ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passverified 2026-10-23 (gpt-6.1-sol)

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Couette shear vorticity and volume flux

Example

Let $L_{1}, L_{2}, h > 0$ and $V \in R$ on the periodic flat channel $Ω = (R / L_{1} Z) \times (0, h) \times (R / L_{2} Z)$ . For the steady incompressible equations $ρ (u \cdot \nabla) u = - \nabla p + η Δ u$ with $ρ, η > 0$ , prescribe no body force, constant pressure, stationary lower wall, and top-wall velocity $(V, 0, 0)$ . The field $u = (V y / h, 0, 0)$ is a solution, its vorticity is $ω = \nabla \times u = (0, 0, - V / h)$ , and its volume flux per unit transverse width, defined for a shear $u = (U (y), 0, 0)$ by $Q = \int_{0}^{h} U (y) d y$ , is $V h / 2$ . Its vorticity is nonzero exactly when $V \neq 0$ . The velocity is unique among solutions $\tilde{u} : \overline{Ω} \to R^{3}$ of the form $\tilde{u} = (\tilde{U} (y), 0, 0)$ with $\tilde{U} \in C^{2} ([0, h]; R)$ , spatially constant scalar pressure $\tilde{p} : \overline{Ω} \to R$ , and wall values $\tilde{U} (0) = 0$ , $\tilde{U} (h) = V$ ; the pressure constant is arbitrary. With SI units, $x, y, z, L_{1}, L_{2}, h$ are metres, $V$ and $u$ are m/s, $ρ$ is kg/m $^{3}$ , $η$ is Pa $\cdot$ s, pressure is Pa, vorticity is s $^{- 1}$ , and flux per unit width is m $^{2}$ /s.

Example references

Flat periodic and infinite channel domains

David Tong, Fluid Mechanics, Chapter 2, §§3.2.1–3.2.3

Verification

Given: The periodic channel, steady momentum equation, and wall velocities in the Example.

[F1] The channel has periodic coordinates $x, z$ and boundary faces $y = 0, h$ ; for the stated $x$ -directed shear, $x$ is streamwise and $z$ is transverse; a prescribed wall velocity is its trace on those faces (Flat periodic and infinite channel domains).

[A1] The steady incompressible momentum equation and zero forcing stated in the Example are the system being checked.

1.1givenF1A1algebra

For $U (y) = V y / h$ , the traces are $U (0) = 0$ , $U (h) = V$ , and $U^{''} = 0$ . The field $u = U e_{x}$ has zero divergence and zero convective term because $u \cdot \nabla u = U \partial_{x} u = 0$ . With constant pressure, $- \nabla p + η Δ u = 0$ , so the equation [A1] holds.

2.1step 1.1A1algebra∎

By the coordinate formula for curl, $\nabla \times (U (y), 0, 0) = (0, 0, - U^{'} (y)) = (0, 0, - V / h)$ . The flux is $\int_{0}^{h} V y / h d y = V h / 2$ . For any competing field in the class stated in the Example, constant pressure and zero convection reduce the streamwise equation to $η {\tilde{U}}^{''} = 0$ . Thus $\tilde{U} - U$ has zero second derivative and zero endpoint values, hence is the zero affine function. Since $h > 0$ , the computed vorticity vanishes exactly when $V = 0$ .

Couette shear vorticity and volume flux

Example

Example references

Verification

Depends on

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Sources