Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck pendingverified 2026-10-23 (gpt-6.1-sol)

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Vortex topology and global-Bernoulli counterexamples — conditional physical prediction

Physical scope

Punctured-axis smooth vortex and solid rotation; prescribed pressures; no global scalar potential or common Bernoulli constant without hypotheses. Worked example is the actual explicit field/parameter selection in that section. B leaf cannot supply any later item. Constant density with spatially varying constraint pressure, not barotropic EOS pressure.

Statement

Let $ρ_{0} > 0$ be constant density in $k g m^{- 3}$ , let $p_{\infty}, p_{0} \in R$ be pressure constants in pascals, let the vortex circulation parameter $κ \neq 0$ be in $m^{2} s^{- 1}$ , and let the angular speed $Ω \neq 0$ be in $s^{- 1}$ . Spatial Cartesian coordinates $x, y, z$ are measured in metres and velocities in $m s^{- 1}$ . Define $e_{r} = (x / r, y / r, 0)$ and $e_{θ} = (- y / r, x / r, 0)$ . In Cartesian coordinates, $div u = \partial_{x} u_{x} + \partial_{y} u_{y} + \partial_{z} u_{z}$ and $curl u = (\partial_{y} u_{z} - \partial_{z} u_{y}, \partial_{z} u_{x} - \partial_{x} u_{z}, \partial_{x} u_{y} - \partial_{y} u_{x})$ . On the punctured region $r = (x^{2} + y^{2})^{1 / 2} > 0$ , set $e_{θ} = (- y / r, x / r, 0)$ , $u = κ r^{- 1} e_{θ}$ , and $p = p_{\infty} - ρ_{0} κ^{2} / (2 r^{2})$ . This steady field solves the selected unforced incompressible Euler model, is divergence-free and curl-free on its domain, and has circulation $2 π κ$ around every positively oriented circle about the axis. A differentiable global potential means a single-valued $C^{1}$ function $ϕ$ on the domain with $u = \nabla ϕ$ ; none exists.

On a connected open region, the steady solid-rotation field $u = Ω (- y, x, 0)$ , $p = p_{0} + ρ_{0} Ω^{2} (x^{2} + y^{2}) / 2$ also solves that model. A streamline here is a $C^{1}$ curve $γ : J \to O$ , where $J \subseteq R$ is an interval and $O \subseteq R^{3}$ is the selected connected open spatial region, satisfying $γ^{'} (s) = u (γ (s))$ . Its Bernoulli function $B = ∣ u ∣^{2} / 2 + p / ρ_{0} = p_{0} / ρ_{0} + Ω^{2} r^{2}$ is constant along each streamline but is not one constant across streamlines at different radii. These are exact conditional model fields, not viscous-flow or empirical claims.

Statement references

Stewart, MIT 8.09, Fluid Mechanics, §6.4, printed pp. 120–123

Fitzpatrick, Fluid Mechanics, UT Austin, Bernoulli and irrotational flow

Facts & Assumptions

Given: $ρ_{0} > 0$ , $κ \neq 0$ , $Ω \neq 0$ , the stated domains and steady fields.

[A1]

Single-component nonpolar continuum and material balances: continuum balances are adopted assumptions, and the calculations are conditional on the selected inviscid branch.

[A2]

Specified wall, thermal, periodic and initial data: no wall data are used for these local fields; data for finite devices are separate.

[F1]

Incompressible inviscid Euler model with pressure multiplier: steady fields satisfy $div u = 0$ and $ρ_{0} (u \cdot \nabla) u + \nabla p = 0$ .

[F2]

Vortex topology and global-Bernoulli counterexamples: the vortex has nonzero circulation despite zero curl, and solid rotation has streamline-wise but non-global Bernoulli values.

[F3]

Formal parameterized fluid systems and observables: for a closed piecewise- $C^{1}$ oriented curve $γ : [a, b] \to Ω$ , circulation is the line integral $\oint_{γ} u \cdot d r = \int_{a}^{b} u (γ (s)) \cdot γ^{'} (s) d s$ . The theorem's circular value uses $γ_{R} (θ) = (R \cos θ, R \sin θ, z_{0})$ with $0 \leq θ \leq 2 π$ and positive orientation about the axis.

Proof

1.1givenA1F1algebra

For $u_{v} = (- κ y / r^{2}, κ x / r^{2}, 0)$ , the Cartesian formulas in the Statement give $div u_{v} = 2 κ x y / r^{4} - 2 κ x y / r^{4} = 0$ and $(curl u_{v})_{z} = \partial_{x} (κ x / r^{2}) - \partial_{y} (- κ y / r^{2}) = κ (y^{2} - x^{2}) / r^{4} - κ (y^{2} - x^{2}) / r^{4} = 0$ ; the first two curl components vanish because $u_{v, z} = 0$ and the field is independent of $z$ . Direct differentiation also gives $(u_{v} \cdot \nabla) u_{v} = - κ^{2} (x, y, 0) / r^{4}$ and $\nabla p = ρ_{0} κ^{2} (x, y, 0) / r^{4}$ , so the steady Euler equation holds for $r > 0$ .

1.2givenF1F2algebra

For solid rotation, div $u = 0$ , acceleration is $- Ω^{2} (x, y, 0)$ , and $\nabla p = ρ_{0} Ω^{2} (x, y, 0)$ . Thus Euler holds and $B = p_{0} / ρ_{0} + Ω^{2} r^{2}$ . For a streamline $γ (s) = (x (s), y (s), z (s))$ , its defining ODE gives $\frac{d}{d s} (x (s)^{2} + y (s)^{2}) = 2 x (- Ω y) + 2 y (Ω x) = 0$ . Hence its radius and Bernoulli value are constant along the curve, and the values differ across distinct radii because $Ω \neq 0$ .

2.1givenF2F3step 1.1algebra

With the positively oriented circle $γ_{R}$ and circulation convention in [F3], $u (γ_{R} (θ)) \cdot γ_{R}^{'} (θ) = κ$ , so $Γ = \int_{0}^{2 π} κ d θ = 2 π κ$ . A gradient has zero integral on a closed curve by the chain rule, so this nonzero integral precludes a single-valued global potential.

3.1A1A2F1step 2.1step 1.2∎

These are conditional deductions inside the adopted model; no wall data are used, and they do not establish viscous behaviour or empirical validity.

Vortex topology and global-Bernoulli counterexamples — conditional physical prediction

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources