148 Chapter 5. Life in the slow lane: the low Reynolds-number world[[Student version, December 8, 2002]]
Chapter 10 that the typical scale of forces inside cells is more like a thousand times smaller, the
piconewton range. Friction rules the world of the cell.
It’s not size per se that counts, but rather force. To understand why, recall the flows of a
Newtonian fluid are completely determined by its mass density and viscosity, and convince yourself
that there is no combination of these two quantities with the dimensions of length. We say that a
Newtonian fluid “has no intrinsic length scale,” or is “scale invariant.” Thus even though we haven’t
worked out the full equations of fluid motion, we already know that they won’t give qualitatively
different physics on scales larger and smaller than some critical length scale, because dimensional
analysis has just told us that there can be no such scale! A large object—even a battleship—will
move in the friction-dominated regime, if we push on it with less than a nanonewton of force.
Similarly, macroscopic experiments, like the one shown in Figure 5.3 on page 149, can tell us
something relevant to amicroscopic organism.
T 2 Section 5.2.1′on page 166 sharpens the idea of friction as dissipation, by reinterpreting viscosity
as a form of diffusion.
5.2.2 The Reynolds number quantifies the relative importance of friction and inertia
Dimensional analysis is powerful, but it can move in mysterious ways. The previous subsection pro-
posed the logic that(i)Two numbers,ρmandη,characterize a simple (that is, isotropic Newtonian)
fluid;(ii)From these quantities we can form another,fcrit,with dimensions of force;(iii)Something
interesting must happen at around this range of externally applied force. Such arguments generally
strike students as dangerously sloppy. Indeed, when faced with an unfamiliar situation a physical
scientistbeginswith dimensional arguments to raise certain expectations, but then proceeds to
justify those expectations with more detailed analysis. In this subsection we begin this process,
deriving a more precise criterion for laminar flow. Even here, though, we will not bother with small
numerical factors like 2πand so on; all we want is a rough guide to the physics.
Let’s begin with an experiment. Figure 5.3 shows a beautiful example of laminar flow past an
obstruction, a sphere of radiusa.Far away, each fluid element is in uniform motion at some velocity
v.We’d like to know whether the motion of the fluid elements is mainly dominated by inertial
effects, or by friction.
Consider a small lump of fluid of size,which comes down the pipe on a collision course with
the sphere (Figure 5.4). In order to sidestep it, the fluid element must accelerate: The velocity
must changedirectionduring the encounter time ∆t≈a/v.The magnitude of the change invis
comparable to that ofvitself, so the rate of change of velocity (that is, the acceleration dv/dt)has
magnitude≈v/(a/v)=v^2 /a.The massmof the fluid element is the densityρmtimes the volume.
Newton’s Law of motion says that our fluid element obeys
ftot≡fext+ffrict=mass×acceleration. (5.7)
Herefextdenotes the external force from the surrounding fluid’s pressure, whileffrictis the net
force on the fluid element from viscous friction. Using the previous paragraph, the right-hand side
of Newton’s Law (the “inertial term”) is
inertial term = mass×acceleration≈(^3 ρm)v^2 /a. (5.8)Wewish to compare the magnitude of this inertial term to that offfrict.Ifone of these terms is
muchlarger than the other, then we can drop the smaller term in Newton’s Law.