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

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 kinematically coupled 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. This correspondence is contrasted with classical Taylor dispersion, where the absence of self-coupling yields only a uniaxial stress. Our results provide an example - not a general theory - of how quadratic gradient stresses can emerge from sub-scale self-coupled gradient-driven flows, without appeal to molecular potentials or variational principles.