickam2025shear} of non-Boussinesq fluids in narrow channels identified a novel shear-induced horizontal buoyancy force that emerges upon depth-averaging the Navier--Stokes equations. This note demonstrates that this force is formally equivalent to the divergence of a Korteweg stress tensor. Unlike classical Korteweg stresses, which are typically attributed to molecular-scale cohesive potentials or implemented through assumed constitutive relations, we show that this emergent stress arises purely from self-coupled transport where the internal Ostroumov flow is "enslaved" to the local density gradient. We derive explicit expressions for the effective stress coefficients, revealing a fundamental dependence on the Prandtl number and Grashof number and identifying a transition in the effective internal pressure at $Pr=1/2$, which marks the crossover between the internal inertia of the shear flow and the hydrostatic tilting induced by the shear. This correspondence is contrasted with classical Taylor dispersion, where the absence of self-coupling yields only a uniaxial stress. Our results suggest that quadratic Korteweg-type stresses may be a universal manifestation of sub-scale transport in gradient-driven flows, providing a rigorous macro-scale origin for capillary-like stresses in miscible fluids.