Fluid Flow Purpose: The purpose of this experiment is to measure ...

Fluid Flow
Purpose:
The purpose of this experiment is to measure water flow through capillary tubes at different
pressures, to study resistance to flow using tubes of different diameter, and to measure series
resistance of capillaries. Refer to appropriate sections of the textbook as necessary.
Apparatus:
- Two constant-level baths,
- 100 ml. graduated cylinder,
- Glass capillary tubes
(A) 1.60 ± 0.05 mm. dia., L ≈ 8 cm.
(B) 1.60 ± 0.05 mm. dia., L ≈ 16 cm.
(C) 0.75 ± 0.05 mm. dia., L ≈ 8 cm.
- watch or timer, tubing, clamps, stand, meter stick, and a beaker.
Introduction:
This experiment will develop skills in graphing and graphical analysis.
Theory:
Fluid flows through a pipe because of the pressure difference between the ends of the pipe.
Q is the fluid flow rate; if V is the volume of water flowing through a pipe in a time t,
fluid flow rate Q = V/t(ml./sec.) (1)
For a given capillary (like arterioles and veinlets in the human body), Q increases as the
pressure difference δP = P 1 − P 2 between the ends of the capillary increases.
This experiment is a study of the relationship between Q and δP for water flow through
capillary tubes. For a sufficiently small flow rate, Q is proportional to δP so we write
δP = RQ
where R is a constant called the capillary’s resistance. Note the analogy to the electrical
case where a potential difference δV causes a flow of charge (current I) through a resistance
R; Ohm’s Law is δV = RI.
δP = RQ is Poiseuille’s Law, valid whenever Q in (ml./sec.) is less than the capillary radius
in mm. If Q exceeds this limit, flow ceases to be orderly (laminar) and becomes turbulent,
and the flow law does not hold because R increases considerably.
1

Fluid Flow
Resistance depends upon the fluid viscosity η and the capillary length L and radius r and
for laminar flow,
R = 8ηL
πr 4 (2)
i. e.: R ∝ L; R ∝ 1/r 4 . Thus, if L is doubled, R is doubled: if r is doubled, R is decreased by
1/16. Consequently, small radius changes will cause relatively large changes in resistance,
and thus also in flow for a given pressure difference.
Don’t confuse R and r in this experiment!
¡¢£¤¥
¦§¨©
Figure 1
Procedure:
The capillary tube is connected by rubber tubes to the drain outlets of two constant–
level baths as in Fig. 1. With baths at different heights there is a pressure difference related
to the difference in height (or “head”) δh = h 1 − h 2 of the upper–surface levels of the two
baths by the relationship
δP = ρgδh
where
ρ = 1000kg/m 3
for water,
g = 9.8m/sec 2
and
1 cc = 1 ml = 10 −6 m 3
2

Fluid Flow
Constant–level baths maintain their upper surface level constant by means of continuous
input and an overflow pipe. Side 1 is the upper level bath, side 2 is the lower level bath.
Capillary tubes must be mounted horizontally and water must flow from the tap continuously
and drain to the sink continuously. With clamp 2 on, fill bath 2 using a graduated
cylinder up to the overflow drain: this is where the flow volume V will be collected. Place a
graduated cylinder under this drain via the drain-tube, open clamp 2 and get a continuous
flow of water into the collecting graduated cylinder by making δh at least several cm. Try
to exclude all bubbles from the tubing by pinching tubes at appropriate positions. Try to
ensure that this flow is continuous for a few minutes, draining the graduated cylinder as
required.
Exercise 1: Capillary Resistance
Use capillary tube A first, and get the apparatus running smoothly as described previously.
Set δh to 20 cm. and again get the system running smoothly by letting flow occur
for a couple of minutes. Drain the graduated cylinder without interrupting water flow (use
a beaker to catch flow while draining cylinder), replace it under flow, and record the time
to the nearest second as water level passes the 15 ml. mark on the cylinder. Record the
time again as the collected water level passes the 25 ml. mark. These collected volumes and
times should be entered in an appropriate table.
If using a stopwatch, the “lap” or ‘split” feature is useful here. The reason for
starting the volume at something other than zero is that it reduces errors due to “getting
started”;ie. you can anticipate the events you are timing, so your reaction time should not
play as big a part.
Still using tube A, repeat measurements with δh = 15, 10, and 5 cm. successively; it is
important to start with highest pressure head first and ”work” downward, having got smooth
flow and bubble-free tubing at higher rates first.
Note: If water being collected splashes into graduated cylinder the level readings are
hard to make, thus, let the drops flow down the side of the graduated cylinder by
touching the flow tubing to the inside wall of the graduated cylinder near the top.
Repeat these measurements for tubes B and C, and use clamps 1 and 2 while making the
change of capillary tubing.
For each tube one must be careful not to exceed the limiting case for laminar flow, hence
the maximum δh is important.
For tube B, use δh roughly 15, 12, 9, and 6 cm.
For tube C, use δh roughly 18, 15, 12, and 9 cm.
Please do not alter tube connections to constant level baths as they are quite delicate.
Capillary diameters are given; measure lengths L with a meter stick.
What is the realistic uncertainty in the length of the tubes, based on how cleanly the ends
are cut Be sure to use this value in your calculations.
3

Fluid Flow
Proceed through the exercise and create a data table with entries for fluid flow and pressure
head causing the flow. When the table is full, plot all the results as pressure δP vs. flow
rate Q, with all data for all 3 tubes on the same graph sheet. (Is the origin a point) Draw
a straight line through the data for each of the tubes.
As δP = RQ, then R = δP/Q which is the slope of the graph. Find R by this graphical method
for each of the capillary tubes. Does the y-intercept agree with what you expect Explain.
As
R = 8ηL
πr 4 ,
then r 4 R/L = 8η/π = constant (3)
Hence, using R as obtained graphically, r 4 R/L is calculated for each of the tubes, and the
average of these three values is obtained. From this average, the viscosity of water at the
top temperature is obtained.
Exercise 2: Resistance of Capillaries in Series
Connect capillary tubes A and B in “series” with a 2 inch piece of tubing. Ensure
somehow that they lie horizontally on the same line. Measure Q for several values of δh and
record in a table like that used in Ex. 1.
Plot the pressure vs. flow results on the same graph as used in Ex. 1, and calculate the
resistance from the graph.
The result R series should equal the sum of the separate resistances:
R series = R A + R B (4)
Compare the result with the sum obtained from Ex. 1. (Note that this is also analogous to
what happens with resistances in series in an electrical circuit.)
4