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Exterior sphere Stokes solution, force and decay-class uniqueness — conditional physical prediction

Physical scope

Exterior r>a smooth Stokes problem, no-slip sphere, constant far field; explicit O(r^-1)/O(r^-2) velocity/gradient/pressure decay class; exact 6πμaU force. Worked example is the actual explicit field/parameter selection in that section. B leaf cannot supply any later item. Inherits ACω.

Statement

Assume ${A C}_{ω}$ . Let $a, μ, ρ_{0} > 0$ , $p_{\infty} \in R$ , $U \in R^{3}$ , $ν = μ / ρ_{0}$ , and $Ω_{a} = {x \in R^{3} : r = ∣ x ∣ > a}$ . Set $s = U \cdot x$ and $u = A (r) U + B (r) s x, A (r) = 1 - \frac{3 a}{4 r} - \frac{a^{3}}{4 r^{3}}, B (r) = - \frac{3 a}{4 r^{3}} + \frac{3 a^{3}}{4 r^{5}}, p = p_{\infty} - \frac{3 μ a}{2} \frac{s}{r^{3}},$ where the force-free, steady creeping model is selected. In Cartesian coordinates, $div u = \sum_{i} \partial_{i} u_{i}$ and $(Δ u)_{i} = \sum_{j} \partial_{j j} u_{i}$ . The field is $C^{2}$ up to the sphere from outside, $p \in C^{1}$ , $u = 0$ at $r = a$ , and $u \to U$ uniformly in direction at infinity. It solves $ρ_{0}^{- 1} \nabla p - ν Δ u = 0$ , $div u = 0$ . For $n = x / a$ on $S_{a}$ , with $σ = - p I + μ (\nabla u + (\nabla u)^{T})$ , the oriented resultant is $\int_{S_{a}} σ n d S = 6 π μ a U .$ Among classical pairs solving the same Stokes system, boundary condition and far-field limits, this pair is unique in the uniform decay class $u - U = O (r^{- 1})$ , $\nabla u = O (r^{- 2})$ , $p - p_{\infty} = O (r^{- 2})$ , with all bounds uniform in direction and pressure normalized by its limit. The classical regularity is $u \in C^{2}$ , $p \in C^{1}$ locally up to $S_{a}$ . Under SI units, $a, r, x$ are metres, $U$ is $m s^{- 1}$ , $ρ_{0}$ is $k g m^{- 3}$ , $μ$ is $P a s$ , $ν$ is $m^{2} s^{- 1}$ , pressure is pascals, and resultant is newtons. This is an exact zero-inertia model result; no error estimate comparing it with the full momentum equation retaining material acceleration, and no exterior Navier–Stokes existence claim, is asserted.

Statement references

Stewart, MIT 8.09, Fluid Mechanics, §6.8, printed pp. 135–139

Fitzpatrick, Fluid Mechanics, UT Austin, Stokes flow around a sphere

Mandre, Fluid Dynamics Lecture Notes, Chapter 9; translating-sphere convention reverses the velocity sign

Facts & Assumptions

Given: $ρ_{0}, μ, a > 0$ , $U, p_{\infty}$ , the exterior domain, exact fields, force-free boundary/far-field data and uniform decay class.

[A1]

Specified wall, thermal, periodic and initial data: no-slip and far-field data are separate model inputs; specifying them alone yields no existence result.

[A2]

Newtonian stress and Fourier conductive closure: in the incompressible constant-viscosity subcase, $σ = - p I + 2 μ D$ .

[F1]

Restricted incompressible creeping-flow Stokes model: the selected force-free model is $div u = 0$ and $div σ = 0$ with $σ = - p I + 2 μ D$ .

[F2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ asserts a choice function for every sequence of nonempty sets; the mathematical uniqueness proof assumes it.

[F3]

Exterior sphere Stokes solution, force and decay-class uniqueness: under ${A C}_{ω}$ , the exact exterior fields solve Stokes, have resultant $6 π μ a U$ , and are unique only in the stated uniform decay class.

Proof

1.1givenA1A2F1F3algebraconstruct

By [F3], the displayed fields are classical on the exterior, solve the force-free Stokes system [A2, F1], satisfy no-slip and the uniform far-field condition, and lie in the stated decay class. To compute the resultant, write $U_{t} = U - (U \cdot n) n$ on $S_{a}$ . From $u (r n) = A (r) U + B (r) r^{2} (U \cdot n) n$ , the values $A (a) = B (a) = 0$ , $A^{'} (a) = 3 / (2 a)$ and $B^{'} (a) = - 3 / (2 a^{3})$ give $u = 0$ and $\partial_{n} u = (3 / (2 a)) U_{t}$ at $r = a$ . The tangential derivatives of the zero boundary trace vanish, hence $\nabla u = (3 / (2 a)) U_{t} \otimes n$ there. Thus $(\nabla u)^{T} n = 0$ , $(\nabla u) n = (3 / (2 a)) U_{t}$ , and $p = p_{\infty} - (3 μ / (2 a)) (U \cdot n)$ on $S_{a}$ , so $σ n = - p_{\infty} n + (3 μ / (2 a)) U$ . Parameterize the sphere by $Φ (θ, φ) = a (\sin θ \cos φ, \sin θ \sin φ, \cos θ)$ ; its area density is $∣ Φ_{θ} \times Φ_{φ} ∣ = a^{2} \sin θ$ , so $Area (S_{a}) = \int_{0}^{2 π} \int_{0}^{π} a^{2} \sin θ d θ d φ = 4 π a^{2}$ . The antipodal parameter change $(θ, φ) \mapsto (π - θ, φ + π m o d 2 π)$ satisfies $Φ \mapsto - Φ$ , has absolute Jacobian one and preserves $\sin θ d θ d φ$ ; thus $\int_{S_{a}} n d S = 0$ . Integrating the stress expression therefore gives $\int_{S_{a}} σ n d S = 6 π μ a U$ .

2.1F2F3step 1.1

Uniqueness uses ${A C}_{ω}$ [F2] and is restricted to the stated regularity, pressure normalization and uniform decay bounds. It gives no uniqueness beyond this class.

3.1F1step 1.1step 2.1∎

The equation was adopted as a zero-inertia model [F1]; the force deduction is conditional and supplies no error estimate against the full material-acceleration equation or experimental measurement.

Exterior sphere Stokes solution, force and decay-class uniqueness — conditional physical prediction

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources