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Exterior sphere Stokes solution, force and uniqueness — formal worked field

Example

Assume the indexed axiom of countable choice ${A C}_{ω}$ : for every sequence $(X_{n})_{n \in N}$ of nonempty sets there is a function $f$ on $N$ with $f (n) \in X_{n}$ for every $n$ . Take $a = μ = ρ_{0} = 1$ , $p_{\infty} = 0$ , and $U = e_{z} = (0, 0, 1)$ on $r > 1$ . With $s = z$ , the exact field and pressure are

$u = A (r) e_{z} + B (r) z x, A (r) = 1 - \frac{3}{4 r} - \frac{1}{4 r^{3}}, B (r) = - \frac{3}{4 r^{3}} + \frac{3}{4 r^{5}}, p = - \frac{3 z}{2 r^{3}} .$

It has zero velocity on the unit sphere, tends uniformly to $e_{z}$ , and its formal stress resultant with $n = x$ is $6 π e_{z}$ . The theorem's uniqueness conclusion applies among fields in its stated uniform decay class.

Example references

Exterior sphere Stokes solution, force and decay-class uniqueness

Facts & Assumptions

Given: Assume ${A C}_{ω}$ as formulated in the Example, and work on the exterior $r > 1$ with constant parameters and far-field/boundary data in [A1]. This supplies the axiom used in the supplier's bounded-annulus proof.

[A1]

Assume ${A C}_{ω}$ as formulated in the Example. Set $a = μ = ρ_{0} = 1$ , $ν = 1$ , $p_{\infty} = 0$ , $U = e_{z}$ , with no-slip trace $u = 0$ at $r = 1$ and uniform limits $u \to e_{z}$ , $p \to 0$ .

[F1]

Exterior sphere Stokes solution, force and decay-class uniqueness: for arbitrary $a, μ, ρ_{0} > 0$ and $U$ , the stated explicit exterior Stokes field has zero sphere trace, constant far field, formal resultant $6 π μ a U$ , and is unique in the explicitly stated decay class, assuming ${A C}_{ω}$ for the annular divergence theorem.

Verification

1.1givenA1F1algebra

Under [A1], $s = U \cdot x = z$ and the general coefficients in [F1] reduce to the displayed $A, B, p$ . At $r = 1$ , $A = 1 - 3 / 4 - 1 / 4 = 0$ and $B = - 3 / 4 + 3 / 4 = 0$ , so $u = 0$ . As $r \to \infty$ , $A \to 1$ and $B z x = O (r^{- 1})$ uniformly because $∣ z ∣ ∣ x ∣ \leq r^{2}$ and $B = O (r^{- 3})$ ; hence $u \to e_{z}$ .

2.1givenA1F1step 1.1algebra∎

The constants meet the positivity and regularity hypotheses of [F1], and [A1] supplies its ${A C}_{ω}$ hypothesis. Its exact resultant specializes to $6 π (1) (1) e_{z} = 6 π e_{z}$ , and its uniqueness statement applies to every competing classical solution with the same data and decay bounds. This is a Stokes-model conclusion, not a finite-Reynolds-number estimate.

Depends on

Exterior sphere Stokes solution, force and decay-class uniqueness