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

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Incompressibility does not imply uniform density

Physical scope

Smooth positive-density, no-mass-source kinematics on a Newtonian spatial domain. The witness satisfies mass transport and incompressibility; no momentum solution, constitutive law, or empirical material claim is asserted.

Statement refuted

For every smooth positive-density material motion satisfying the no-source mass balance, material constancy of density $D_{t} ρ = 0$ forces spatially uniform density $\nabla ρ = 0$ , and incompressibility $div u = 0$ forces irrotationality $curl u = 0$ .

Statement references

David Tong, Fluid Mechanics, Part I, §§1.1.1–1.1.3 (pathlines, material derivative, and mass conservation)

Andrew Hazel, MATH45061 Continuum Mechanics, Chapter 4, §4.1 (mass conservation)

Facts & Assumptions

Given: A smooth no-source material motion, a positive density transported by the adopted mass balance, and the kinematic meanings of $D_{t} ρ$ , divergence, and curl.

[F1]

Classical frames, primitive quantities and SI units: $D_{t}$ is the material derivative, $div u$ is the velocity divergence, and $curl u = (\partial_{y} u_{z} - \partial_{z} u_{y}, \partial_{z} u_{x} - \partial_{x} u_{z}, \partial_{x} u_{y} - \partial_{y} u_{x})$ is the vorticity vector.

[F2]

Mass density reconstruction and incompressibility: for a smooth positive-density material motion under the adopted mass balance, $\partial_{t} ρ + div (ρ u) = 0$ , $D_{t} ρ = 0$ is equivalent to $div u = 0$ , and $ρ (t, χ (t, a)) J (t, a) = ρ_{0} (a)$ . Its complete conditional proof is authored; the theorem and its transport supplier remain draft dependencies, and authoring readiness is not proof acceptance.

Counterexample

Given: Let $I = R$ , choose $R > 0$ metres, $ρ_{*} > 0$ kg m $^{- 3}$ , $0 < ε < 1$ , and a nonzero shear rate $γ$ in s $^{- 1}$ . Work on the open slab $O = {(x, y, z) \in R^{3} : ∣ y ∣ < 2 R}$ and the bounded reference ball $B_{R} = {(X, Y, Z) : X^{2} + Y^{2} + Z^{2} < R^{2}}$ . Define $χ_{t} (X, Y, Z) = (X + γ t Y, Y, Z)$ , $u (t, x, y, z) = (γ y, 0, 0)$ , and $ρ_{0} (X, Y, Z) = ρ_{*} (1 + ε Y / (2 R))$ .

1.1givenF1algebra

The motion $χ_{t}$ is a smooth diffeomorphism on $O$ with $F = (\begin{array}{ccc} 1 & γ t & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array})$ and $J = 1$ ; it preserves the slab because it leaves $y$ unchanged. Its Eulerian velocity is $u (t, χ_{t} (X, Y, Z)) = \partial_{t} χ_{t} (X, Y, Z) = (γ Y, 0, 0)$ , which agrees with the stated field since $y = Y$ . The initial density is positive on $B_{R}$ because $∣ Y ∣ < R$ and $ε < 1$ .

2.1F1F2givenstep 1.1algebra

The current field $ρ (t, x, y, z) = ρ_{*} (1 + ε y / (2 R))$ is positive throughout $O$ , nonconstant in $y$ , and satisfies $ρ (t, χ_{t} (X, Y, Z)) J = ρ_{0} (X, Y, Z)$ . Direct differentiation gives $\partial_{t} ρ = 0$ , $div u = \partial_{x} (γ y) = 0$ , and $u \cdot \nabla ρ = 0$ , hence $D_{t} ρ = 0$ by [F1] and the local continuity equation. The material mass balance follows from [F2].

3.1F1givenstep 2.1algebra∎

The same field has $\partial_{y} ρ = ρ_{*} ε / (2 R) \neq 0$ and $curl u = (0, 0, - γ) \neq 0$ by the curl convention [F1]. It is thus an incompressible, mass-conserving motion with nonuniform density and nonzero vorticity, refuting both asserted implications. The example is purely kinematic and is not claimed to solve a momentum equation.

Incompressibility does not imply uniform density

Statement refuted

Statement references

Facts & Assumptions

Counterexample

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