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Exterior potential sphere and zero ideal force

Statement

Let $a, ρ_{0} > 0$ , $p_{\infty} \in R$ , $U \in R^{3}$ , and on $Ω_{a} = {x \in R^{3} : r = ∣ x ∣ > a}$ define $n = x / r$ , $ϕ (x) = (U \cdot x) (1 + a^{3} / (2 r^{3}))$ , $u = \nabla ϕ$ , and $p = p_{\infty} + ρ_{0} (∣ U ∣^{2} - ∣ u ∣^{2}) / 2$ . Then $u, p$ are smooth in the exterior and up to $r = a$ , $u \to U$ and $p \to p_{\infty}$ uniformly in direction at infinity, and $u \cdot n = 0$ on $r = a$ . The pair solves $(u \cdot \nabla) u + ρ_{0}^{- 1} \nabla p = 0$ , $div u = 0$ , the steady formal constant-density incompressible Euler system. Its boundary condition is impermeable slip/no-through: $u$ is generally nonzero tangent to the sphere, so this is not a no-slip solution. For the formal Euler stress $σ_{E} = - p I_{3}$ and sphere normal $n = x / a$ pointing from the center into the exterior, the exact resultant is

$\int_{∣ x ∣ = a} σ_{E} n d S = - \int_{∣ x ∣ = a} p n d S = 0.$

The theorem concerns only this explicit inviscid mathematical model; it asserts neither a viscous zero-drag result nor an observed force. Under SI interpretation, $a, r, x$ are in metres, $U, u$ in metres per second, $ρ_{0}$ in kilograms per cubic metre, and $p, p_{\infty}$ in pascals.

Statement references

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

David Tong, Fluid Mechanics lecture notes, Chapter 1, §§2.3.2–2.3.3

Formal constant-density incompressible Euler system and pressure-stress definition: Formal constant-density incompressible Euler system.

Facts & Assumptions

Given: Use the constants, exterior domain, potential, velocity and pressure in the Statement. For the surface calculation, extend $ϕ, u, p$ by the same displayed formulas to the open set $D = {x \in R^{3} : ∣ x ∣ > 0}$ , as time-independent fields on $R \times D$ , and evaluate their spatial sections at a fixed time. All equations are formal pointwise model equations; the surface resultant uses the stated orientation.

[F1]

Formal parameterized fluid systems and observables: $\nabla$ and $Δ$ are Cartesian operators, and smooth scalar fields have commuting mixed partials.

[F2]

Formal constant-density incompressible Euler system: the steady inviscid equations are $(u \cdot \nabla) u + ρ_{0}^{- 1} \nabla p = 0$ , $div u = 0$ , and the formal stress is $σ_{E} = - p I_{3}$ . On a compact oriented regular surface patch $S \subset Ω$ in the open spatial domain of the fields, its resultant is $\int_{S} σ_{E} n_{S} d S$ ; here the fields used for this integral are stationary spatial sections.

[F3]

Barotropic Kelvin and Bernoulli with exact hypotheses: for every $C^{2}$ vector field, $(u \cdot \nabla) u = \nabla (∣ u ∣^{2} / 2) - u \times curl u$ . This is the exact algebraic clause of that theorem used here; no barotropic Bernoulli conclusion is invoked.

[F4]

The first fundamental form, Gram matrix, and area density of a surface patch: the surface density is the nonnegative square root of the determinant of the Gram matrix of the parameter tangents.

[F5]

Surface area and scalar surface integrals on a regular patch: a continuous scalar field is integrated against that density on the parameter region; area is the integral of the density, and changing values on a content-zero parameter boundary does not change the integral. Vector integrals are evaluated componentwise.

