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A thermodynamical suspension model for blood

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Pith's one-line read A two-temperature, two-velocity suspension model for blood is shown to satisfy the second law of thermodynamics, with all constitutive restrictions solved explicitly in one dimension.

desk verdict New two-temperature blood suspension model with internal variables; the 1D solution is plausible but the omitted Cq computation makes it uncheckable as printed. read the letter →

arxiv 2506.00067 v1 pith:6TF5IAWU submitted 2025-05-29 physics.flu-dyn math-phmath.MPphysics.bio-ph

M. Gorgone , C.F. Munafo' , A. Palumbo , P. Rogolino This is my paper
classification physics.flu-dynmath-phmath.MPphysics.bio-ph
keywords bloodsuspensionmixturetheorytwo-temperatureinternalvariablesClausius-DuheminequalityextendedColeman-Nollprocedurenon-localconstitutiveequationsthermodynamiccompatibility
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper constructs a thermodynamically consistent model of blood as a two-component suspension: red blood cells in plasma, each component carrying its own temperature, velocity, and an internal scalar variable for additional dissipation. The authors impose the Clausius-Duhem entropy inequality on the whole mixture and exploit it through the extended Coleman-Noll procedure, in which the balance equations and their gradient extensions act as constraints. Their central result is an explicit one-dimensional solution of all the restrictions that the second law places on the constitutive equations. The solution yields temperature-dependent viscosity-like stress terms, Fourier-like heat fluxes, and entropy fluxes with a cubic extra-flux in the gradients. If the derivation is correct, the model provides a second-law-compatible starting point for blood flow simulations that include thermal and non-local (gradient) effects.

What carries the argument

The load-bearing mechanism is the extended Coleman-Noll entropy exploitation: the Clausius-Duhem inequality for the mixture is constrained not only by the balance equations of mass, momentum, energy, and internal-variable evolution, but also by their first-order gradient extensions, because the state space contains first gradients. After eliminating time derivatives, the entropy inequality takes a form linear in the highest derivatives (third spatial derivatives) and quadratic in the higher derivatives (second spatial derivatives), which the procedure treats as arbitrarily assignable. Annihilating their coefficients yields $A_p=0$, $C_q=0$, the positive-semidefiniteness of $B_{qr}$, and a residual dissipation inequality. The paper then solves these conditions in one dimension using polynomial constitutive ansatze and a first-order gradient expansion of the Helmholtz free energies, with the algebra carried out by a computer algebra system.

What would settle it

Compute the full Clausius-Duhem entropy production (13) along a smooth one-dimensional flow satisfying the balance equations (9) with boundary data that fix a third spatial derivative, such as $\rho^{(1)}_{,xxx}$, at a value the extended procedure treated as arbitrary; if the production can be negative for data that also satisfy the solved constraints (35)-(45), the arbitrariness premise behind $A_p=0$ fails.

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Extended reading notes

Core claim

The paper's claim is that for a binary mixture of red blood cells and plasma, with two temperatures $\theta^{(A)}$, two velocities $v^{(A)}$, two internal variables $\gamma^{(A)}$, and a first-order gradient state space, the second law does not force the constitutive theory back to a purely local form. Under the extended Coleman-Noll procedure, requiring the entropy inequality to hold for arbitrary highest and higher spatial derivatives leads to the conditions $A_p=0$, $C_q=0$, positive semidefiniteness of $B_{qr}$, and a residual dissipation inequality. The authors solve these conditions in one dimension under assumed polynomial constitutive forms, obtaining explicit representations for the free energies, entropy fluxes, stresses, source terms, and heat fluxes. The residual inequality reduces to the quadratic form (32) with the inequalities (35)-(45). The structural outcomes are an entropy extra-flux cubic in the gradients and temperature-dependent viscosity terms, while a squared velocity-gradient term in the red-blood-cell stress used in an earlier mechanical model is shown to be incompatible with the second law.

