CBE30355 - Lecture Notes - Sept. 21, 2023

Announcements

Class notes

• Plane Couette Flow and Flow Down an Inclined Plane

Goals:

• Set up and solve unidirectional flow problems using the Navier-Stokes equations.

Reading

• The class notes.
• BS&L, chapter 3

Additional Readings:

In today's lecture we talked about the phenomenon of gravity driven flow down an inclined plane. Note that while simple, this technique is still actively used for studying non-Newtonian rheology. In particular, if you flow a concentrated suspension down a semi-circular trough the normal stress differences lead to a measureable deflection of the upper surface. This is actually the most sensitive way I know of to measure N2, the rheological function which leads to particle migration, meniscus accumulation, resuspension, and all sorts of other effects. A nice paper describing this effect (and with a pretty picture of the deflection) is given here.

There are many examples of flow down an incline, both in the laboratory and in nature. Perhaps the most spectacular, however, (and certainly the most deadly) are the pyroclastic flows associated with volcanic activity. A description of such flows may be found here which includes an incredible movie of a dome collapse and resulting pyroclastic flow in Japan. A longer documentary with some pretty amazing videos is given here.

Demonstration:

A flow which is closely related to gravitational flow down an incline is the "tears of wine" phenomenon. In this case, however, the interesting part of the flow is driven by surface tension gradients drawing a film of fluid upwards (e.g., the evaporation of alcohol from the wine induces a surface tension gradient that imposes a stress at the surface of a thin film). In this demonstration we will both visualize the phenomenon and calculate the upper bound on the film thickness - where the surface tension gradient is balanced by a downward gravitationally driven flow. This phenomenon has been studied by many wine drinkers over the years, dating back to Thomson in 1855. A recent paper describing the flow is by Venerus and Simavilla, which may be found here.