arxiv: 2604.26846 · v1 · submitted 2026-04-29 · ⚛️ physics.flu-dyn · cond-mat.stat-mech· math-ph· math.MP

Recognition: unknown

Constitutive Modelling of Korteweg Fluids Using Liu's Method

Zagorka Mati\'c , Srboljub Simi\'c , Peter V\'an

Authors on Pith no claims yet

Pith reviewed 2026-05-07 11:32 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cond-mat.stat-mechmath-phmath.MP

keywords Korteweg fluidsLiu's methodconstitutive modellingthermodynamic consistencycapillary effectsgeneralized Gibbs relationentropy balance

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The pith

Korteweg fluids remain thermodynamically consistent when the material parameter in their stresses depends on temperature.

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

The paper applies Liu's method of multipliers to enforce compatibility between constitutive relations for Korteweg fluids and the entropy balance law. Appropriate assumptions allow capillary effects to enter the specific entropy, from which equilibrium conditions then determine the form of the Korteweg stresses. The key material parameter in those stresses is permitted to vary with temperature. This choice produces cross-coupling between mechanical and thermal fields and yields a generalized Gibbs relation that carries the capillary contributions.

Core claim

Equilibrium conditions obtained by requiring vanishing entropy production and its minimization at equilibrium give Korteweg stresses whose material coefficient may depend on temperature. Liu's multipliers enforce the constraints imposed by the balance laws, and the resulting constitutive structure is consistent with kinetic-theory results while generating a generalized Gibbs relation that inherits the capillary effects in the entropy.

What carries the argument

Liu's method of multipliers, which introduces auxiliary fields to enforce the entropy inequality subject to the local balance laws.

If this is right

The material parameter in the Korteweg stresses can depend on temperature without violating thermodynamic consistency.
Mechanical and thermal effects become cross-coupled through the temperature dependence.
A generalized Gibbs relation is obtained that includes capillary contributions to the entropy.
The constitutive model remains compatible with results from kinetic theory.

Where Pith is reading between the lines

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

The temperature dependence opens a route to continuum descriptions of fluids whose surface tension varies with heat.
The same multiplier technique may be used to derive gradient theories for other non-classical continua while preserving the second law.

Load-bearing premise

Capillary effects can be incorporated into the specific entropy through suitable constitutive assumptions.

What would settle it

A measurement of entropy or free-energy changes in a temperature-gradient Korteweg fluid that deviates from the derived generalized Gibbs relation would falsify the model.

read the original abstract

The paper studies constitutive modelling of Korteweg fluids. Thermodynamic consistency, i.e. compatibility with entropy balance law, is achieved using Liu's method of multipliers. Appropriate constitutive assumptions facilitated inclusion of the capillary effects in the specific entropy. Korteweg stresses are derived from the equilibrium conditions -- vanishing of the entropy production and its minimization in equilibrium. Material parameter in Korteweg stresses is allowed to depend on temperature, which turns out to be consistent with kinetic-theory results and leads to cross-coupling of mechanical and thermal effects. The generalized Gibbs' relation, which inherits the capillary effects, is derived as consequence, which is a peculiar feature of the Liu's method.

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.

Desk Editor's Note private letter to a colleague

Liu's method on Korteweg fluids produces temperature-dependent stresses and a generalized Gibbs relation that follows directly from equilibrium conditions.

read the letter

The core result is that allowing the material parameter in the Korteweg stresses to depend on temperature is thermodynamically consistent, produces mechanical-thermal cross terms, and yields a generalized Gibbs relation that carries the capillary effects. This comes out of applying Liu multipliers to the entropy inequality under constitutive assumptions that fold density gradients into the specific entropy, then imposing vanishing entropy production at equilibrium and its minimization there. The temperature dependence matches existing kinetic-theory checks, which is a useful external anchor rather than an internal fit. The derivation is straightforward once the entropy assumption is granted, and the paper avoids the usual circularity traps by treating the parameter dependence as permitted rather than imposed. The main limitation is that the whole construction rests on those entropy assumptions being physically reasonable; if they turn out to be overly permissive or miss some constraint from kinetic theory, the cross-coupling and generalized relation could be narrower than presented. No internal contradictions appear in the logic, and the citation pattern to prior Liu-method work and Korteweg literature looks standard. This is a specialized theoretical note for people already working on non-equilibrium thermodynamics of fluids with interfaces or capillary forces. A reader who needs a clean, multiplier-based route to consistent constitutive relations for such models will get something usable here. It is worth sending to peer review; the central steps are reproducible from the stated assumptions and the kinetic-theory consistency provides a concrete check that can be examined in detail.

