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

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Exact Reynolds and Mach parameter substitution

Example

Choose the formal SI parameter values $ρ_{0} = 2 k g m^{- 3}$ , $μ = 2 P a s$ , $U = 2 m s^{- 1}$ , and $L = 3 m$ . For the barotropic branch take $P (ρ) = 16 ρ$ , with the coefficient carrying units so that $P^{'} (ρ_{0}) = 16 m^{2} s^{- 2}$ and $c = 4 m s^{- 1}$ . The kinematic viscosity and dimensionless groups are $ν = 1 m^{2} s^{- 1}$ , $Re = 6$ , $Ma = 1 / 2$ , and the transformed equation coefficients are ${Re}^{- 1} = 1 / 6$ and ${Ma}^{- 2} = 4$ . With $ρ = 2 ρ^{*}$ , the normalized pressure is $p^{*} = ρ^{*} - 1$ because $P (ρ_{0}) = 32 P a$ and $ρ_{0} c^{2} = 32 P a$ . These are formal parameter substitutions, not measurements or a claim that any selected fields solve the fluid equations.

Example references

Exact Reynolds and Mach scaling identities: Exact Reynolds and Mach nondimensionalization.

Facts & Assumptions

Given: The parameter assignments in the Example and the exact scaling result of thm-fd-x01.

[F1]

Under the positive scale substitutions, the incompressible coefficient is ${Re}^{- 1} = ν / (U L)$ and the barotropic momentum coefficient is ${Ma}^{- 2} = c^{2} / U^{2}$ (Exact Reynolds and Mach nondimensionalization).

Verification

1.1givenF1algebra

The assigned density and dynamic viscosity give $ν = μ / ρ_{0} = 2 / 2 = 1 m^{2} s^{- 1}$ . Hence $Re = U L / ν = (2) (3) / 1 = 6$ and ${Re}^{- 1} = 1 / 6$ , agreeing with [F1].

2.1givenF1algebra∎

Since $P^{'} (ρ_{0}) = 16 m^{2} s^{- 2}$ , its positive square root is $c = 4 m s^{- 1}$ . Thus $Ma = U / c = 2 / 4 = 1 / 2$ and ${Ma}^{- 2} = c^{2} / U^{2} = 16 / 4 = 4$ . Also $P (2) = 32 P a$ , $P (2 ρ^{*}) = 32 ρ^{*} P a$ , and $p^{*} = (P (2 ρ^{*}) - P (2)) / (ρ_{0} c^{2}) = ρ^{*} - 1$ . By [F1] these are the exact transformed coefficients; the arithmetic adds no limiting or empirical assertion.

Depends on

Exact Reynolds and Mach nondimensionalization