Load-bearing premise

The decisive premise is that the highest and higher spatial derivatives appearing in the constrained entropy inequality can be varied freely across admissible thermodynamic processes, so their coefficients must vanish independently; if the balance equations and their gradient extensions tie any of those derivatives to lower-order data, the derived restrictions are not necessary.

Editorial extensions

If this is right

  • The second law is compatible with first-order non-local constitutive equations for a two-temperature blood suspension, so thermal and microstructure-gradient effects can be kept in a thermodynamically admissible model.
  • The model supplies explicit constitutive forms: temperature-dependent viscosity terms in the partial stresses, Fourier-like heat fluxes, internal-variable source terms, and an entropy extra-flux cubic in the density and internal-variable gradients.
  • In this framework the squared velocity-gradient term in the red-blood-cell stress from the earlier purely mechanical model is ruled out by the entropy inequality.
  • The remaining free material functions can in principle be specialized to experimental rheological and thermal data, giving predictive one-dimensional blood flow simulations at fixed hematocrit and temperature.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Inference: The same state space and exploitation machinery could be applied to other two-phase biological suspensions, such as platelet-rich plasma or cell culture flows, where two temperatures and two internal variables are physically relevant; the one-dimensional coefficient solution would carry over with different material functions.
  • Inference: The solved condition $\Gamma^{(2)}_4=0$, which removes a density-gradient coupling in the plasma internal-variable source, suggests an asymmetry between the red-blood-cell and plasma phases that could be probed in particle-resolved or lattice-Boltzmann simulations of suspension rheology.
  • Inference: The cubic extra-flux in the entropy flux is a distinctive non-local signature; numerical arterial flow studies based on this model could quantify whether that term is clinically significant at physiological hematocrit and temperature gradients.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper develops a thermodynamic model of blood as a binary mixture of red blood cells and plasma, allowing different temperatures and velocities for the two constituents and introducing two scalar internal variables to model additional dissipative effects. Using the extended Coleman-Noll procedure, the authors derive entropy-inequality restrictions on the constitutive equations and, in one space dimension, propose polynomial constitutive ansatzes for stresses, heat fluxes, internal-variable sources, and Helmholtz free energies. They report that the resulting constraints can be solved explicitly with the computer algebra system Reduce, yielding explicit expressions for the partial Cauchy stresses, entropy fluxes, a residual dissipation inequality, and a list of sufficient conditions for its non-negativity.

Significance. If the computations are correct, the paper offers a nontrivial contribution to the thermodynamics of nonlocal mixture models: it exhibits a second-law-admissible constitutive class for a two-temperature, two-velocity blood suspension with first-order gradient dependence, including temperature-dependent viscosity terms and entropy extra-fluxes. The model extends the purely mechanical framework of Massoudi et al. and is directly relevant to continuum thermodynamics and hemorheology. The main strength is the explicit residual inequality (32) and the closed-form expressions (26)-(30), which are falsifiable and could be fitted to experimental data. However, the claimed admissibility rests on a verification gap: the Cq = 0 restrictions are omitted and the CAS computations are not reproducible from the manuscript.

