ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-23 (gpt-6.1-sol)

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Plane and pipe Couette–Poiseuille verification — formal worked fields

Example

Set $h = 2$ , $V_{0} = 0$ , $V_{1} = 1$ , $G = 4$ , $μ = 2$ , $ρ_{0} = 1$ , and $p_{0} = 0$ in the plane-channel model. The profile is $v (y) = \frac{5}{2} y - y^{2}$ for $0 \leq y \leq 2$ , with flux $Q_{c h} = 7 / 3$ and fixed-coordinate wall shear components $σ_{x y} (0) = 5$ and $σ_{x y} (2) = - 3$ . In the pipe, set $a = 1$ , $G = 8$ , $μ = 2$ , $ρ_{0} = 1$ , and $p_{0} = 0$ ; the profile is $w (r) = 1 - r^{2}$ on the unit disk and its volume flux is $Q_{p i p e} = π / 2$ . These are exact infinite or fully developed formal solutions within the theorem's one-dimensional ansatz.

Example references

Plane and pipe Couette–Poiseuille verification

Facts & Assumptions

Given: Use the exact constant-density equations and the plane-channel and circular-pipe ansatz, domains, endpoint data, and formulas in [A1].

[A1]

Plane channel: $h = 2$ , $V_{0} = 0$ , $V_{1} = 1$ , $G = 4$ , $μ = 2$ , $ρ_{0} = 1$ , $p = p_{0} - G x$ with $p_{0} = 0$ . Pipe: $a = 1$ , $G = 8$ , $μ = 2$ , $ρ_{0} = 1$ , $w (1) = 0$ , and radial velocity field on the full disk.

[F1]

Plane and pipe Couette–Poiseuille verification: the channel profile is $V_{0} + (V_{1} - V_{0}) y / h + G y (h - y) / (2 μ)$ with the stated flux and shear formulas; the full-disk pipe profile is $G (a^{2} - r^{2}) / (4 μ)$ with flux $π G a^{4} / (8 μ)$ . Both fields solve the constant-density Navier–Stokes/Stokes equations exactly within their ansatz.

Verification

1.1givenA1F1algebra

Substituting the channel parameters into the formula in [F1] gives $v (y) = y / 2 + y (2 - y) = 5 y / 2 - y^{2}$ . It obeys $v (0) = 0$ , $v (2) = 1$ , and $μ v^{''} = 2 (- 2) = - 4 = - G$ . Integrating gives $\int_{0}^{2} v (y) d y = 5 - 8 / 3 = 7 / 3$ . Also $v^{'} (0) = 5 / 2$ and $v^{'} (2) = - 3 / 2$ , so $μ v^{'} (0) = 5$ and $μ v^{'} (2) = - 3$ .

2.1givenA1F1step 1.1algebra∎

Substituting the pipe parameters into [F1] gives $w (r) = 1 - r^{2}$ , which as a Cartesian radial field is $1 - y^{2} - z^{2}$ , smooth on the full disk and zero at $r = 1$ . Polar integration gives $2 π \int_{0}^{1} r (1 - r^{2}) d r = π / 2$ . The parameter values are positive where required, so the exact-solution and restricted-uniqueness conclusions of [F1] apply to both displayed fields.

Depends on

Plane and pipe Couette–Poiseuille verification