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PHYSICS 121 Experiment 9

Fluid Flow The purpose of this lab is to study viscous fluid flow .The resistance to flow of single capillaries, 2 capillaries in a series and 2 capillaries in a parallel configurations is measured and compared to predictions. Equipment 2 cups 1 panel with 3 glass capillaries (Fig. 1) 1 panel with 4 glass capillaries, 2 in series, 2 in parallel (Fig. 3) tubes with valves 1 beaker 1 ruler

Fig. 1

valves top middle bottom (unused)

glass capillaries, 1, 2, 3 mm diameter

cup B

cup A

plastic tubes calibrated beaker scale faint!!

ruler

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Introduction We study the flow of viscous fluids (see Ch11 sheet 35) through glass capillaries. The flow rate Q (Ch11 sheet 25) through the capillary is proportional to the pressure difference P across the capillary (Ch11 sheet 36), so that we can write

P = RQ (1) where R is the resistance of the capillary to the fluid flow. Poiseuille’s Law tells us that the flow rate

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Q =Vt

=πr4P8ηL

(2)

V is the volume of fluid which flows in a time t. r is the radius and L is the length of the capillary and η is the fluid’s viscosity. The applicability of this law to water flow through glass capillary tubes is investigated here. The apparatus is very simple. It is sketched in Figure 2 below.

Fig. 2 The capillary is connected to the bottom of cups A and B. Cup A is positioned a height h

above cup B. The capillary is kept horizontal, and the water in cups A and B is kept up to the overflow tubes. Then the pressure difference on the capillary is P = ρ g h (see Ch11, sheet 5 and 11’), where ρ is the density of water. As water flows from A to B it will collect in the calibrated beaker. By measuring the collection time, the flow rate Q can be computed. It is important to keep adding water to cup A so that the water level stays at the top of the overflow tube, thus keeping h constant during the measurement. Make sure you measure h correctly as drawn in Figure 2.

Cup A

h Cup B

Overflow Tube

Calibrated Beaker

Capillary

Overflow closed

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Part I Measurement of the Dependence of the Resistance R on the radius r: You will not use the 1 mm capillary (bottom) in this part. Simply keep it shut at all times.

Enter the length of the capillaries, L = 10 cm, into your worksheet. In this Part you will measure the flow rate Q for 2 different pressure differences P across the capillary and get the resistance R of the 2 mm diameter capillary. Then you repeat the same for a 3 mm diameter capillary and compare the two. Procedure 2 mm diameter capillary (middle) : Enter the radius into your worksheet. 1) You first have to establish steady flow in the setup. Connect the 2mm capillary by opening the red valve for it. Add water to cup A and pinch the rubber connecting tubes until the air bubbles are removed and water commences to flow. It is helpful to incline any tube or capillary you want to remove the bubbles from, such that the bubbles percolate upwards to the top. Note it is very important that you get rid of as many of the air bubbles as you can because they are a source of error and will cause your data to be inaccurate. Keep adding water to cup A so that the water is always at the same level in it. Once the air bubbles have been removed, close the valve and empty your calibrated beaker so you can start collecting your data. Note! Removing bubbles from all lead tubes and capillaries is a tedious job. Do it for all capillaries to be used together (not for the bottom 1mm capillary in Part I). Then make sure that you keep adding fluid to cup A as long as flow occurs and only when all valves are shut stop adding water. Otherwise bubbles will form in the lead from cup A, migrate into the apparatus and you have to start with the bubble removal again for the next measurement.

You will take data setting the height h in Figure 2 above to the 2 values of ~0.1, ~0.2 m.

Make your measurement with the top and bottom valves closed and the middle valve opened. Enter the valve positions –open/closed – for the top, middle and bottom valves into your worksheet. 2) Make sure you have enough water in cup A (the water level reaching the top of the overflow tube), and that your drain tube is straight and empties into the calibrated beaker. Adjust the height h to ~ 0.1 m, measure it (see Figure 2!) and record it in Table 1 in your worksheet. Simultaneously open the 1mm capillary valve and start the timer. Stop the time and close the valve when ~50-100 ml of water have been collected in your calibrated beaker. While the water is flowing, make sure that you keep the water level in cup A constant at the top of the overflow tube. Record the elapsed time t and the volume V of water in your calibrated beaker in Table 1.

