 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-2/37a/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Write

d_{+} = σ + h

and

d_{-} = h - σ

for the lower and upper layer thicknesses. A steady unidirectional velocity

u = u (y) e_{x}

u^{''} = 0

in each layer. Continuity of velocity and tangential stress at the interface gives one constant shear stress

τ = μ_{+} u_{+}^{'} = μ_{-} u_{-}^{'} = \frac{U _{-} - U _{+}}{d _{+} / μ _{+} + d _{-} / μ _{-}} .

u_{+} (y) = U_{+} + \frac{τ}{μ _{+}} (y + h), - h \leq y \leq σ,

u_{-} (y) = U_{-} - \frac{τ}{μ _{-}} (h - y), σ \leq y \leq h .

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