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arxiv: 2604.12058 · v1 · submitted 2026-04-13 · ⚛️ physics.flu-dyn

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

Prabakaran Rajamanickam This is my paper

Pith reviewed 2026-05-10 15:05 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords Korteweg stressesshear-induced buoyancynon-Boussinesq fluidsnarrow channelsOstroumov flowPrandtl numberdepth averagingmiscible fluids
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The pith

A shear-induced horizontal buoyancy force in narrow channels equals the divergence of a Korteweg stress tensor

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper demonstrates that the horizontal buoyancy force generated by shear in non-Boussinesq fluids inside narrow channels can be rewritten exactly as the divergence of a Korteweg stress tensor. The equivalence follows from depth-averaging the Navier-Stokes equations once the internal flow pattern is locked to the local density gradient. A sympathetic reader would care because the result supplies a purely fluid-mechanical source for stresses that behave like capillary forces in miscible fluids, without any reference to molecular cohesion potentials. The coefficients of the effective stress are shown to depend explicitly on the Prandtl and Grashof numbers, with a change in the character of the internal pressure when the Prandtl number reaches one half.

Core claim

The shear-induced horizontal buoyancy force identified upon depth-averaging the Navier-Stokes equations for non-Boussinesq fluids in narrow channels is formally equivalent to the divergence of a Korteweg stress tensor. This tensor emerges from self-coupled transport in which the internal Ostroumov flow remains enslaved to the local density gradient. Explicit expressions for the effective stress coefficients depend on the Prandtl number and Grashof number, and the effective internal pressure undergoes a transition at Pr equal to one half that separates the regime dominated by shear-flow inertia from the regime dominated by shear-induced hydrostatic tilting.

What carries the argument

Depth-averaged Navier-Stokes equations that incorporate an internal Ostroumov flow enslaved to the local density gradient, which upon averaging yields an effective Korteweg stress tensor.

If this is right

  • The magnitude and form of the effective stresses vary with the Prandtl and Grashof numbers.
  • The effective internal pressure changes character at a Prandtl number of one half.
  • Quadratic Korteweg-type stresses appear as a general consequence of sub-scale transport in any gradient-driven flow that possesses self-coupling.
  • The same averaging procedure applied to flows without self-coupling, such as classical Taylor dispersion, produces only uniaxial stress instead of the full Korteweg tensor.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same enslaved-flow mechanism could generate analogous stresses in other confined gradient-driven systems such as thin liquid films or flow through porous media.
  • Experiments that vary the Prandtl number across one half while holding the Grashof number fixed would directly test the predicted change in effective pressure.
  • Many macro-scale capillary-like effects observed in miscible fluids may ultimately trace back to unresolved small-scale transport rather than to any assumed molecular constitutive relation.

Load-bearing premise

The internal flow pattern inside the channel remains directly slaved to the local density gradient at every horizontal location.

What would settle it

A laboratory measurement of the horizontal body force in a narrow channel with a controlled vertical density gradient, checked against the predicted divergence of the derived stress tensor for fluids whose Prandtl numbers straddle one half.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript is a short note claiming that the shear-induced horizontal buoyancy force identified in prior work on non-Boussinesq fluids in narrow channels is formally equivalent to the divergence of a Korteweg stress tensor. The equivalence is obtained by depth-averaging the Navier-Stokes equations under the assumption that the internal Ostroumov flow is enslaved to the local density gradient, producing self-coupled transport. Explicit expressions for the effective stress coefficients are derived, showing dependence on Prandtl and Grashof numbers, with a transition in effective internal pressure at Pr = 1/2 that marks the crossover between shear-flow inertia and hydrostatic tilting. The result is contrasted with classical Taylor dispersion, which lacks self-coupling and produces only uniaxial stress.

Significance. If the derivations and enslavement assumption hold with the stated generality, the work supplies a hydrodynamic, macro-scale origin for Korteweg-type stresses arising from sub-scale transport in gradient-driven flows. This could explain capillary-like effects in miscible fluids without molecular-scale potentials and would distinguish such stresses from uniaxial Taylor dispersion. The explicit Pr- and Gr-dependence and the Pr = 1/2 transition provide testable parameter regimes. The significance is limited by the absence of a clear validity range for the central enslavement assumption.

major comments (3)
  1. [Abstract and §2] Abstract and §2 (derivation outline): The formal equivalence to the divergence of a Korteweg tensor is asserted to follow from depth-averaging under enslavement of the Ostroumov flow to the local density gradient. The manuscript does not supply the intermediate steps showing how the quadratic stress term is generated specifically by this self-coupling (as opposed to higher-order corrections or the uniaxial term already present in the prior buoyancy derivation), making it impossible to verify that the result is not reducible to the Taylor-dispersion case explicitly contrasted.
  2. [Section deriving stress coefficients] Section deriving stress coefficients (presumably §3–4, around the Pr = 1/2 transition): The transition at Pr = 1/2 is identified as the crossover between internal inertia and hydrostatic tilting. This identification rests on the enslavement assumption holding uniformly across the relevant Pr–Gr range; the manuscript provides no regime-of-validity analysis (e.g., in terms of channel aspect ratio or Gr) to confirm that the assumption remains valid when inertial terms are retained or when the density gradient varies spatially.
  3. [Equation for effective stress coefficients] Equation for effective stress coefficients: The explicit expressions are stated to depend on Pr and Gr. Because the underlying buoyancy force is taken from the cited prior work, it is unclear whether these coefficients are independently derived or reduce by construction once the enslavement relation is imposed; the manuscript should show the algebra that isolates the Korteweg-like quadratic term without circular reference to the earlier definitions.
minor comments (2)
  1. [Notation and assumptions] The term 'enslaved' is used repeatedly but never given a precise mathematical statement (e.g., an explicit relation between transverse velocity and density gradient). Adding this definition would clarify the assumption.
  2. [Figures and equations] Figure captions and equation numbering should be checked for consistency with the depth-averaging procedure described in the text.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address each major comment below, indicating where revisions will be made to improve clarity and completeness of the derivations.

