146 Chapter 5. Life in the slow lane: the low Reynolds-number world[[Student version, December 8, 2002]]
liquid.
The preceding scenario sounds good for corn syrup. But it doesn’t address one key question:
Whydoesn’twater behave this way? When you stir cream into your coffee, it immediately swirls
into a complex,turbulentpattern. Nor does the fluid motion stop when you stop stirring; the
coffee’s momentum continues to carry it along. In just a few seconds, an initial blob of cream gets
stretched to a thin ribbon only a few molecules thick; diffusion can then quickly and irreversibly
obliterate the ribbon. Stirring in the opposite direction won’t reassemble the blob. It’s easy to mix
anonviscous liquid.
5.2 Low Reynolds number
Tosummarize, the last two paragraphs of the previous subsection served to refocus our attention,
awayfrom the striking observed distinction between mixing and nonmixing flows and onto a more
subtle underlying distinction, between turbulent and laminar flows. To make progress, we need
some physical criterion that explains why corn syrup (and other fluids like glycerine and crude oil)
will undergo laminar flow, while water (and other fluids like air and alcohol) commonly exhibit
turbulent flow. The surprise will be that the criterion depends not only on the nature of the fluid,
but also on thescaleof the process under consideration. In the nanoworld, water will prove to be
effectivelymuch thickerthan the corn syrup in your experiment, and hence essentially all flows in
this world are laminar.
5.2.1 A critical force demarcates the physical regime dominated by fric-
tion
Because viscosity certainly has something to do with the distinction between mixing and nonmixing
flows, let’s look a bit more closely at what it means. The planar geometry sketched in Figure 5.2b
is simpler than that of a spherical ball, so we use it for our formal definition of viscosity. Imagine
twoflat parallel plates separated by a layer of fluid of thicknessd.Wehold one plate fixed while
sliding the other sideways (thezdirection in Figure 5.2b) at speedv 0 .This motion is calledshear.
Then the dragged plate feels a resisting frictional force directed opposite tov 0 ;the stationary plate
feels an equal and opposite force (called an “entraining force”) parallel tov 0.
The forcefwill be proportional to the areaAof each plate. It will increase with increasing
speedv 0 ,but decrease as we increase the plate separation. Empirically, for small enoughv 0 many
fluids indeed show the simplest possible force rule consistent with these expectations:
f=−ηv 0 A/d. viscous force in a Newtonian fluid, planar geometry (5.4)The constant of proportionalityηis the fluid’sviscosity.Equation 5.4 separates out all the situation-
dependent factors (area, gap, speed), exposingηas the one factor intrinsic to the type of fluid. The
minus sign reminds us that the drag force opposes the imposed motion. (You can verify that the
units work out in Equation 5.4, using your result in Your Turn 5c(a) on page 143.)
Any fluid obeying the simple rule Equation 5.4 is called aNewtonian fluidafter the ubiqui-
tous Isaac Newton. Most Newtonian fluids are in additionisotropic(the same in every direction;
anisotropic fluids will not be discussed in this book). Such a fluid is completely characterized by
its viscosity and its mass densityρm.