major comments (3)
  1. [Section 3 (after Eq. (22)) and Section 4 (first paragraph)] The central load-bearing step is the derivation and solution of the Cq = 0 restrictions, but these restrictions are never displayed. Section 3 states that their expressions are 'rather long' and omits them, and Section 4 reports that they were solved with CAS Reduce without including the script, the output, or a reproducible transcript. Since Cq = 0 is necessary in Eq. (16) for the entropy inequality to hold for arbitrary higher derivatives Y_q, the displayed solution (26)-(30) cannot be checked from the manuscript alone. Please provide the full Cq = 0 system and the Reduce computations, or an appendix with sufficient intermediate algebra, as supplementary material. Without this, the claim that the model is thermodynamically admissible is not verifiable.
  2. [Section 4, Eqs. (33)-(34)] The 'explicit solution of all thermodynamic constraints' is conditional on assumptions that are not consequences of the second law. Eq. (33) is explicitly acknowledged not to be a thermodynamic restriction, and Eq. (34) sets the non-Fourier heat-flux coefficients to zero. As a result, what is exhibited is a second-law-admissible subfamily of the original ansatz (23)-(25), not a complete solution of the entropy-inequality constraints. The abstract and Section 5 should be reworded to state this clearly, and the residual inequality (32) should be presented as holding under (33)-(34) rather than as the general outcome of the entropy principle.
  3. [Section 4, Eqs. (35)-(45)] The statement that the residual inequality (32) is fulfilled 'if and only if' conditions (35)-(45) hold is not derived in the text. In particular, conditions (43)-(45) mix sign restrictions on derivatives of Gamma(A)_0 with heat-flux coefficients, and it is not obvious that the list is exhaustive. Please provide the derivation or a Reduce transcript showing that these conditions are both necessary and sufficient for (32). This is part of the verification gap already noted, but it should be addressed explicitly because the 'if and only if' claim is stronger than the rest of the exposition supports.
minor comments (5)
  1. [Section 2.1, after Eq. (3)] The text calls epsilon(A) the 'partial internal energies per unit volume', but the balance equations multiply epsilon(A) by the mass density rho(A), which suggests that epsilon(A) is the specific internal energy per unit mass. Please correct this terminology or clarify the intended meaning.
  2. [Section 3, Eq. (22)] The sentence 'the thermodynamical restrictions (18)-(21) are satisfied provided that (22) holds' would benefit from a short explanation of how the condition (22) annihilates the coefficients in (18)-(21). As written, the reader must reconstruct the algebra to see why the weighted sum of free energies controls all these coefficients.
  3. [Section 4, Eq. (26)] The notation psi(A)_0 is used both as a function appearing in the free-energy expansion and as a coefficient in the relation bpsi(A)_0 = epsilon(A) - theta(A) psi(A)_0. Please use distinct symbols to avoid ambiguity.
  4. [Section 4, Eqs. (26), (29), and (31)] The result Gamma(2)_4 = 0 is derived only in Eq. (31), but Eqs. (26) and (29) are written for general A before this specialization. It would be clearer to state Gamma(2)_4 = 0 immediately after Eq. (26) or to explain that the formulas are valid with Gamma(2)_4 set to zero in the second constituent.
  5. [Section 3, after Eq. (15)] The claim that the highest and higher derivatives 'may assume arbitrary values' is essential for the necessity of Ap = 0 and Cq = 0. This follows from the cited extended Coleman-Noll framework, but a one-sentence justification in the present nonlocal setting would make the paper more self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: derivations are from entropy inequality, not assumed; the 1D solution is a constructed example, not a repackaged prediction.

full rationale

The derivation chain is not circular. The state space (8) and ansatze (23)-(25) are explicit assumptions; the entropy inequality (13) is exploited to derive restrictions (17)-(21), (26)-(31), and residual inequality (32). This proves compatibility of the chosen ansatz with the second law, a scope limitation, not circularity. The paper does not claim necessity of the ansatz, only that a thermodynamically admissible solution is exhibited. Constraint (33) is explicitly labeled non-thermodynamical and assumed for technical reasons, so it is not disguised as a derived output. Citations [29], [41], [42] support the general Coleman-Noll exploitation method and the polynomial structure of (16); they do not supply the blood-specific coefficients or the 1D solution. The main weaknesses are verification gaps: Cq=0 restrictions are omitted, the Reduce computation is not included, and the displayed conditions (35)-(45) are sufficient rather than shown necessary for the residual inequality. These are correctness/reproducibility concerns, not circularity. No fitted parameter is renamed as a prediction, and no conclusion is equivalent by definition to an input.

