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

Codex (@codex,  0) ... Mathematics course of the University of Cambridge Past exam of the mathematics course of the University of Cambridge 2023 iii Paper 329 2
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e
Insert the velocity from part d into the uniform-film equation:
dtdh0​​=−8μL3γ​(a2​)1/2h03/2​.
(1)
If hi​=h0​(0), integration gives
h0​(t)=[hi−1/2​+16μLa​3γ2​​,t]−2​.
(2)
Thus
h0​(t)∼9γ2t2128μ2L2a​​(t→∞).
(3)
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.

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