164 Chapter 5. Life in the slow lane: the low Reynolds-number world[[Student version, December 8, 2002]]
do seem to attract each other with a gravitational force described (approximately!)
byNewton’s Law of gravitation.- Though they are general, physical Lawsneed not be, and generally cannot be, exact.
Thus as people discovered more and deeper layers of physical reality, Newton’s Law of
motion had to be replaced by a quantum-mechanical version; his Law of gravitation
wassuperseded by Einstein’s, and so on. The older, approximate laws remain valid
and useful in the very large domain where they were originally discovered, however. - Physical Laws all seem to beintrinsically mathematical in their expression.This char-
acteristic may give them an air of mystery, but it is also the key to their great
simplicity. There is very little room in the terse formula “f=ma”tohide any
sleight-of-hand, little room to bend a simple formula to accommodate a new, dis-
crepant experiment. When a physical theory starts to acquire too many complicating
features, added to rescue it from various new observations, physicists begin to suspect
that the theory was false to begin with. - Yetout of the simplicity of a Law there always emerge subtle, unexpected, and true
conclusions revealed by mathematical analysis.Word-stories are often invented later
to make these conclusions seem natural, but generally the clearest, most direct route
to get them in the first place is mathematical.
An appreciation of these ideas may not make you a more productive scientist. But many people
have drawn inspiration, even sustenance, from their wonder at the fact that Nature should have
any such unifying threads at all.
The big picture
Returning to the Focus Question, the key difference between the nanoworld and our everyday life
is that viscous dissipation completely dominates inertial effects. A related result is that objects in
the nanoworld are essentially unable to store any significant, nonrandom kinetic energy—they don’t
coast after they stop actively pushing themselves (see Problem 5.4). These results are reminiscent
of the observation in Chapter 4 that diffusive transport, another dissipative process, is fast on small
length scales; indeed, we saw in Section 5.3.2 that diffusion beats stirring in the submicron world.
Wesaw how to express the distinction between dissipative and nondissipative processes in a
very concise form by describing the invariance properties of the appropriate equations of motion:
frictionless Newtonian physics is time-reversal invariant, whereas the friction-dominated world of
low Reynolds number is not (Section 5.2.3).
Hiding in the background of all this discussion has been the question ofwhymechanical energy
tends to dissipate. Chapter 1 alluded to the answer—the Second Law of thermodynamics. Our task
in the next chapter is to make the Second Law, and its cardinal concept of entropy, more precise.
Key formulas
- Viscosity: Suppose a wall is perpendicular to thexdirection. The force per area in thez
direction exerted by a fluid on a wall is−ηdvz/dx(Equation 5.9). The kinematic viscosity
is defined asν=η/ρm(Equation 5.20), whereρmis the fluid mass density, and has the units
of a diffusion constant.