166 Chapter 5. Life in the slow lane: the low Reynolds-number world[[Student version, December 8, 2002]]
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5.2.1′
- Section 4.1.4′on page 132, point (2), pointed out that our simple theory of frictional drag would
break down when the force applied to a particle was too great. We have now found a precise
criterion: The inertial (memory) effects neglected in Section 4.1.4 will indeed be significant for
forces greater thanfcrit. - The phenomenon of viscosity actually reflects yet another diffusion process. When we have small
indestructible particles, so that the number of particles is conserved, we found that random thermal
motion leads to a diffusive transport of particle number via Fick’s law, Equation 4.18 on page 117.
Section 4.6.4 on page 127 extended this idea, showing that when particles carry electric charge,
another conserved quantity, their thermal motion again leads to a diffusive transport of charge
(Ohm’s law). Finally, since particles carry energy, yet another conserved quantity, Section 4.6.4
argued for a third Fick-type transport rule, called thermal conduction. Each transport rule had its
owndiffusion constant, giving rise to the electrical and thermal conductivity of materials.
One more conserved quantity from first-year physics is the momentump. Random thermal
motion should also give a Fick-type transport rule for each component ofp.
Figure 5.2b shows two flat plates, each parallel to theyz-plane, separated bydin thexdirection.
Letpzdenote the component along thezdirection of momentum per unit volume. If the top plate
is dragged atvzin the +zdirection while the bottom is held stationary, we get a nonuniform
density ofpz,namelyρm·vz(x), whereρmdenotes the mass density of fluid. We expect that this
nonuniformity should give rise to afluxofpzwhose component in thexdirection is given by a
formula analogous to Fick’s law (Equation 4.18 on page 117):
(jpz)x=−ν
d(ρmvz)
dx. planar geometry (5.20)The constantνappearing above is a new diffusion constant, called thekinematic viscosity.Check
its units.
But the rate of loss of momentum is just aforce;similarly thefluxof momentum is a force per
unit area. Hence the flux of momentum (Equation 5.20) leaving the top plate exerts a resisting drag
force opposingvz;when this momentum arrives at the bottom plate, it exerts an entraining force
alongvz.Wehavethus found the molecular origin of viscous drag. Indeed it’s appropriate to name
νakind of “viscosity,” since it’s related simply toη: Comparing Equation 5.4 to Equation 5.20
shows thatν=η/ρm.
- We now have two empirical definitions of viscosity, namely the Stokes formula (Equation 4.14 on
page 107) and our parallel-plates formula (Equation 5.4). They look similar, but there’s a certain
amount of work to prove that they are equivalent. One must write down the equations of motion
for a fluid, containing the parameterη,solve them in both the parallel-plate and moving-sphere
geometries, and compute the forces in each case. The math can be found in (Landau & Lifshitz,
- or (Batchelor, 1967), for example. But theformof the Stokes formula just follows from
dimensional analysis. Once we know we’re in the low-force regime, we also know that the mass
densityρmcannot enter into the drag force (since inertial effects are insignificant). For an isolated
sphere the only length scale in the problem is its radiusa,sothe only way to get the proper