Proof

1.1givenF1algebra

For $r > 0$ , $\partial_{i} (1 / r) = - x_{i} / r^{3}$ and $\partial_{i}^{2} (1 / r) = - 1 / r^{3} + 3 x_{i}^{2} / r^{5}$ for $i = 1, 2, 3$ , so $Δ (1 / r) = - 3 / r^{3} + 3 r^{2} / r^{5} = 0$ . Since $(U \cdot x) / r^{3} = - U \cdot \nabla (1 / r)$ , commuting smooth derivatives gives $Δ ((U \cdot x) / r^{3}) = 0$ . Also $Δ (U \cdot x) = 0$ , so $Δ ϕ = 0$ . Therefore $div u = Δ ϕ = 0$ and $curl u = 0$ by equality of the mixed second partials of $ϕ$ . Direct differentiation yields $u = U + \frac{a^{3}}{2 r^{3}} (U - 3 (U \cdot n) n)$ . This gives uniform $u \to U$ at infinity. At $r = a$ , $u = (3 / 2) (U - (U \cdot n) n)$ , which is tangent to the sphere and has zero normal component.

2.1givenF2F3step 1.1algebra

By [F3] and $curl u = 0$ , $(u \cdot \nabla) u = \nabla (∣ u ∣^{2} / 2)$ . The pressure definition gives $ρ_{0}^{- 1} \nabla p = - \nabla (∣ u ∣^{2} / 2)$ , so the steady momentum equation in [F2] holds; step 1.1 supplies its divergence condition. The explicit formula gives $u - U = O (r^{- 1})$ , hence $∣ u ∣^{2} \to ∣ U ∣^{2}$ and $p \to p_{\infty}$ uniformly in direction.

3.1givenF2F4F5step 1.1constructalgebra∎

The displayed formulas extend smoothly to $D$ since their denominators are powers of $r > 0$ . Thus $S_{a} = {∣ x ∣ = a} \subset D$ , and [F2] applies to these stationary spatial fields on this sphere; their restrictions there equal the exterior traces by continuity. On the sphere, $u = (3 / 2) U_{t}$ with $U_{t} = U - (U \cdot n) n$ , so $∣ u ∣^{2} = (9 / 4) (∣ U ∣^{2} - (U \cdot n)^{2})$ . Thus $p ∣_{r = a}$ depends on $n$ only through $(U \cdot n)^{2}$ and is unchanged by $n \mapsto - n$ . For $Φ (θ, φ) = a (\sin θ \cos φ, \sin θ \sin φ, \cos θ)$ , $0 \leq θ \leq π$ , $0 \leq φ \leq 2 π$ , direct differentiation gives tangent Gram coefficients $E = a^{2}$ , $F = 0$ , $G = a^{2} \sin^{2} θ$ , so [F4] gives density $\sqrt{E G - F^{2}} = a^{2} \sin θ$ (also $∣ Φ_{θ} \times Φ_{φ} ∣$ ). The chart is regular in the parameter interior, and its poles and seam lie on the content-zero parameter boundary allowed in [F5]; those boundary values do not affect the integral. The substitution $(θ, φ) \mapsto (π - θ, φ + π m o d 2 π)$ has absolute Jacobian one on each half of the longitude interval, preserves the density because $\sin (π - θ) = \sin θ$ , and induces the antipodal map, which reverses $n$ , so the vector integrand $p n$ has zero integral over the sphere. By the stress convention in [F2], $\int_{∣ x ∣ = a} σ_{E} n d S = - \int_{∣ x ∣ = a} p n d S = 0$ .

Remarks

The diagram is a two-dimensional cross-section of the sphere. The radial arrow is the outward unit normal $n$ ; the tangent arrow indicates the direction of the boundary velocity $u = (3 / 2) U_{t}$ . The drawing is schematic and is not used in the field-equation or resultant proof.

diagram failed to render:
\begin{tikzpicture}[scale=1.25]
  \draw[thick] (0,0) circle (1);
  \fill (0,0) circle (0.025);
  \node[below left] at (0,0) {center};
  \fill (0.5,0.866) circle (0.035);
  \draw[->] (0,0) -- (0.5,0.866) node[midway,left] {$n$};
  \draw[->,thick] (0.5,0.866) -- (1.15,0.49) node[right] {$u=\tfrac32U_t$};
  \node at (0.72,0.98) {$r=a$};
  \draw[->] (2.0,0.0) -- (3.0,0.0) node[right] {$U$ at infinity};
\end{tikzpicture}

Exterior potential sphere and zero ideal force

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

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