5.2. Low Reynolds number[[Student version, December 8, 2002]] 147
Table 5.1:Density, viscosity and viscous critical force for some common fluids at 25◦C.Fluid ρm(kg m−^3 ) η(Pa·s) fcrit(N)
Air 1 2 · 10 −^54 · 10 −^10
Water 1000 0.0009 8 · 10 −^10
Olive oil 900 0.080 4 · 10 −^6
Glycerine 1300 1 0.0008
Corn syrup 1000 5 0.03Weare pursuing the suggestion that simple, laminar flow ensues whenηis “large,” whereas
weget complex, turbulent flow when it’s “small.” But the question immediately arises: Large
compared to what? The viscosity is not dimensionless, so there’s no absolute meaning to saying
that it’s large (see Section 1.4.1 on page 15). Nor can we form any dimensionless quantity by
combining viscosity (dimensionsML−^1 T−^1 )with mass density (dimensionsML−^3 ). No fluid can
bedeemed “viscous” in an absolute sense. But wecanform a characteristic quantity with the
dimensions offorce:
fcrit=η^2 /ρm. viscous critical force (5.5)The motion of any fluid will have two physically distinct regimes, depending on whether we
apply forces bigger or smaller than that fluid’s critical force. Equivalently, we can say that:
a. There’s no dimensionless measure of viscosity, and hence nointrinsicdis-
tinction between “thick” and “thin” fluids, but...
b. Still there is asituation-dependentcharacterization of when a fluid’s mo-
tion will be viscous, namely when the dimensionless ratiof/fcritis small.(5.6)
Foragiven applied forcefwecan get a large ratiof/fcritbychoosing a fluid with a large mass
density or small viscosity. Then inertial effects (proportional to mass) will dominate over frictional
effects (proportional to viscosity), and we expect turbulent flow (the fluid keeps moving after we
stop applying force). In the opposite case, friction will quickly damp out inertial effects and we
expect laminar flow.
Summarizing the discussion so far, the previous subsection began with the distinction between
mixing and nonmixing flows. This subsection first rephrased the issue as the distinction between
turbulent and laminar flow, then finally as a distinction between flows dominated by inertia or
viscous friction, respectively. We found a criterion for making this distinction in a given situation
using dimensional analysis.
Let’s examine some rough numbers for familiar fluids. Table 5.1 shows that if we pull a marble
in corn syrup with a force less than 0. 01 N,then we may expect the motion to be dominated by
friction. Inertial effects will be negligible, and indeed in the corn-syrup experiment there’s no
swirling after we stop pushing the stirring rod. In water, on the other hand, even a millinewton
push puts us well into the regime dominated by inertia, not friction; indeed turbulent motion then
ensues.
What’s striking about the table is that it predicts that water will appear just as viscous to a
tiny creature exerting forces less than a nanonewton as glycerine does to us! Indeed, we’ll see in