ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck pendingverified 2026-10-23 (gpt-6.1-sol)

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Plane and pipe Couette–Poiseuille verification — worked interpretation

Physical scope

Constructed SI parameter substitution into the exact infinite-geometry profiles; illustrative model inputs, not measurements.

Example

In the exact channel and fixed-wall pipe models, choose $h = 2 m$ , $a = 1 m$ , $ρ_{0} = 1000 k g m^{- 3}$ , $μ = 2 P a s$ , $G = 12 P a m^{- 1}$ , and for the channel $V_{0} = 0, V_{1} = 1 m s^{- 1}$ . Define dimensionless coordinates $\overset{ˉ}{y} = y / (1 m)$ and $\overset{ˉ}{r} = r / (1 m)$ . The model profiles and observables are $v (y) = (\frac{1}{2} \overset{ˉ}{y} + 3 \overset{ˉ}{y} (2 - \overset{ˉ}{y})) m s^{- 1}, Q_{c h} = 5 m^{2} s^{- 1}, (σ_{x y} (0), σ_{x y} (h)) = (13, - 11) P a;$ $w (r) = \frac{3}{2} (1 - {\overset{ˉ}{r}}^{2}) m s^{- 1}, Q = \frac{3 π}{4} m^{3} s^{- 1}, σ_{r x} (a) = - 6 P a .$ These are calculated values for the selected models, not experimental data.

Example references

Stewart, MIT 8.09, Fluid Mechanics, §6.7, printed pp. 130–133

Fitzpatrick, Fluid Mechanics, UT Austin, parallel plates

Fitzpatrick, Fluid Mechanics, UT Austin, pipe flow

Facts & Assumptions

Given: The SI parameters and exact channel/pipe formulas stated above.

[F1]

Plane and pipe Couette–Poiseuille verification — conditional physical prediction: channel flux and wall shear and fixed-pipe flux and wall shear follow from the exact stated profiles in the infinite one-dimensional model.

Proof

1.1givenF1algebra

With $\overset{ˉ}{y} = y / (1 m)$ , substitution gives $v (y) = (\overset{ˉ}{y} / 2 + 3 \overset{ˉ}{y} (2 - \overset{ˉ}{y})) m s^{- 1}$ . Integrating the profile over $0 \leq y \leq 2 m$ gives $Q_{c h} = 5 m^{2} s^{- 1}$ .

2.1givenF1step 1.1algebra∎

The channel shear formula gives $2 (1 / 2 + 6) = 13$ Pa at $y = 0$ and $2 (1 / 2 - 6) = - 11$ Pa at $y = 2$ . For the pipe, with $\overset{ˉ}{r} = r / (1 m)$ , $w = (3 / 2) (1 - {\overset{ˉ}{r}}^{2}) m s^{- 1}$ , $Q = 3 π / 4 m^{3} s^{- 1}$ , and $- G a / 2 = - 6 P a$ . These are model calculations only.

Depends on

Plane and pipe Couette–Poiseuille verification — conditional physical prediction