172 Chapter 5. Life in the slow lane: the low Reynolds-number world[[Student version, December 8, 2002]]
mentum is a vector quantity whereas concentration is a scalar. For example, Section 5.2.2 noted
that the viscous force law (Equation 5.9 on page 149) needs to be modified for situations other than
planar geometry. The required modification really matters if we want to get the correct answer for
the spinning-rod problem (Figure 5.11b on page 160).
Weconsider a long cylinder of radiusRwith its axis along theˆzdirection and centered at
x=y=0.Some substance surrounds the cylinder. First suppose that this substance issolid ice.
When we crank the cylinder, everything rotates as a rigid object with some angular frequencyω.
The velocity field is thenv(r)=(−ωy,+ωx,0). Certainly nothing is rubbing against anything, and
there should be no dissipative friction—the frictional transport of momentum had better be zero.
And yet if we examine the pointr 0 =(r 0 , 0 ,z)wefind a nonzero gradient ddvxy
∣∣
∣r
0=ω.Evidentlyour formula for the flux of momentum in planar geometry (Equation 5.20 on page 166) needs some
modification for the non-planar case.
Wewantamodified form of Equation 5.20 that applies to cylindrically symmetric flows and
vanishes when the flow is rigid rotation. Lettingr=‖r‖=
√
x^2 +y^2 ,wecan write a cylindrically
symmetric flow as
v(r)=(−yg(r),xg(r),0).
The case of rigid rotation corresponds to the choiceg(r)=ω.Youare about to findg(r)for a
different case, the flow set up by a rotating cylinder. We can think of this flow field as a set of
nested cylinders, each with a different angular velocityg(r).
Near any point, sayr 0 ,letu(r)=(−yg(r 0 ),xg(r 0 ))) be the rigidly rotating vector field that
agrees withv(r)atr 0 .Wethen replace Equation 5.20 by
(jpy)x(r 0 )=−η(
dvy
dx∣∣
∣∣
r 0−
duy
dx∣∣
∣∣
r 0)
. cylindrical geometry (5.21)
In this formulaη≡νρm,the ordinary viscosity. Equation 5.21 is the proposed modification of
the momentum-transport rule. It says that we compute ddvxy
∣∣
∣
r 0
andsubtract offthe corresponding
quantity withu,inorder to ensure that rigid rotation incurs no frictional resistance.
a. Each cylindrical shell of fluid exerts a torque on the next one, and feels a torque from the
previous one. These torques must balance. Show that therefore the tangential force per area across
the surface at fixedris 2 τ/Lπr 2 ,whereτ is the external torque on the central cylinder andLis the
cylinder’s length.
b. Set your result from (a) equal to Equation 5.21 and solve for the functiong(r).
c. Findτ/Las a constant timesω.Hence find the constantCin Equation 5.18 on page 163.
5.10 T 2 Pause and tumble
In between straight-line runs,E. colipauses. If it just turned off its flagellar motors during the
pauses, eventually the bacterium would find itself pointing in a new, randomly chosen direction,
due to rotational Brownian motion. How long must the pause last for this to happen? [Hint: see
Problem 4.9 on page 140.] Compare to the actual measured pause time of 0. 14 s.Doyou think the
bacterium shuts down its flagellar motors during the pauses? Explain your reasoning.