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Numerical checks for channel flows: $h = η = 1$ , $G = 12$ gives flux $1$ and centre speed $3 / 2$ , and $V = 2$ gives shear $2$

Example

Take $h = 1$ , $η = 1$ in the plane channel of Exact plane Poiseuille and Couette flows of the incompressible Newtonian model, with flux and ansatz uniqueness.

Poiseuille, $G = 12$ . The profile is $v (y) = 6 y (1 - y)$ ; its flux per unit width is $1$ and its maximal speed is $v (1 / 2) = 3 / 2$ .
Couette, $V = 2$ . The profile is $v (y) = 2 y$ ; the only nonzero entry of $D u$ is $\partial_{y} u_{1} = 2$ , so the antisymmetric part of the velocity gradient has magnitude $1$ in the relevant off-diagonal slots and the flow is not irrotational.

Facts & Assumptions

Given: The channel flows of Exact plane Poiseuille and Couette flows of the incompressible Newtonian model, with flux and ansatz uniqueness with $h = η = 1$ , $G = 12$ , $V = 2$ .

[L1]

Poiseuille: $v (y) = \frac{G y (h - y)}{2 η}$ has flux $\frac{G h^{3}}{12 η}$ (Exact plane Poiseuille and Couette flows of the incompressible Newtonian model, with flux and ansatz uniqueness).

[L2]

Couette: $v (y) = \frac{V y}{h}$ solves the equations with constant pressure, and for $V \neq 0$ the velocity gradient is not symmetric (Exact plane Poiseuille and Couette flows of the incompressible Newtonian model, with flux and ansatz uniqueness, Incompressible Newtonian and barotropic inviscid fluid constitutive models, Euclidean spatial calculus: gradient, divergence and material derivative, The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The vector space $M_{m \times n} (F) : = F^{m \times n}$ of $m$ by $n$ matrices over a field, with entrywise operations, Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes).

Verification

technique · direct substitution into the closed formulas

1.1givenL1algebra

With $h = η = 1$ and $G = 12$ : $v (y) = \frac{12 y (1 - y)}{2} = 6 y (1 - y)$ , so $v (1 / 2) = 6 \cdot \frac{1}{2} \cdot \frac{1}{2} = \frac{3}{2}$ ; the flux is $\frac{12 \cdot 1^{3}}{12 \cdot 1} = 1$ .

1.2givenL2algebra

With $h = 1$ and $V = 2$ : $v (y) = 2 y$ , so $\partial_{y} u_{1} = 2$ and all other first partial derivatives of $u$ vanish; the antisymmetric part of $D u$ is the matrix with entries $\pm 1$ in positions $(1, 2), (2, 1)$ and zeros elsewhere, nonzero.

2.1step 1.1step 1.2algebra∎

The two substitutions reproduce the flux and centre-speed numbers of the Poiseuille profile and the nonzero shear of the Couette profile; no boundary-value problem beyond the explicit profiles is solved.

Remarks

The numbers $h = η = 1$ and $G = 12$ , $V = 2$ are chosen to give exact arithmetic; they are not measurements and carry no empirical claim about any fluid.

Depends on

Dependency tree · 2 levels

Sources

K. Likharev, Essential Graduate Physics, Classical Mechanics (September 1, 2026 complete volume), Section 8.5 (the Couette and Poiseuille channel-flow examples and the discharge law)

Numerical checks for channel flows: h=η=1h=η=1, G=12G=12 gives flux 11 and centre speed 3/23/2, and V=2V=2 gives shear 22