144 Chapter 5. Life in the slow lane: the low Reynolds-number world[[Student version, December 8, 2002]]
Your Turn 5c
a. Work out the dimensions ofηfrom the Stokes formula. Show that they can be regarded as
those of pressure times time, and that hence the SI unit for viscosity isPa·s.
b. Your Turn 5a raised a paradox: Our equilibrium formula suggested that milk should separate,
and yet we don’t normally observe this happening. Use the Stokes formula to estimate howfast
this should happen in homogenized milk. Then compare the situation in raw milk, whose fat
droplets are over 5μmin diameter, and comment.It’s worth memorizing the value ofηfor water at room temperature:^1 ηw≈ 10 −^3 kg m−^1 s−^1 =
10 −^3 Pa·s.
Wecan use the above remarks to look once again at the sizes of polymer coils. Combining
Equation 5.3 with the Stokes formula and the scaling law for random coils (Idea 4.16 on page 111)
gives thats=(m−Vρm)/(6πηa). If we assume that the polymer displaces a volume of water
proportional to the number of monomers, and that its coil size is a constant timesmp,then we find
s∝m^1 −p.Our simple picture of random walks gave usp=1/2, and indeed Figure 4.7b on page
111 shows that this scaling is roughly observed. (More precisely, Figure 4.7a gavep=0.57, while
Figure 4.7b shows an exponent of 0.44, quite close to 1−p.)
5.1.3 It’s hard to mix a viscous liquid
Section 5.2 will argue that in the nanoworld of cells, ordinary water behaves as a very viscous liquid.
Since most people have made only limited observations in this world, though, it’s worthwhile first
to notice some of the spooky phenomena that happen there.
Pour a few centimeters of clear corn syrup into a clear cylindrical beaker or wide cup. Set aside
some of the syrup and mix it with a small amount of ink to serve as a marker. Put a stirring
rod in the beaker, then inject a small blob of marked syrup somewhere below the surface, far from
both the rod and the walls of the beaker. (A syringe with a long needle helps with this step, but
amedicine dropper will do; remove it gently to avoid disturbing the blob.) Now try moving the
stirring rod slowly. One particularly revealing experiment is to hold the rod against the wall of the
beaker, slowly run it around the wall once clockwise, then slowly reverse your first motion, running
it counterclockwise to its original position.
Among the phenomena you’ll note are that:
- It’s very hard to mix the marked blob into the bulk.
- The marked blob actually seems to take evasive action when the stirring rod approaches.
- In the clockwise-counterclockwise experiment, the blob will smear out in the first step. But if
you’re careful in the second step to retrace the first step exactly, you’ll see the blob magically
reassemble itself into nearly its original position and shape! That’s not what happens when
youstir cream into your coffee.
Figure 5.1 shows the result of a more controlled experiment. A viscous liquid sits between two
concentric cylinders. One cylinder is rotated through several full turns, smearing out the marker
blob shown (Figure 5.1b). Upon rotation through an equal and opposite angle, the blob reassembles
itself (Figure 5.1c).
(^1) Some authors express this result in units ofpoise(abbreviationP), defined aserg·s/cm (^3) =0. 1 Pa·s;thusηwis
about one centipoise. Some values ofηfor other biologically relevant fluids appear in Table 5.1 on page 147.