Assumptions & free parameters 4 free parameters · 6 assumptions · 1 invented entities

The proof rests on standard entropy-inequality machinery plus a series of freely chosen constitutive functions. No experimental data enter, so the contribution is a consistency framework: it shows a wide class of gradient-dependent constitutive equations can satisfy the second law in 1D, but it does not determine the material coefficients.

free parameters (4)
  • κ(1), κ(2) = arbitrary constants
    Introduced in (26) and fixed by the technical assumption (33), bΓ(A)_3 = κ(A)(ρ(A))^3(ψ(A)_3)^2, to make the reduced entropy inequality a homogeneous quadratic form; no physical meaning or data fit.
  • ψ(A)_3(ρ(A)) = unspecified free functions
    Coefficients of quadratic gradient terms in the Helmholtz free energy (25); constrained only by ψ(A)_3 ≤ 0, not determined by data.
  • Γ(A)_0 and Γ(A)_4 = unspecified free functions
    Source and kinetic coefficients for internal-variable dynamics (5), (26); satisfy partial differential constraints (36)-(38) but are not fixed by experiment.
  • q(A)_1, q(A)_2, τ(A)_4, τ(A)_5 = unspecified free functions
    Thermal conductivity and viscosity coefficients in the constitutive forms (23); only inequality constraints (39)-(42) are imposed.
assumptions (6)
  • domain assumption The total entropy production of the mixture is the sum of partial productions and is nonnegative, σ = σ(1)+σ(2) ≥ 0 (Eq. 7).
    The second-law criterion used to eliminate constitutive relations. It allows a single phase to have negative production as long as the mixture total is nonnegative.
  • domain assumption The principle of phase separation: T(1) and T(2) depend only on kinematical quantities of the corresponding constituent (Section 2.1, [32,33]).
    Restricts interaction coupling in the stress tensors before the entropy analysis.
  • domain assumption Partial balances of momentum and energy neglect interaction terms between RBCs and plasma (Section 2.1, Eq. (3)).
    A physical simplification admitted by the authors; including drag or energy exchange would change the balance equations and hence the derived restrictions.
  • domain assumption The state space is first-order nonlocal: Z contains ρ(A), θ(A), γ(A), D(A), and first gradients of ρ(A), θ(A), γ(A) (Eq. (8)).
    Defines which gradients the constitutive functions may depend on; the main conclusion is limited to this state space.
  • standard math The highest and higher derivatives (14)-(15) may assume arbitrary values in admissible processes, justifying annihilating their coefficients (Sec. 3 after Eq. (15)).
    This is the key lemma of the extended Coleman-Noll procedure from [29]; if it fails, the necessity of Ap=0 and Cq=0 is not established.
  • ad hoc to paper Constitutive ansatz for stresses, heat fluxes, and internal-variable sources (23), free-energy expansion (25), and technical assumption (33) with q(A)_k=0 for k=3,4 (34).
    These choices are not derived from the second law; the paper states (33) is assumed for technical reasons and (34) is a simple-case ansatz, so the solved constraints apply only within this class.
invented entities (1)
  • Internal variables γ(1) and γ(2)
    purpose: Scalar fields whose evolution equations (5) represent additional dissipative microstructure effects in each blood constituent.
    No direct physical interpretation or independent measurement is given; their dynamics are governed by free functions Γ(A) left uncalibrated.

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Pith. "Pith review of A thermodynamical suspension model for blood." pith.science (2026). https://pith.science/paper/6TF5IAWU

@misc{pith2026250600067,
  author       = {Pith},
  title        = {Pith review of: A thermodynamical suspension model for blood},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6TF5IAWU}},
  note         = {Machine review of arXiv:2506.00067}
}
read the original abstract

A complete thermodynamical analysis for a blood model, based on mixture theory, is performed. The model is developed considering the blood as a suspension of red blood cells (solid component) in the plasma (fluid component), and taking into account the temperature effects. Furthermore, two independent scalar internal variables are introduced accounting for additional dissipative effects. Using Clausius-Duhem inequality, the general thermodynamic restrictions and residual dissipation inequality are derived. The thermodynamic admissibility with the second law of thermodynamics is assessed by means of the extended Coleman-Noll procedure; in one space dimension we exhibit a solution of all the thermodynamical constraints.

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