170 Chapter 5. Life in the slow lane: the low Reynolds-number world[[Student version, December 8, 2002]]
motion with the Stokes drag force. How far does it travel before it stops? Comment.
b. Our discussion of Brownian motion assumed that each random step was independent of the
previous one; thus for example we neglected the possibility of a residual drift speed left over from
the previous step. In the light of (a), would you say that this assumption is justified for a bacterium?
5.5Blood flow
Your heart pumps blood into your aorta. The maximum flow rate into the aorta is about 500cm^3 s−^1.
Assume the aorta has diameter 2.5cm,the flow is laminar (not very accurate), and that blood is
aNewtonian fluid with viscosity roughly equal to that of water.
a. Find the pressure drop per unit length along the aorta. Express your answer in SI units. Compare
the pressure drop along a 10cmsection of aorta to atmospheric pressure (10^5 Pa).
b. How much power does the heart expend just pushing blood along a 10cmsection of aorta?
Compare to your basal metablism rate, about 100W,and comment.
c. The fluid velocity in laminar pipe flow is zero at the walls of the pipe and maximum at the center.
Sketch the velocity as a function of distancerfrom the center. Find the velocity at the center.
[Hint: The total volume flow rate, which you are given, equals
∫
v(r)2πrdr.]5.6 T 2 Kinematic viscosity
a. Thoughνhas the same dimensionsL^2 /Tas any other diffusion constant, its physical meaning
is quite different from that ofD,and its numerical value for water is quite different from the value
Dfor self-diffusion of water molecules. Find the value ofνfromηand compare toD.
b. Still, these values are related. Show, by combining Einstein’s relation and the Stokes drag
formula, that taking the radiusaof a water molecule to be about 0. 2 nmleads to a satisfactory
order-of-magnitude prediction ofνfromD,a,and the mass density of water.
5.7 T 2 No going back
Section 5.2.3 argued that the motion of a gently sheared, flat layer would retrace its history if we
reverse the applied force. When the force is large, so that we cannot ignore the inertial term in
Newton’s Law of motion, where exactly does the argument fail?
5.8 T 2 Intrinsic viscosity of a polymer in solution
Section 4.3.2 argued that a long polymer chain in solution would be found in a random-walk
conformation at any instant of time.^5 This claim is not so easy to verify directly, so instead in
this problem we approach the question indirectly, by examining theviscosityof a polymer solution.
(Actually this is an important issue in its own right for biofluid dynamics.)
Figure 5.2 on page 145 shows two parallel plates separated by distanced,with the space filled
with water of viscosityη.Ifone plate is slid sideways at speedv,then both plates feel viscous force
ηv/dperunit area. Suppose now that a fractionφof the volume between plates is filled withsolid
objects,taking up space previously taken by water. We’ll supposeφis very small. Then at speed
vthe shear strain rate in the remaining fluid must begreaterthan before, and so the viscous force
will be greater too.
a. To estimate the shear strain rate, imagine that all the rigid objects are lying in a solid layer of
thicknessφdattached to the bottom plane, effectively reducing the gap between the plates. Then
what is the viscous force per area?
b. We can express the result by saying the suspension has an effective viscosityη′bigger thanη.
(^5) This problem concerns a polymer under “theta condtions” (see Section 4.3.1′on page 133).