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

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Exterior potential sphere and zero ideal force — formal worked field

Example

Choose $a = ρ_{0} = 1$ , $p_{\infty} = 0$ , and $U = e_{x} = (1, 0, 0)$ . For $r > 1$ let $ϕ = x (1 + 1 / (2 r^{3}))$ , $u = \nabla ϕ$ , and $p = (1 - ∣ u ∣^{2}) / 2$ . On the unit sphere, $u = (3 / 2) (e_{x} - n_{x} n)$ and $p = - 5 / 8 + (9 / 8) n_{x}^{2}$ . The pressure is unchanged by $n \mapsto - n$ , so the pressure-only formal resultant is zero.

Example references

Exterior potential sphere and zero ideal force

Facts & Assumptions

Given: Use the exterior unit sphere, potential flow, constant density and pressure definition in [A1]. The boundary condition is slip/no-through.

[A1]

$a = ρ_{0} = 1$ , $p_{\infty} = 0$ , $U = e_{x}$ , $ϕ = x (1 + 1 / (2 r^{3}))$ , $u = \nabla ϕ$ , $p = (1 - ∣ u ∣^{2}) / 2$ , and $n = x$ on $r = 1$ .

[F1]

Exterior potential sphere and zero ideal force: this potential produces a smooth steady formal incompressible Euler field with $u \cdot n = 0$ ; the sphere velocity is $(3 / 2) (U - (U \cdot n) n)$ , and the pressure-only formal resultant is zero by antipodal symmetry.

Verification

1.1givenA1F1algebra

The parameters satisfy the theorem's positivity assumptions. Substituting $a = 1$ , $U = e_{x}$ into its potential gives $ϕ = x (1 + 1 / (2 r^{3}))$ ; the displayed pressure is the theorem's pressure formula, and on $r = 1$ the velocity is $(3 / 2) (e_{x} - n_{x} n)$ , whose dot product with $n$ is $(3 / 2) (n_{x} - n_{x}) = 0$ .

2.1givenA1F1step 1.1algebra∎

On $r = 1$ , $∣ u ∣^{2} = (9 / 4) (1 - n_{x}^{2})$ , so $p = \frac{1}{2} (1 - \frac{9}{4} (1 - n_{x}^{2})) = - 5 / 8 + (9 / 8) n_{x}^{2}$ . This is even under $n \mapsto - n$ ; spherical area is invariant and $n$ changes sign, hence $\int_{∣ x ∣ = 1} p n d S = 0$ . Therefore $\int σ_{E} n d S = - \int p n d S = 0$ , as claimed.

Depends on

Exterior potential sphere and zero ideal force