Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 38C / b

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- Solution b

Solution

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.

At the solid plane

y = 0

, no slip and no penetration give

u = v = 0

. At the free surface

y = h (x, t)

, neglecting surface tension and ambient viscous stress gives

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

to leading order. The kinematic boundary condition for a free-surface graph is

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

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