CBE 30355 - Lecture Notes - Oct. 29, 2024

Announcements

Class notes

• The Reynolds Lubrication Equation

Goals:

• Use the Reynolds Lubrication Equation to calculate the forces in 2-D lubrication flows.

Reading

• The class notes.
• BS&L, chapter 4

Additional Readings:

One of the most interesting examples of lubrication flows is that found in ice- skating. In that winter sport, the friction between the blade and the ice melts a small quantity of water, forming a thin lubricating layer. The old story that it's the pressure of the blade which melts the ice isn't true - and the later theory that it's the frictional heating that does the trick isn't quite right either. Instead, the current theory is that there is a very thin layer of water at the surface of ice even well below freezing due to thermodynamic effects. A good discussion of the effects of a thin layer of water on ice (taken from a Physics Today article) is given here.

Demonstration:

In class today we demonstrated the Benard-Marangoni convective instability. This occurs when a thin layer of fluid with a free surface is heated from below. Heat transfer at the upper surface leads to a vertical temperature gradient. If the gradient is large enough, it will cause convection cells to form. The idea is that fluid convects heat from the center of a cell to the surface, leading to a locally higher surface temperature. Because surface tension decreases with increasing temperature, this "hot spot" has a lower tension than the surrounding fluid, causing a surface convection away from the hot spot - a self-sustaining process. The conditions under which such an instability forms is called the Marangoni number Ma. The critical Marangoni number can be interpreted as a ratio of thermal diffusion time to surface tension driven convection time. If the convection is fast relative to thermal diffusion the fluid becomes unstable. Surface tension driven instabilities are important in many systems ranging from drying paint to particle self-assembly in thin films. The classic 1958 paper by Pearson describing this phenomenon may be found here.