Referee Report

1 major / 3 minor

Summary. The manuscript develops a constitutive theory for Korteweg fluids by applying Liu's method of Lagrange multipliers to the entropy inequality, ensuring thermodynamic consistency with the balance laws of mass, momentum, and energy. Appropriate constitutive assumptions are introduced to incorporate density-gradient (capillary) dependence into the specific entropy. Equilibrium conditions—vanishing entropy production together with its minimization—are then used to obtain the Korteweg stress tensor. The material coefficient multiplying the capillary terms is allowed to depend on temperature; this dependence is shown to be consistent with kinetic-theory results and produces mechanical-thermal cross-coupling. As a direct consequence, a generalized Gibbs relation that inherits the capillary contributions is derived.

Significance. If the derivation is correct, the work supplies a systematic, multiplier-based route to thermodynamically admissible Korteweg models that naturally accommodates temperature-dependent capillary coefficients and the associated cross-effects. The emergence of a generalized Gibbs relation as an equilibrium consequence is a distinctive feature of the Liu procedure and may prove useful for extending classical thermodynamic relations to fluids with non-local stresses. The consistency check against kinetic theory adds credibility to the temperature dependence, which could improve modeling of thermocapillary and phase-change phenomena.

major comments (1)

The central claim rests on the constitutive assumption that the specific entropy may depend on density gradients (stated in the abstract and presumably detailed in the constitutive-equation section). The manuscript should explicitly state the precise functional dependence adopted for the entropy (e.g., linear or quadratic in the gradient) and demonstrate that this choice is the weakest sufficient assumption compatible with the entropy inequality; without that specification it is difficult to judge whether the resulting Korteweg stresses are uniquely determined or merely one possible family.

minor comments (3)

The abstract and introduction refer to “Liu’s method” without a concise recap of the multiplier procedure; a short paragraph or appendix outlining the steps (multipliers for mass, momentum, energy) would improve accessibility.
Notation for the material coefficient in the Korteweg stress (often denoted c or α) should be introduced once and used consistently; its possible temperature dependence should be indicated in the final expression for the stress tensor.
The kinetic-theory comparison is asserted but not shown in detail; a brief table or paragraph contrasting the derived temperature dependence with the corresponding kinetic result would strengthen the consistency claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation of minor revision. We address the single major comment below and will incorporate the requested clarification in the revised manuscript.

read point-by-point responses

Referee: The central claim rests on the constitutive assumption that the specific entropy may depend on density gradients (stated in the abstract and presumably detailed in the constitutive-equation section). The manuscript should explicitly state the precise functional dependence adopted for the entropy (e.g., linear or quadratic in the gradient) and demonstrate that this choice is the weakest sufficient assumption compatible with the entropy inequality; without that specification it is difficult to judge whether the resulting Korteweg stresses are uniquely determined or merely one possible family.

Authors: We appreciate the referee's request for greater precision on the entropy constitutive assumption. In the constitutive theory section we postulate that the specific entropy depends on density, temperature and the density gradient, with the gradient dependence taken to be quadratic. This is the weakest assumption compatible with the entropy inequality because (i) a linear term in the gradient would be odd under reversal and would not survive the equilibrium minimisation condition used to extract the Korteweg stress, and (ii) higher-order gradient terms are unnecessary for recovering the standard first-gradient capillary stresses while still satisfying non-negative entropy production. The quadratic form is therefore both sufficient and minimal; it uniquely determines the admissible Korteweg stresses within the Liu-multiplier framework once the equilibrium conditions are imposed. We will add an explicit statement of this functional dependence together with the above justification in the revised constitutive-assumptions paragraph. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via standard Liu procedure

full rationale

The paper applies Liu's multiplier method to the entropy inequality under explicit constitutive assumptions that place capillary (density-gradient) dependence inside specific entropy. Equilibrium conditions (vanishing entropy production and its minimization) then yield the Korteweg stresses and a generalized Gibbs relation as direct consequences. Temperature dependence of the material parameter is permitted rather than imposed by the framework and is only checked for consistency against independent kinetic-theory results. No step reduces by construction to a fitted input, self-definition, or load-bearing self-citation; the central claims remain independent of the target outputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard thermodynamic assumptions and specific constitutive choices for entropy; no new entities are postulated and no free parameters are explicitly fitted to data in the abstract.

free parameters (1)

material parameter in Korteweg stresses
Allowed to depend on temperature as part of the constitutive assumption; its explicit functional form is not given as a fit in the abstract.

axioms (2)

domain assumption Entropy balance law must hold for thermodynamic consistency
Invoked to apply Liu's method of multipliers.
domain assumption Equilibrium requires vanishing entropy production and its minimization
Used to derive the Korteweg stresses from the constitutive assumptions.

pith-pipeline@v0.9.0 · 5424 in / 1339 out tokens · 57461 ms · 2026-05-07T11:32:49.275295+00:00 · methodology

discussion (0)

Reference graph

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