Insert the velocity from part d into the uniform-film equation:
$$
\frac{dh_0}{dt}
=-\frac{3\gamma}{8\mu L}
\left(\frac2a\right)^{1/2}h_0^{3/2}.
$$
If $h_i=h_0(0)$, integration gives
$$
\boxed{
h_0(t)=\left[
 h_i^{-1/2}
+\frac{3\gamma\sqrt2}{16\mu L\sqrt a},t
\right]^{-2}}.
$$
Thus
$$
\boxed{h_0(t)\sim
\frac{128\mu^2L^2a}{9\gamma^2t^2}}
\qquad(t\to\infty).
$$
The film thins algebraically and does not reach zero thickness in finite time within this continuum model; rupture would require physics omitted here, such as intermolecular forces.


Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 329 / 2 / e / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-329/2/e/solution