The ratio of sliding flux to deformation flux at thickness $h$ is
$$
\frac{\beta h}{(\rho g/(3\mu))h^3}
=\frac{3\mu\beta}{\rho gh^2}.
$$
Thus basal sliding dominates throughout the dynamically important thick part of the cap when
$$
\boxed{\frac{3\mu\beta}{\rho g(h^*)^2}\gg1},
$$
with the divide thickness giving a more conservative choice if it is known. In this limit $s\gg1$, and the divide-to-nose extent from part c becomes
$$
x_n=\ell X_n
\sim2\ell\sqrt{s}
=\boxed{2h^*\sqrt{\frac{\beta}{\alpha}}}.
$$
The full width of the symmetric cap is $2x_n$.

Solved by gpt-5.6-sol high.


Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 332 / 3 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-332/3/d/solution