5.4. Track 2[[Student version, December 8, 2002]] 167
dimensions for a viscous friction coefficient is to multiply the viscosity byato the first power.
That’s what the Stokes formula says, apart from the dimensionlass prefactor 6π.
5.2.2′
- The physical discussion in Section 5.2.2 may have given the impression that the Reynolds-number
criterion is not very precise—Ritself looks like the ratio of two rough estimates! A more mathemat-
ical treatment begins with the equation of incompressible, viscous fluid motion, the “Navier–Stokes
equation.” The Navier–Stokes equation is essentially a precise form of Newton’s Law, as we used
it in Equation 5.7. Expressing the fluid’s velocity fieldu(r)interms of the dimensionless ratio
u ̄≡u/v,and the positionrin terms of ̄r≡r/a,one finds thatu ̄( ̄r)obeys a dimensionless equa-
tion. In this equation the parametersρm,η,v,andaenter in only one place, via the dimensionless
combinationR(Equation 5.11). Two different flow problems of the same geometrical type, with the
same value ofR,will therefore beexactly the samewhen expressed in dimensionless form, even if
the separate values of the four parameters may differ widely! (See for example (Landau & Lifshitz,
1987,§19).) This “hydrodynamic scaling invariance” of fluid mechanics is what lets engineers test
submarine designs by building scaled-down models and putting them in bathtubs. - Section 5.2.2 quietly shifted from a discussion of flow around an obstruction to Reynolds’ results
onpipeflow. It’s important to remember that the critical Reynolds number in any given situation is
alwaysroughlyone, but that this estimate is only accurate to within a couple of orders of magnitude.
The actual value in any specified situation depends on the geometry, ranging from about 3 (for exit
from a circular hole) to 1000 (for pipe flow, whereRis computed using the pipe radius), or even
more.
5.2.3′
- Section 5.2.3 claimed that the equation of motion for a purely elastic solid has no dissipation.
Indeed a tuning fork vibrates a long time before its energy is gone. Mathematically, if we shake the
top plate in Figure 5.2b back and forth, ∆z(t)=Lcos(ωt), then Equation 5.14 on page 152 says
that for an elastic solid the rate at which we must do work isfv=(GA)(Lcos(ωt)/d)(ωLsin(ωt)),
which is negative just as often as it’s positive: All the work we put in on one half-cycle gets
returned to us on the next one. In a fluid, however, multiplying the viscous force byvgives
fv=(ηA)(Lωsin(ωt)/d)(ωLsin(ωt)), which is always positive. We’re always doing work, which
gets converted irreversibly to thermal energy. - There’s no reason why a substance can’t displaybothelastic and viscous response. For example,
when we shear a polymer solution there’s a transient period when its individual polymer chains
are starting to stretch. During this period if the applied force is released, the stretched chains can
partially restore the original shape of a blob. Such a substance is calledviscoelastic.Its restoring
force is in general a complicated function of the frequencyω,not simply a constant (as in a solid)
nor linear inω(as in a Newtonian fluid). The viscoelastic properties of human blood, for example,
are important in physiology (Thurston, 1972). - As mentioned in the beginning of Section 5.2.3, it’s not necessary to apply the exact time-
reversed force in order to return to the starting configuration. That’s because the left side of
Equation 5.13 is more special than simply changing sign under time-reversal: Specifically, it’s of
first-orderin time derivatives. More generally, the viscous force rule (Equation 5.4 on page 146)
also has this property. Applying a time-dependent force to a particle in fluid then gives a total