 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ii/paper-1/36a/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

The signed tangential stress component is

σ_{x y} = μ u^{'} = C - G y

. Thus

σ_{x y} (- a) = \frac{G a ( μ _{+} + 3 μ _{-} )}{2 ( μ _{+} + μ _{-} )}, σ_{x y} (a) = - \frac{G a ( 3 μ _{+} + μ _{-} )}{2 ( μ _{+} + μ _{-} )} .

When

μ_{+} / μ_{-} \to 1

, these approach

+ G a

and

- G a

, the usual symmetric plane Poiseuille flow stresses. When

μ_{+} / μ_{-} \to \infty

, they approach

+ G a /2

and

- 3 G a /2

. The upper fluid's velocity tends to zero, but its viscosity times velocity gradient has a finite limit. These are tensor components; traction on a chosen wall normal includes that normal's sign.

 New to topics? Read the docs here!