Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 329 / 3 / a

Table of contents
- Solution a

Solution

Take

x

downslope,

y

across the slope, and

z

normal to the plane. The leading normal momentum balance gives

p = p_{a} + ρ g cos α (h - z) .

The tangential lubrication theory equations, with no slip at

z = 0

and zero tangential stress at

z = h

, then give the depth-integrated flux

q = \frac{ρ g h ^{3}}{3 μ} (sin α e_{x} - cos α \nabla h) .

For

α ≪ 1

h = \frac{h}{h _{0}}, (x, y) = \frac{α}{h _{0}} (x, y), q = \frac{3 μ}{ρ gα h _{0}^{3}} q .

Dropping tildes, steady mass conservation

\nabla \cdot q = 0

becomes

\nabla \cdot (h^{3} e_{x}) = \nabla \cdot (h^{3} \nabla h),

with

h \to 1

x \to - \infty

.

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