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

Table of contents
- Solution a

Solution

Balancing the downslope gravitational force density

ρ g sin α

against the viscous term

μ u_{yy}

gives the velocity scale

U \sim \frac{ρ g h ^{2} sin α}{μ} .

If the streamwise variation scale is

L \sim h /∣ h_{x} ∣

, the lubrication theory requirement is first

∣ h_{x} ∣ ≪ 1 .

The ratio of streamwise inertia

ρ U^{2} / L

to transverse viscous stress

μU / h^{2}

is

\frac{ρ U h ^{2}}{μL} \sim \frac{ρ ^{2} g h ^{3} s i n α}{μ ^{2}} ∣ h_{x} ∣.

Thus self-consistency also requires

\frac{ρ ^{2} g h ^{3} sin α}{μ ^{2}} ∣ h_{x} ∣ ≪ 1 .

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version