Fluid Mechanics Laboratory Peter Love, Physics Department, Haverford College © 2011 Introduction Fluids – the generic term for substances that can flow (i.e., gases and liquids) – are everywhere in the world around us: from the atmosphere that sustains us, to the plumbing systems in our homes and workplaces, to the blood that flows through our circulatory systems. Analyzing fluid behavior with high accuracy is a great challenge, but a good general understanding of the behavior of fluids is possible using relatively simple theoretical constructs and experimental apparatuses. In this lab, we’ll explore the basic properties of fluids at rest (the field of hydrostatics) and in motion (hydrodynamics). Experiment 1: Equations of State In fluids, pressure, temperature and volume are related to each other by an equation of state. If you have studied basic chemistry, you may be familiar with the equation of state of an “ideal gas” – a gas whose molecules interact only by (rare) direct collisions rather than through long-range forces. The equation of state for such a gas is known as the ideal gas law, which appears as PV = nRT = nkT. (1) Here P is pressure. The unit of pressure is Newtons per square meter (N/m2), and one Newton per square meter is called a Pascal, and 1000 Pascals is 1 Kilopascal, or kPa. These are the units measured by the pressure sensor. V is volume (in m3), n is the number of moles of atoms or molecules in the volume (where 1 mole = 6.02 × 1023 molecules = Na, which is Avogadro’s number), N is the actual number of atoms or molecules, T is the temperature (in Kelvin, K), k is “Boltzmann’s constant” (k = 1.38 × 10- 23 Joules/K) and R is the “gas constant.” The gas constant is related to Boltzmann’s constant by R = Na k. [The Joule (J) is a unit of energy, related to force by 1 J = 1 N · 1 m = 1 N· m.] The ideal gas law shows, for example, that if a fixed amount of gas is compressed at constant temperature then PV is a constant. Processes at constant temperature are called isothermal. When you compress a gas you perform work on it and its temperature will tend to rise (which is why bicycle pumps heat up when you use them). An isothermal expansion or compression must be performed slowly, so that the heat added to the gas can leak away and the temperature of the gas remains the same as its surroundings. On your desk you should find the “Gas Law Apparatus” that will allow you to explore the equation of state of air. This Apparatus is shown above. We don’t know the number of moles or molecules of air in the piston, so n and N are unknowns. But they should remain constant during the compression process. Start with the valve open and raise the plunger to near the top of its range; then close the valve. Slowly turn the screw to move the plunger down, and record the pressure and volume at convenient points. If you change things gradually enough, the temperature should remain essentially constant (at the ambient air temperature; variation by a few tenths of a degree isn’t significant). You can determine whether the air is accurately modeled by the ideal gas law by confirming that the product PV is a constant. Plot P as a function of 1/V and perform a power law fit (an Allometric fit in Origin under non-linear curve fitting). Print out this plot showing the fit. Complete the section in the report form for Experiment 1. Experiment 2: Hydrostatic pressure in Fluids The hydrostatic pressure at the bottom of a column of fluid of height h is p = ρgh where p is the pressure, g is the gravitational acceleration on the surface of the earth and ρ is the fluid density. In this experiment you will verify this relationship by measuring the pressure at varying heights in a narrow column of fluid. The apparatus to do this is shown at the left. Connect this apparatus to the fluid resevoir – first making sure that all the valves are closed. The valves in their open and closed configuration are shown below: Valve open Valve Closed Open the valve at the very bottom of the tube to fill the column with water, being careful that water that flows out goes into a graduated cylinder, then close the valve again. Next take six pressure measurements using the pressure sensor – attach the pressure sensor to the valve, open the valve, take the pressure reading from Logger Pro (you can just read it off, no need to collect data), then close the valve and remove the pressure sensor. Be careful not to get water in the pressure sensors. Measure the height from the surface of the water to the T junction in the column. Plot P against h, attach the plot and complete the section in the report form for Experiment 2. Experiment 3: Measuring viscosity in Poiseuille Flow As you have seen in the case of terminal velocity and various experiments where the effects of air resistance are evident, fluids resist motion through them. This tendency to resist motion varies from fluid to fluid. Oil and honey are harder to move through than water. Viscosity is a material property that is a measure of the internal friction of fluids, how much they resist flow. For flow in a pipe, a pressure difference is required between the ends of the pipe in order to produce a flow. The fluid flows fastest in the center of the pipe, the flow velocity is zero next to the walls and the flow profile is a parabola, as shown below. The flow rate is the total volume of fluid per unit time that comes out of one end of the pipe. As the pressure difference increases, the