ns/exterior — the heat exterior

Beyond the annulus the flow is purely azimuthal and independent of height: fluid circles the axis, pressure supplies the centripetal force, and the swirl evolves by viscous diffusion alone. The paper chooses it to solve the radial heat equation for the azimuthal velocity exactly, so its momentum residual is identically zero and no force is needed there. This computes that solution from the paper's integral formula and checks the equation numerically at every point drawn.

exponent h

highlight τ

K(r,τ) = c_∞·(r²/2)^−A·H(4τ/r²), H(Z) = Γ(1+h)⁻¹∫₀^∞ e^−vv^h(1+Zv)^−h dv, −∂_τK = ∂_r²K + r⁻¹∂_rK − r⁻²K (4.29), (3.5), (10.7)

Don't take the formula's word for it. Start a plain finite-difference solver from the exact profile at τ = 1, march the swirl heat equation forward in time on an annulus with the exact values as boundary data, and compare with the formula at a later time. The error should shrink like the grid spacing squared — and it does.

march from τ = 1 down to

grid points on r ∈ [0.4, 8]

Explicit Euler, dt = 0.4·dr², second-order centred differences. Nothing clever — that is the point.

exact K(r, τ₁) marched solution initial profile K(r, 1)

The exterior formula is only used where X = r²/2q ≥ X_b, outside the pulse annulus — and that region swallows everything but the origin as τ → 0. Fix any radius r₀ > 0: once τ < r₀²/2X_b the point (r₀, z = 0) is in the exterior, where the velocity and pressure have smooth limits, with every derivative, at τ = 0 (10.8). That is what makes it possible to cut the flow off in space and time without touching the core (§3.5).

annulus radii (illustrative)

X_a

X_b

probe radius r₀

core and annulus (X < X_b) at z = 0 exterior X ≥ X_b: exact heat swirl, zero residual the probe r₀ and the time it enters the exterior

§2.3 · §3.5 · Theorem 3.1(iii) · Theorem 4.6(v) · Lemmas A.6, 10.2 · (10.7)–(10.8) of the paper

An exact solution as the outer boundary condition

For a purely azimuthal, z-independent velocity u = K(r,t)e_θ, the Navier–Stokes equations reduce to two statements: the radial equation ∂_rp = K²/r, which just defines the pressure, and the azimuthal equation ∂_tK = ∂_r²K + r⁻¹∂_rK − r⁻²K, the heat equation for the first angular harmonic. The nonlinear term drops out entirely — (u·∇)u is pure centripetal acceleration. So any solution of this linear equation is an exact solution of Navier–Stokes with zero force. The paper's choice is

K(r,τ) = c_∞·s^−A·H(2τ/s), s = r²/2, A = ½ + h, H(Z) = Γ(1+h)⁻¹∫₀^∞ e^−vv^h(1+Zv)^−h dv

with τ = 1 − t. In profile variables this is E = c_∞X^−AH(2d/X) for X ≥ X_b, where d = 1 − η² (Theorem 4.6(v)); Lemma A.6 verifies the heat equation and that H is positive and smooth for Z ≥ 0 with K_r < 0. Its two limits are worth knowing:

τ → 0 at fixed r: H(0) = 1, so K → c_∞(r²/2)^−A = c_∞2^Ar^−1−2h. At the singular time the exterior is a pure power-law swirl, and every τ-derivative is bounded by C_jr^{−1−2h−2j} (10.8), uniformly in τ ≥ 0. Nothing singular happens at any positive radius.
r → 0 at fixed τ: Z = 4τ/r² → ∞ and H(Z) ~ Z^−h/Γ(1+h), so K ~ τ^−h·r⁻¹·const: a potential vortex with slowly growing circulation. The formula is never used there — inside X_b the annulus and core take over — but it shows the exterior is the far field of a vortex whose circulation at the joining radius grows like τ^−h, matching Re_θ ≍ τ^−h in the core.

Why the paper needs it

Joining the collapsing core to any smooth outer swirl leaves a residual in between; the paper arranges for that residual to be a compactly supported stress divergence (see stress), which needs the outer flow to be exactly balanced so nothing leaks past the annulus. The five cumulative radial integrals of (4.15) — the angular moment made convergent by subtracting the exterior power law H_pow = √(2X)c_∞X^−A — are matched at every join precisely so that the prescribed exterior fields are preserved (Lemma 4.4, Lemma A.8). Replacing a plain power-law tail by the exact heat flow changes the pressure integral, and Proposition A.7 restores it at smaller radii.

The second role is localisation. Theorem 3.1(ii) gives bounded derivatives on compact sets where q stays positive; but in the remaining region — r bounded below, q → 0 — the vector potential vanishes and the velocity is exactly Ke_θ, with the pressure −∫_r^∞K²/ρ dρ. Dominated convergence on the integral (the majorant ρ^{−3−4h−2j} is integrable) gives uniform one-sided limits of every derivative there (Lemma 10.2). Together with the flatness of the residual near the origin, that is what makes the cut-off force converge, with all its derivatives, as t ↑ 1 — and what Lemma 10.3 then extends smoothly past t = 1.

What the canvases compute

the swirl — H(Z) by Simpson's rule after the substitution v = u², which removes the v^h endpoint singularity (without it the quadrature is only 1% accurate; H(0) should be exactly 1 and is). The PDE residual is a centred finite difference in r and τ at eleven sample points, reported relative to |∂_r²K|; the remaining 10⁻⁴ is the finite difference, not the formula.
march it — explicit Euler on the swirl heat equation with the exact solution as Dirichlet data at r = 0.4 and r = 8, compared at τ₁. Doubling the grid divides the error by about four.
where it applies — the regions at z = 0 in the (r, τ) half-plane, with q = τ there, and the pressure integral evaluated by quadrature with its ρ^−3−4h tail added analytically.