Thursday, March 10, 2011

Captain Meteo: Sandstorms and All That

One of those little puzzles with which fluids professors like to annoy and edify their students goes like this: the velocity of a fluid drops to zero at a solid interface, so how do winds propel, pick up and fling dust motes, sand grains, pebbles and larger objects?

Captain Meteo will offer two looks at this. If we consider this purely as a Newtonian fluids problem, we need to remember that stress depends not on velocity but on velocity gradient, so that even though the velocity drops to zero at the surface, the velocity gradient doesn’t.

Does this still sound a bit funky to you? Should it? Consider a molecular point of view. The molecules of that fluid are busy jostling each other at pretty high speeds – hundreds of meters per second at room temperature. If the fluid is in motion, there is superposed on that jostling a general drift, and when that fluid flows over a solid surface, we know from experiment that there is a velocity gradient in the mean flow near the surface.

The molecules have motions transverse to the flow as well as parallel to it. Consequently, they are constantly wandering from their “lanes” to collide with other objects. Those closer to the wall collide with it more frequently, transferring momentum with each collision. These collisions tend to equalize momentum between fluid and wall. Of course these molecules are busy colliding with their neighbors, with the net effect of transferring momentum from the flow to the wall. Note that this transfer depends on the fact that the molecules actually hitting the wall come from a place where there actually is momentum along the wall – in other words from somewhere the fluid velocity is not zero. This is because actual molecules have finite size and spend most of their time at least a small distance from the wall. The average longitudinal velocity of molecules goes to zero as one approaches the wall, but the actual molecules hitting the wall *do* have average longitudinal velocity – and that’s the funky point.

Of course we have talked only about longitudinal forces only. For transverse velocities, not only the velocities but the gradients go to zero at the wall. So how do you get that sand grain into the air? You push it along the ground until some collision bounces it up high enough so that it can get caught up in some general motion.

For the more ambitious student: how do the foregoing arguments help us understand why superfluids have zero viscosity?

[Hint: think about transverse momentum]