flow rate increases. For Poiseuille flow, the flow rate is related to the pressure difference by the Hagen-Poiseuille equation: (3) Where is the flow rate, P is the pressure difference between the ends of the pipe, L is the length of the pipe, r is the radius of the pipe and is the viscosity. The units of viscosity are Pa s, or N s /m2. The viscosity of water is 10-3 Pa s at 20o C. Notice that for lower viscosity you get a larger flow rate for the same pressure difference. On your lab bench there is a Poiseuille viscometer – a pipe connected to a reservoir of fluid (the fluid is water with food coloring in it). There are two points at the end of the pipe where you can attach the pressure sensor so that you can measure P. You will measure flow rate by taking a video of the fluid filling a graduated cylinder and analyzing the video in VideoPoint, just as you did for the collisions lab. Make these measurements one after the other – don’t try and do everything at once. Procedure – Pressure measurement 1. In Logger Pro set the data collection parameters to collect ten samples per second for twenty seconds. 2. Connect the gas pressure sensors to the measurement points and open the valves to connect the pressure sensors to the flow line. 3. With the end of the flow line pointing in the bucket, open the valve at the end of the flow line so that the fluid begins to flow. 4. Collect a set of pressure data. 5. Stop the flow by closing the valve at the end of the flow line 6. Create a new calculated column that gives the pressure difference. 7. Record the mean and standard deviation of the pressure difference using the “statistics” option under analysis in logger pro. Procedure – Flow rate measurement 1. Make a video of the graduated cylinder filling up with fluid. 2. Edit the video in Videopoint Capture so that you have the graduated cylinder filling over a 30mL range 3. Select one frame out of every 15 so that you have around 50 frames. 4. Save the video and open in VideoPoint 5. Track the height of the liquid in VideoPoint and use the ruler function to convert to mL using the scale visible in the video (there is no mL unit available so just choose meters) 6. Import your data into Origin and perform a linear fit to find the flow rate. Once you have your flow rate and pressure difference and errors, measure the distance between the pressure measurement points. The radius of the pipe is 3mm. Download the data analysis spreadsheet from blackboard – this will propagate the errors for you and give you a viscosity measurement and standard deviation. Complete the section of the report form for Experiment 3 Experiment 4: Low pressure experiments Perhaps the most obvious characteristic of a fluid (besides its density) is its pressure, P, defined as force per unit area: P = F/A. The unit of pressure is Newtons per square meter, and one Newton per square meter is called a Pascal. The layer of atmosphere above us creates a pressure at ground level of roughly 101000 (N/m2), or 101 kPa. That means that each square inch of your body’s surface essentially is supporting approximately 14.7 pounds worth of air. So why don’t we feel the pressure? In certain situations we are aware of atmospheric pressure – for example, when we use a suction cup to hold up some object. In expelling the air from the inside of the cup when it is compressed, we create an imbalance between the pressure of the atmosphere outside and the little remaining air inside. If we could remove all of the air from inside the cup, and it had an area of one square inch, then it could support an object weighing up to 14.7 pounds. On your lab bench you should see the setup shown above – you can connect the pressure sensor and use the syringe to pump air out of the chamber. You should be able to get the pressure down to less than 10 kPa (for comparison, at an altitude of 100km, the edge of space, atmospheric pressure falls to 1 Pa). Demo A – Inflation When you breathe in your diaphragm lowers, expanding your chest cavity and lowering the pressure. Air is forced into your lungs from the atmosphere. To familiarize yourself with the vacuum chamber, perform the following demo. Place a balloon with only a small amount of air in, and with the end tied off, in the pressure chamber. Then evacuate air from the chamber using the syringe. Observe the balloon – what you are seeing is exactly the same physics that causes your lungs to inflate when you breathe in. Demo B – Suction cups don’t suck Suction is simply a pressure differential between one place and another – its better understood in terms of a higher pressure pressing onto something rather than a low pressure ``sucking’’ onto something. To show this, lets remove the higher pressure and observe what happens. Stick the suction cup to the side of the vacuum chamber – then evacuate some of the air. What happens to the suction cup? Demo C – Vapor pressure of water The temperature at which a liquid boils is dependent on pressure. In the report form is a plot showing how the boiling point of water depends on pressure. In this experiment you will find the pressure at which water that is below 100 degrees Celcius boils. Take some hot water in the small plastic cup and measure its temperature. Then place it in the vacuum chamber with the pressure sensor attached and pump out the air. Record the pressure as a function of time and mark when the water starts to boil. Complete the section in the report form for Experiment 4.