Biological Physics: Energy, Information, Life

150 Chapter 5. Life in the slow lane: the low Reynolds-number world[[Student version, December 8, 2002]]

everything together, we find
friction term =ffrict≈η^3 v/a^2. (5.10)
Weare ready to compare Equations 5.8 and 5.10. Dividing these two expressions yields a
characteristic dimensionless quantity:^3

R=vaρm/η. theReynolds number (5.11)

WhenRis small, friction dominates. Stirring produces the least possible response, namely laminar
flow, and the flow stops immediately after the external forcefextstops. (Engineers often use the
synonym “creeping flow” for low Reynolds-number flow.) WhenRis big, inertial effect dominate,
friction is negligible, the coffee keeps swirling after you stop stirring, and the flow is turbulent.
As a first application of Equation 5.11, consider the flow of fluid down a pipe of radiusa.In
aseries of careful experiments in the 1880s, O. Reynolds found that generally the transition to
turbulent flow occurs aroundR≈1000. Reynolds varied all the variables describing the situation
(pipe size, flow rate, fluid density and viscosity) and found that the onset of turbulence always
depended on just one combination of the parameters, namely the one given in Equation 5.11.
Let’s connect Reynolds’ result to the concept of critical force discussed in the previous subsection:

Example Suppose that the Reynolds number is small,R1. Compare the external force

needed to anchor the obstruction in place to the viscous critical force.

Solution: Atlow Reynolds number the inertial term is negligible, sofextis essen-

tially equal to the frictional force (Equation 5.10). To estimate this force, take the

fluid element sizeto be that of the obstruction itself; then

ffrict

fcrit ≈

ηa^3 v

a^2

1

η^2 /ρm,w=

vaρm,w

η =R.

So indeed the force applied to the fluid is much smaller thanfcritwhenRis small.

Your Turn 5d

Suppose that the Reynolds number is big,R1. Compare the external force needed to anchor

the obstruction in place to the viscous critical force.

As always, we need to make some estimates. A 30mwhale, swimming in water at 10ms−^1 ,has
R≈300 000 000. But a 1μmbacterium, swimming at 30μms−^1 ,hasR≈ 0 .000 03! Section 5.3.1
will show that the very meaning of the word “swim” will be quite different for these two organisms.
T 2 Section 5.2.2′on page 167 outlines more precisely the sense in which fluids have no characteristic
length scale.

5.2.3 The time reversal properties of a dynamical law signal its dissipa-

tive character

Now that we have a criterion for laminar flow, we can be a bit more explicit in our understanding
of the mixing/umixing puzzle (Section 5.1.3).

(^3) Notice that the sizeof our fluid element dropped out of this expression, as it should: Our fluid element was
arbitrarily defined.