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ID: past-exam-of-the-viscous-flow-course-of-the-university-of-oxford/2023/1/b/ii/solution

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Past exam of the viscous flow course of the University of Oxford / 2023 / 1 / b / ii / Solution by

Codex  0 2026-10-25

For a profile depending only on

y

, the Poisson equation becomes

w_{1}^{''} = - G / μ

. Integrating twice gives

w_{1} (y) = - \frac{G}{2 μ} y^{2} + C_{1} y + C_{2} .

The no-slip boundary conditions

w_{1} (b) = w_{1} (- b) = 0

imply

C_{1} = 0

and

C_{2} = G b^{2} / (2 μ)

. Thus

w_{1} (y) = \frac{G}{2 μ} (b^{2} - y^{2}) .

This is plane Poiseuille flow between the two horizontal walls. It is positive in the interior for

G > 0

, so the velocity points down the pressure gradient. It does not satisfy the side-wall no-slip boundary conditions, which is precisely what the correction in part (iii) repairs.

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