 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-4/38c/b/solution

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

The leading lubrication momentum equations and incompressibility condition are

0 = - p_{x} + ρ g sin α + μ u_{yy}, 0 = - p_{y} - ρ g cos α, u_{x} + v_{y} = 0.

y = 0

, no slip and no penetration give

u = v = 0

y = h (x, t)

, neglecting surface tension and ambient viscous stress gives

p = p_{atm}, u_{y} = 0,

v = h_{t} + u h_{x} at y = h .

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