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Vortex topology and global-Bernoulli counterexamples

Statement

Let $D = {(x, y, z) \in R^{3} : r = (x^{2} + y^{2})^{1 / 2} > 0}$ , let $ρ_{0} > 0$ , $p_{0} \in R$ , and let $κ \neq 0$ . On $D$ define

$u_{v} (x, y, z) = (- \frac{κ y}{r^{2}}, \frac{κ x}{r^{2}}, 0), p_{v} (x, y, z) = p_{0} - \frac{ρ_{0} κ^{2}}{2 r^{2}} .$

For every $R > 0$ and fixed $z_{0}$ , the positively oriented circle $γ_{R} (θ) = (R \cos θ, R \sin θ, z_{0})$ , $0 \leq θ \leq 2 π$ , has circulation $\oint_{γ_{R}} u_{v} \cdot d r = 2 π κ$ . Consequently no single-valued $C^{1}$ function $ϕ : D \to R$ satisfies $u_{v} = \nabla ϕ$ on all of $D$ , even though $curl u_{v} = 0$ there. Both $div u_{v} = 0$ and $Δ u_{v} = 0$ on $D$ , and the pointwise identity

$\partial_{t} u_{v} + (u_{v} \cdot \nabla) u_{v} + ρ_{0}^{- 1} \nabla p_{v} - ν Δ u_{v} = 0$

holds for every $ν > 0$ , with $\partial_{t} u_{v} = 0$ .

Separately, for $Ω_{0} \neq 0$ set $u_{s} = (- Ω_{0} y, Ω_{0} x, 0)$ and $p_{s} = p_{0} + ρ_{0} Ω_{0}^{2} r^{2} / 2$ on $R^{3}$ . This smooth steady field is divergence-free and satisfies the same constant-density equation for every $ν > 0$ . For this steady field, a streamline is a $C^{1}$ curve $γ : J \to R^{3}$ , where $J \subseteq R$ is an interval, satisfying $γ^{'} (s) = u_{s} (γ (s))$ . Its Bernoulli quantity $B = ∣ u_{s} ∣^{2} / 2 + p_{s} / ρ_{0} = p_{0} / ρ_{0} + Ω_{0}^{2} r^{2}$ is constant along each streamline and takes different values at distinct radii. Thus a common Bernoulli constant is not implied for all steady constant-density flows. This example does not contradict a Bernoulli theorem that assumes an irrotational field or a separate barotropic model. Under SI interpretation, $r$ is in metres, the velocities in metres per second, $κ$ in square metres per second, $Ω_{0}$ in inverse seconds, $ρ_{0}$ in kilograms per cubic metre, and $p, p_{0}$ in pascals.

Statement references

Iain W. Stewart, MIT 8.09/8.309, Chapter 6, §6.4

David Tong, Fluid Mechanics lecture notes, Chapter 1, §§2.2.1 and 2.4.1

Facts & Assumptions

Given: The domain excludes the entire $z$ -axis in the vortex example; the constants and fields are those in the Statement. The equations are pointwise mathematical model equations.

[F1]

Formal parameterized fluid systems and observables: divergence, curl and componentwise Laplacian are defined by Cartesian partial derivatives; the constant-density Navier–Stokes equation is pointwise, with $ρ_{0}, μ > 0$ and $ν = μ / ρ_{0}$ .

[A1]

For $u_{v}$ , $r > 0$ and the pressure is the stated scalar multiplier. For $u_{s}$ , the spatial domain is all of $R^{3}$ and $Ω_{0} \neq 0$ .

Proof

1.1givenF1A1algebra

Put $r^{2} = x^{2} + y^{2}$ . Direct differentiation gives $div u_{v} = \partial_{x} (- κ y / r^{2}) + \partial_{y} (κ x / r^{2}) = 2 κ x y / r^{4} - 2 κ x y / r^{4} = 0$ . The only potentially nonzero curl component is $\partial_{x} (u_{v})_{y} - \partial_{y} (u_{v})_{x} = κ (y^{2} - x^{2}) / r^{4} + κ (x^{2} - y^{2}) / r^{4} = 0$ . For the Laplacian, $Δ (x / r^{2}) = - 4 x / r^{4} + 4 x (x^{2} + y^{2}) / r^{6} = 0$ , and interchanging $x, y$ gives $Δ (y / r^{2}) = 0$ ; hence $Δ u_{v} = 0$ for $r > 0$ . The operator $u_{v} \cdot \nabla = - κ y r^{- 2} \partial_{x} + κ x r^{- 2} \partial_{y}$ gives $(u_{v} \cdot \nabla) (u_{v})_{x} = - κ^{2} x / r^{4}$ and $(u_{v} \cdot \nabla) (u_{v})_{y} = - κ^{2} y / r^{4}$ . Thus $(u_{v} \cdot \nabla) u_{v} = - κ^{2} (x, y, 0) / r^{4}$ .

2.1A1step 1.1constructalgebra

Along $γ_{R}$ , $u_{v} (γ_{R} (θ)) \cdot γ_{R}^{'} (θ) = κ$ , so the circulation is $\int_{0}^{2 π} κ d θ = 2 π κ \neq 0$ . If $u_{v} = \nabla ϕ$ for a single-valued $C^{1}$ function on $D$ , the chain rule gives $\oint_{γ_{R}} \nabla ϕ \cdot d r = ϕ (γ_{R} (2 π)) - ϕ (γ_{R} (0)) = 0$ , contradicting the computed value.

2.2F1A1step 1.1algebra

Differentiation gives $\nabla p_{v} = ρ_{0} κ^{2} (x, y, 0) / r^{4}$ . This cancels $ρ_{0} (u_{v} \cdot \nabla) u_{v}$ ; together with $Δ u_{v} = 0$ and $\partial_{t} u_{v} = 0$ , the displayed Navier–Stokes expression vanishes for every $ν > 0$ .

3.1F1A1algebra∎

For the solid-rotation field, $div u_{s} = 0$ , $Δ u_{s} = 0$ , $(u_{s} \cdot \nabla) u_{s} = - Ω_{0}^{2} (x, y, 0)$ , and $\nabla p_{s} = ρ_{0} Ω_{0}^{2} (x, y, 0)$ , so its steady Navier–Stokes expression vanishes. Writing a streamline as $γ (s) = (x (s), y (s), z (s))$ , its defining equation gives $γ^{'} (s) = (- Ω_{0} y (s), Ω_{0} x (s), 0)$ and therefore $\frac{d}{d s} (x (s)^{2} + y (s)^{2}) = 2 x (- Ω_{0} y) + 2 y (Ω_{0} x) = 0$ . Thus $r$ is fixed along the streamline and $B = p_{0} / ρ_{0} + Ω_{0}^{2} r^{2}$ is constant there; because $Ω_{0} \neq 0$ , distinct radii give distinct values.

Remarks

The sketch shows only the punctured $(x, y)$ cross-section. The dot marks the excluded axis, the circle is positively oriented as viewed from $+ z$ , and the tangent arrows indicate the direction of $u_{v}$ when $κ > 0$ . It is schematic and is not used in the circulation or no-potential proof.

Depends on

Formal parameterized fluid systems and observables