read point-by-point responses

  1. Referee: [Abstract and §2] The formal equivalence to the divergence of a Korteweg tensor is asserted to follow from depth-averaging under enslavement of the Ostroumov flow to the local density gradient. The manuscript does not supply the intermediate steps showing how the quadratic stress term is generated specifically by this self-coupling (as opposed to higher-order corrections or the uniaxial term already present in the prior buoyancy derivation), making it impossible to verify that the result is not reducible to the Taylor-dispersion case explicitly contrasted.

    Authors: We agree that the outline in §2 would benefit from explicit intermediate algebra. In the revised manuscript we will expand the derivation to show the substitution of the enslavement relation (velocity perturbation proportional to local density gradient) into the depth-averaged horizontal momentum balance, followed by rearrangement that isolates the quadratic term as the divergence of a symmetric stress tensor. This step-by-step expansion will demonstrate that the quadratic contribution arises directly from the self-coupled product and is distinct from the uniaxial Taylor-dispersion term retained from the prior work. revision: yes

  2. Referee: [Section deriving stress coefficients] The transition at Pr = 1/2 is identified as the crossover between internal inertia and hydrostatic tilting. This identification rests on the enslavement assumption holding uniformly across the relevant Pr–Gr range; the manuscript provides no regime-of-validity analysis (e.g., in terms of channel aspect ratio or Gr) to confirm that the assumption remains valid when inertial terms are retained or when the density gradient varies spatially.

    Authors: The Pr = 1/2 transition emerges algebraically from the signs of the derived coefficients once the enslavement relation is imposed. We acknowledge that an explicit discussion of the assumption’s validity range is absent. In the revision we will add a dedicated paragraph (new §4.1) that recalls the scaling arguments from the cited prior work and states the conditions (narrow-channel aspect ratio ≪ 1 and moderate Gr) under which the internal Ostroumov flow remains enslaved to the local gradient even when weak inertial corrections are retained. revision: yes

  3. Referee: [Equation for effective stress coefficients] The explicit expressions are stated to depend on Pr and Gr. Because the underlying buoyancy force is taken from the cited prior work, it is unclear whether these coefficients are independently derived or reduce by construction once the enslavement relation is imposed; the manuscript should show the algebra that isolates the Korteweg-like quadratic term without circular reference to the earlier definitions.

    Authors: The coefficients are obtained by direct substitution of the enslavement relation into the depth-averaged equations and subsequent identification of the divergence form; they are not imposed by construction. In the revised §3 we will display the intermediate algebraic steps that begin from the buoyancy term of the prior work, apply the enslavement, collect the quadratic contributions, and arrive at the effective stress tensor without presupposing its Korteweg structure. revision: yes

Circularity Check

1 steps flagged

Korteweg equivalence relies on self-cited shear-induced buoyancy force and enslavement assumption from prior work

specific steps
  1. self citation load bearing [Abstract]
    "Recent study <cit>rajamanickam2025shear</cit> 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. ... we show that this emergent stress arises purely from self-coupled transport where the internal Ostroumov flow is enslaved to the local density gradient."

    The central demonstration of equivalence to the Korteweg tensor takes the existence and form of the shear-induced buoyancy force as an established input from the author's prior publication. The self-coupling that produces the Korteweg-like term upon averaging is enabled by the enslavement assumption, which is introduced in the context of that prior identification rather than derived from first principles within this note alone.

full rationale

The derivation starts from the shear-induced horizontal buoyancy force identified in the author's prior self-cited work. The formal equivalence to the divergence of a Korteweg stress tensor is then shown by depth-averaging under the assumption that the internal Ostroumov flow is enslaved to the local density gradient. This enslavement and self-coupling are presented as enabling the quadratic stress term, but they originate in the prior paper rather than being re-derived independently here. The explicit coefficients depending on Pr and Gr, and the Pr=1/2 transition, are new calculations, so the circularity is partial and load-bearing only on the input force and enslavement concept. No self-definitional loops, fitted predictions, or ansatz smuggling are present in the text.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on the Navier-Stokes equations, the validity of depth-averaging in narrow channels, and the enslavement of internal flow to the density gradient; no new free parameters are introduced beyond the standard dimensionless groups Pr and Gr.

axioms (3)
  • standard math The Navier-Stokes equations govern the flow of non-Boussinesq fluids
    Starting point for the depth-averaging procedure that identifies the horizontal buoyancy force.
  • domain assumption Depth-averaging is valid and applicable in narrow channels
    Enables emergence of the shear-induced buoyancy upon averaging.
  • ad hoc to paper The internal Ostroumov flow is enslaved to the local density gradient
    This self-coupling assumption is required to generate the Korteweg-like stress from transport.

pith-pipeline@v0.9.0 · 5514 in / 1711 out tokens · 54720 ms · 2026-05-10T15:05:23.257432+00:00 · methodology

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Reference graph

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