CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pendingverified 2026-10-23 (gpt-6.1-sol)

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Negative bulk viscosity violates entropy admissibility

Physical scope

Algebraic counterexample in the Newtonian/Fourier residual family with negative bulk coefficient. This coefficient lies outside the adopted nonnegative-bulk postulate; the example is not a material observation or claim about a realized fluid.

Statement refuted

Let $Sym (3) = {D \in R^{3 \times 3} : D^{T} = D}$ and let $I_{3}$ be the $3 \times 3$ identity matrix. For $D \in Sym (3)$ set $θ = tr D$ , $D_{0} = D - θ I_{3} / 3$ , and $∣ D_{0} ∣^{2} = \sum_{i, j = 1}^{3} (D_{0})_{i j}^{2}$ . For $μ \geq 0$ , $ζ \in R$ , $K \in R^{3 \times 3}$ , $g \in R^{3}$ , and $T > 0$ , define the pointwise Newtonian/Fourier residual $G (D, g, T) = \frac{2 μ ∣ D_{0} ∣^{2} + ζ θ^{2}}{T} + \frac{g^{T} sym (K) g}{T^{2}}, sym (K) = \frac{K + K^{T}}{2} .$ Here $g$ is the temperature-gradient value; in SI units $D$ is in $s^{- 1}$ , $μ, ζ$ in $P a s$ , $K$ in $W m^{- 1} K^{- 1}$ , $g$ in $K m^{- 1}$ , and $G$ in $W m^{- 3} K^{- 1}$ . The false assertion is that $G (D, g, T) \geq 0$ for every such $D, g, T$ whenever $μ \geq 0$ , independently of the value of $ζ$ .

Statement references

David Tong, Fluid Mechanics, Part II lecture notes, §3.1.1, printed pp. 60–61

An Introduction to Continuum Mechanics, Second Edition, Chapter 6, §§6.1.4 and 6.5.3, printed pp. 224–225 and 244–245

Facts & Assumptions

Given: The Newtonian/Fourier residual formula and coefficient test, with the algebraic coefficient family extended to values outside the adopted physical coefficient range.

[L1]

Continuum entropy residual and Newtonian admissibility: for the smooth balance/state hypotheses and the closure $τ = 2 μ D_{0} + ζ θ I_{3}, q = - K g$ , the residual is $(2 μ ∣ D_{0} ∣^{2} + ζ θ^{2}) / T + g^{T} sym (K) g / T^{2}$ . Its exact algebraic test says nonnegativity for every independent $D, g, T > 0$ requires $μ, ζ \geq 0$ and $sym (K) ⪰ 0$ . This is the parent theorem’s explicitly qualified algebraic coefficient-family test; the negative witness lies outside the separately adopted nonnegative-bulk physical subfamily.

[A1]

The negative- $ζ$ coefficient choice below is a pointwise algebraic extension of the closure. It is excluded by the adopted post-fd-newtonian-fourier sign postulate and is not asserted to describe a measured material.

Counterexample

Given: Fix a local state with $T > 0$ , choose $μ > 0$ and $ζ < 0$ , set $K = 0$ , choose any $θ \neq 0$ , and let $D = (θ / 3) I_{3}$ and $g = 0$ . Define $λ = ζ - 2 μ / 3$ if the stress is written $2 μ D + λ (tr D) I_{3}$ .

1.1A1L1givenalgebra

The trace is $tr D = θ$ , so $D_{0} = D - θ I_{3} / 3 = 0$ . By [L1], the residual for these data is $G = ζ θ^{2} / T$ .

2.1A1L1step 1.1algebra∎

Since $ζ < 0$ , $θ^{2} > 0$ , and $T > 0$ , one has $G < 0$ . Thus a positive shear coefficient alone does not guarantee entropy admissibility for compressible strain. This is a coefficient counterexample outside the adopted sign range, not an observation about nature.

Depends on

Continuum entropy residual and Newtonian admissibility