Viscous Flow©98

Experiment 12

Objective: To measure some properties of fluid flows and interpret the results using the continuity equation, Bernoulli’s equation, and Poiseuille’s Law.

DISCUSSION:

Steady flows are driven by forces that are balanced by resisting forces. For instance, the amount of water coming out of a shower depends on the water pressure as provided by private or municipal water systems, and the resistance to flow of the many small holes in the shower head.

A graph of the height, h, of the water level above the head of the pipe vs. time, t, illustrates the variation in the flow rate as a function of time. The volume flow, Vf, during a time t is given by Poiseuille's Law

(1)

where L is the length of the pipe, d is the diameter of the pipe, is the coefficient of viscosity, P2 (= Patm), is the atmospheric pressure at the open end of the pipe, and P1 is the pressure at the opening of the pipe inside the bottle, as seen in Figure 1. Neglect for the moment the contribution the pressure differential makes towards increasing the kinetic energy of the fluid. Then P2=Patm, atmospheric pressure, while P1 is the pressure at the head of the pipe. This pressure is given by

(2)

where is the density of water and h is the height of the water level above

the hole. Thus P1 - P2 = gh. Note the rate of change of height is negative, . The

rate of decrease of volume inside the bottle is given by.

(3)

where A is the cross-sectional area of the bottle. The continuity equation states that the rate of decrease of fluid volume in the bottle must equal the negative of the rate of fluid volume that flows out of the pipe, i.e.

(4)

Using Eq. (3) for the flow in the bottle, and Eq. (1) for the flow in the pipe, Eq. (4) becomes

12-1

Integrating this equation gives

(5)

so,

(6)

where

(7)

So the viscosity can also be found from the slope of the length vs. time constant graph,

(8)

EXERCISES FOR VISCOUS FLOW:

Apparatus Setup:You have a cylindrical plastic bottle, 3 beakers, 3 lengths of tubing and one

aluminum pan and some plumber’s putty. You should also have a, thermometer, and a stopwatch. The set up should be done inside the pan to help prevent water spillage. Place the bottle on top of one of the beakers, placed upside down. Insert the appropriate length of tubing into the spigot at the bottom of the bottle. Arrange the tubing so that it will empty into another beaker.

Data Acquisition:

12-2

You should use the attached blank data sheets with their six columns. There is space for recording the length of the tube, diameter of the tube, room temperature, water temperature, diameter of the bottle, and notes. There is one column for the head (water level), three columns for three sets of time measurements, one for the calculated average time, and one for the standard deviation of the average time. Record your estimate for the error in measuring each type of quantity in the space for notes.

1. Fill the remaining beaker with water as close to room temperature as possible as indicated on your thermometer.

2. Record room and water temperatures. 3. Measure and record the diameter of the

bottle. The tube’s inner diameter is 2mm.

4. Fill the bottle and observe what happens as the water drops.

5. Once the water has stopped flowing refill the bottle.

6. Call out as the water level passes the top mark and each of the lower marks.

7. Write down the time to the nearest second as the level passes each mark.

8. Repeat this 2 more times, re-using water from the collecting cup each time.

Note: At some level the nature of the flow changes; it becomes a series of drops, a dribble, and you can stop timing.

Repeat the procedure two more times using the two other lengths of plastic tubing.

Data Analysis:

You should now have three sets of data, each with three time measurements for each of the water levels. For each separate set of data, do the following:

1. Enter all the data into an Excel spreadsheet. Convert to SI units (mks)!

2. Find the average time, tave, for each water level, h.

3. Plot the head, h, vs. average time tave.

4. Fit a decreasing exponential to each data set and display the equation on the graph (your TA can help you with this if you don’t know how). This graph and its

12-3

equation correspond to Eqs. (8a) and (8b). The decay time is a fit parameter and each data set will give you a different value for .

5. Be sure to repeat the previous steps (1 through 4) for all three straw lengths. You should have 3 separate graphs with three different values of .

6. Now, make a list of the tube lengths, L and another list of the corresponding values. Make a fourth graph, plotting L vs. . Fit a line to these points.

7. Use Eq. 8, the slope of the L vs. graph and the other necessary values to calculate the viscosity.

8. Compare this value to the theoretical value based on the temperature dependent information shown in the table below.

T(oC) - Viscosity of Water (kg m/s)10 1.31 x 10-3

20 1.006 x 10-3

Final Note: This lab, developed at MIT, is from an NSF Faculty Enhancement Conference, Harvard University, June 15-26, 1998.

12-4

Data Sheet for Viscosity

Trial 1

Length of tube = ______________ Diameter of tube = _______________

Diameter of bottle: __________________

Room Temperature: Start______ Finish______

Water Temperature: Start______ Finish______

WaterLevel

T1 (sec) T2 (sec) T3 (sec)

Notes:

12-5

Data Sheet for Viscosity

Trail 2

Length of tube = ______________ Diameter of tube = _______________

Diameter of bottle: __________________

Room Temperature: Start______ Finish______

Water Temperature: Start______ Finish______

WaterLevel

T1 (sec) T2 (sec) T3 (sec)

Notes:

12-6

Data Sheet for Viscosity

Trial 3

Length of tube = ______________ Diameter of tube = _______________

Diameter of bottle: __________________

Room Temperature: Start______ Finish______

Water Temperature: Start______ Finish______

WaterLevel

T1 (sec) T2 (sec) T3 (sec)

Notes:

12-7