Repeat this procedure for the other h setting and enter all values into Table 1.

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Complete Table 1 with your calculations of the 2 flow rates Q, pressure differences P, and resistances R for the 2mm capillary: For the 2 mm capillary, calculate the rate of flow Q from the expression Q = V/t, and record your values in Table 1 on your worksheet. For the 2mm capillary, calculate the pressure P for each of the 2 heights using the equation

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P = ρgh (3) Use for the density of water ρ = 103 kg/m3, the density of water at 20oC. Enter the values into Table 1. Calculate the resistance R of the 2mm capillary to water flow using equation (1) above. Enter all values into Table 1. Take the average of the two measurements as the resistance of the 2 mm capillary and enter it into your worksheet. You’ll notice that the glass capillaries are not the only “tubes” in the setup. There are plastic tubes connecting to the cups, plastic tubes connecting the capillaries to the black tubing, and the black tubing itself. But you attributed the measured resistance to the 2 mm capillary only. Which part of equation (2) allows you to neglect the other tubes ? (Hint: consider the radius of the other tubes compared to the radius of the capillary. What does the strong dependence of the resistance R on the radius r, , tell you about the contribution of the other tubes to the total resistance in the setup, compared to the contribution of the capillary? 3 mm diameter capillary (bottom): Enter the radius into your worksheet.

Repeat the above for the 3 mm capillary. Make your measurement with the bottom and middle valves closed and the top valve opened. Enter the valve positions –open/closed – for the top, middle and bottom valves into your worksheet. Complete Table 2 with your calculations of the 2 flow rates Q, pressure differences P, and resistances R for the 3 mm capillary. Take the average of the two measurements as the resistance of the 3 mm capillary and enter it into your worksheet. Use equation (2) to predict the ratio of the resistances from the ratio of the radii . For this calculation simply assume that all quantities in equation (2) cancel except the radii.

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Part II Series vs. Parallel connections: In this part you will observe what the resistance R of series and parallel connections of capillaries are and compare them with predictions. See Figure 3 below.

Fig. 3 Procedure Change to the panel with two 2 mm diameter capillaries in a series connection and two 2mm diameter capillaries in a parallel connection as shown in Figure 3 above. All capillaries have a length of 5 cm.. Enter the radius and length into your worksheet. Measure the resistance R for the middle capillary first, then the series connection at the top, then the parallel connection at the bottom. You choose the three configurations simply by opening the appropriate valve. You do the measurements as you did it in Part I, with the exception that here you only do it for one pressure difference P i.e. for one height difference h ~ 0.15 m. Again, make sure that air bubbles are removed! Enter your data in Table 3 and complete it with the calculated values for the flow rate Q, the pressure difference P and the resistance R.

2 mm capillaries in series, length L = 5 cm

2 mm capillaries in parallel, length L=5 cm

valves top middle bottom

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The 2 mm capillary alone (middle): Make your measurement with the top and bottom valves closed and the middle valve opened. Enter the valve positions –open/closed – for the top, middle and bottom valves, and the lengths of the capillaries into your worksheet. Enter your data and the calculated values for flow rate Q, pressure difference P, and resistance R into Table 3. The 2 mm capillaries in series (top): Make your measurement with the middle and bottom valves closed and the top valve opened. Enter the valve positions –open/closed – for the top, middle and bottom valves into your worksheet. Enter your data and the calculated values for flow rate Q, pressure difference P, and resistance R into Table 4. For the series connection, the theoretical effective resistance is as follows:

(4) We assume that all capillaries in this setup are identical, equal to R . Thus and equation (4) becomes

(4’) Calculate from equation (4’) and compare it with your value from Table 4. The 2 mm capillaries in parallel (middle and bottom): Make your measurement with the top valve closed and the middle and bottom valves opened. Enter the valve positions –open/closed – for the top, middle and bottom valves into your worksheet. Enter your data and the calculated values for flow rate Q, pressure difference P, and resistance R into Table 5. For the parallel connection, for the theoretical effective resistance holds: (5)

We assume again that all capillaries in this setup are identical, equal to R . Thus and equation (5) becomes

(5’)

Calculate RP from equation (5’) and compare it with , your value from Table 5.