5.1. Friction in fluids[[Student version, December 8, 2002]] 145
Figure 5.1:(Photographs.) An experiment showing the peculiar character of low Reynolds-number flow. (a)A
small blob of colored glycerine is injected into clear glycerine in the space between two concentric cylinders. (b)The
inner cylinder is turned through four full revolutions, apparently mixing the blob into a thin smear. (c)Uponturning
the inner cylinder back exactly four revolutions, the blob reassembles, only slightly blurred by diffusion. The finger
belongs to Sir Geoffrey Taylor.[Copyrighted figure; permission pending.][From (Shapiro, 1972).]
a
b
d zxyFigure 5.2:(Schematics.) Shearing motion of a fluid in laminar flow, in two geometries. (a)Cylindrical (ice-cream
maker) geometry, viewed from above. The central cylinder rotates while the outer one is held fixed. (b)Planar
(sliding plates) geometry. The top plate is pushed to the right while the bottom one is held fixed.
What’s going on? Have we stumbled onto some violation of the Second Law? Not necessarily.
If you just leave the marked blob alone, it does diffuseaway, butextremely slowly. That’s because
the viscosityηis large, and the Einstein and Stokes relations giveD=kBT/ζ=kBT/(6πηa)
(Equations 4.15 and 4.14). Moreover, diffusion initially only changes the density of ink near the
edgesof the blob (see Figure 4.16 on page 136), so a compact blob cannot change much in a short
time. One could imagine that stirring causes an organized motion, in which successive layers of
fluid simply slide over each other and stop as soon as the stirring rod stops (Figure 5.2). Such a
stately fluid motion is calledlaminar flow.Then the motion of the stirring rod, or of the container
walls, would just stretch out the blob, leaving it still many billions of molecules thick. The ink
molecules are spread out, but still not random, because diffusion hasn’t yet had enough time to
randomize them fully. When we slide the walls back to their original configuration, the fluid layers
could then each slide right back and reassemble the blob. In short, we could explain the reassembly
of the blob by arguing that it never “mixed” at all, despite appearances. It’s hard to mix a viscous