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ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck pendingverified 2026-10-23 (gpt-6.1-sol)
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Exterior sphere Stokes solution, force and decay-class uniqueness — worked interpretation

Physical scope

Constructed substitution into the exact zero-inertia Stokes sphere result, conditional on its uniform decay class and ACω; no comparison error bound or measurement is asserted.

Example

Assume ACω and use the exact uniform-decay Stokes model for a sphere of radius a=1 m with μ=1 Pa s, ρ0=1 kg m−3, and uniform far-field velocity U=(10−3,0,0) m s−1. The exact model resultant with sphere normal directed into the fluid is 6π⋅10−3ex N. Choose characteristic speed Uref=∣U∣ and length Lref=a, and define Re⁡a=ρ0UrefLref/μ=10−3. This is only a parameter calculation and supplies no error estimate comparing this zero-inertia model with the full equation retaining material acceleration.

Facts & Assumptions

Given: ACω, a=μ=ρ0=1 in the stated SI units, U=(10−3,0,0), and the exact uniform-decay class.

[F1]

Exterior sphere Stokes solution, force and decay-class uniqueness — conditional physical prediction: under ACω, the fixed-sphere Stokes resultant is 6πμaU in the stated orientation and decay class.

[F2]

The Axiom of Countable Choice (ACω): ACω asserts that for every sequence (Xn)n∈N of nonempty sets there is a function f with domain N such that f(n)∈Xn for every n. It is assumed here, as in [F1].

Proof

1.1givenF1algebra

Substituting μ=a=1 and U=10−3ex m s−1 into 6πμaU gives 6π×10−3ex N.

2.1givenF1F2step 1.1algebra∎

The dimensionless ratio is (1)(10−3)(1)/(1)=10−3. It does not change the Stokes-model hypotheses or create an error bound for Navier–Stokes.

Depends on

Dependency tree · 2 levels

Sources