(ta initials) first name (print) last name (print) brock id (ab17cd) (lab date) Experiment 5 Viscosity and drag In this Experiment you will learn that the motion of an object through a fluid is governed by a complex set of parameters; that the Reynolds number R and drag coefficient C d are central to the physics of fluid dynamics; to determine the drag coefficient of a sphere by performing a computer analysis of the velocity curve; to visualize data in different ways in order to improve the understanding of a physical system; to apply different methods of error analysis to experimental results. Prelab preparation Print a copy of this Experiment to bring to your scheduled lab session. The data, observations and notes entered on these pages will be needed when you write your lab report and as reference material during your final exam. Compile these printouts to create a lab book for the course. To perform this Experiment and the Webwork Prelab Test successfully you need to be familiar with the content of this document and that of the following FLAP modules (www.physics.brocku.ca/pplato). Begin by trying the fast-track quiz to gauge your understanding of the topic and if necessary review the module in depth, then try the exit test. Check off the box when a module is completed. FLAP PHYS 1-1: Introducing measurement FLAP PHYS 1-2: Errors and uncertainty FLAP PHYS 1-3: Graphs and measurements FLAP MATH 1-1: Arithmetic and algebra FLAP MATH 1-2: Numbers, units and physical quantities WEBWORK: the Prelab Viscosity Test must be completed before the lab session! Important! Bring a printout of your Webwork test results and your lab schedule for review by the TAs before the lab session begins. You will not be allowed to perform this Experiment unless the required Webwork module has been completed and you are scheduled to perform the lab on that day.! Important! Be sure to have every page of this printout signed by a TA before you leave at the end of the lab session. All your work needs to be kept for review by the instructor, if so requested. CONGRATULATIONS! YOU ARE NOW READY TO PROCEED WITH THE EXPERIMENT! 32
33 Viscosity and drag Drag force arises when an object moves through a fluid or, equivalently, when fluid flows past an object. In general, the drag force grows larger with increased flow velocity, but viscosity is a complex phenomenon that cannot be reduced to the simple relationship drag force is proportional to velocity. The origin of the drag force F d lies in the need to displace the particles of the fluid out of the way of a moving object. At low velocities the movement of the fluid is smooth (laminar) and the faster the object moves, the greater is the amount of fluid that has to get out of the way, giving rise to the linear relationship F (viscous) d = aηv where v is the velocity of the object relative to the fluid, η is the coefficient of viscosity, and a is the size of the object; for a sphere of radius r, a = 6πr. The larger the size, the greater is the amount of fluid that needs to get out of the way, hence it is not surprising that the relationship is linear with respect to a as well. The story gets far more complicated for higher velocities, unusual object shapes, of unusual fluids. At higher velocities or around oddly-shaped objects the flow stops being smooth and turns turbulent, which requires more energy and causes the drag force to switch to the quadratic regime, where F d v 2, F (inertial) d = S ρ 0v 2. 2 Note that this expression represents an inertial rather than a viscous force, and instead of the the viscosity, η, the fluid density, ρ 0, enters the formula. S is the cross-sectional area of the moving object. An empirical parameter called the Reynolds number determines which of the above two terms dominates: R = dρv η where for a simple sphere, the diameter d(= 2r) is the obvious definition of the size of the particle 1. For R < 1, the viscous drag dominates (and thus there is no inertial coasting), while for R > 1000 the viscous drag is completely negligible as compared to the inertial drag. At higher velocities still, in compressible fluids one needs to account for the compression waves that get created (e.g. the sonic boom of a supersonic aircraft), and in incompressible fluids the surface layer of the fluid may lose contact with the rapidly moving object creating local vacuum (e.g. cavitation on the surface of the blades of naval propellers). Unusual fluids such as solutions of long polymers may experience flow in directions other than the direction of motion, as the entangled polymer molecules transmit the movement of the object to remote areas of the fluid and back. The fluid constrained into finite-size channels and tubes also behaves differently from an infinitely-broad medium through which a small-size sphere is traveling. There are corrections that must account for all of these details. A convenient way to express just such a correction is through an empirical factor called the coefficient of drag, C d, which can be measured experimentally and account for the details of shape, size, and the nature of the media through which an object is traveling. This dimensionless coefficient is well-known and its tabulated values can be found in handbooks. As a function of the Reynolds number, the experimentally measuredvaluesofthedragcoefficient, C d, constitutetheso-called standarddragcurve showninfig. 5.1. For a small sphere traveling through an infinitely broad fluid, this C d provides an excellent agreement with the experimental drag force: F d = K f C d S ρ 0v 2 2, C d = C d (R) (5.1) Thefinite-sizecorrection coefficient, K f, describeshowthisequationismodifiedbytheboundaryconditions of a fluid confined in a finite-size vessel. For a sphere of diameter d traveling axially through a fluid channel 1 For other shapes, an equivalent hydrodynamic radius r h is used instead of r
34 EXPERIMENT 5. VISCOSITY AND DRAG Figure 5.1: Drag coefficient on a sphere in an infinite fluid, as a function of the Reynolds number (apipe)ofdiameter D, adimensionlessratio x = d/d canbeusedtoapproximatethiscorrection coefficient to within 6% via 2 1 K f = 1 αxα, α = 1.60 for x 0.6, (5.2) in the range of Reynolds number values of 10 2 < R < 10 5. For an object at rest, there is no drag force. As we apply a force to an object, it experiences an acceleration, its velocity grows, and with it grows the drag force that is directed opposite the velocity. This drag force counteracts the applied force and reduces the net force and the object s acceleration. Eventually, velocity increases to the point where the drag force is exactly matched to the applied force, bringing the net force to zero. From this point on, the object is in equilibrium, there is no acceleration, and the velocity remains constant, equal to its steady-state value called the terminal velocity, or v t. For example, if an object is in a free fall through a fluid, this equilibrium condition corresponds to the weight of the object exactly equal to the drag force: F d = W. When the fluid in question is a gas such as air, the buoyant force F b equal to the weight of the displaced gas is tiny and can be neglected. When the fluid is a more dense substance such as water, the above equation may need to be modified to include the buouyant correction due to the displacement of the fluid by the object F d = W F b = m object g m fluid g = m object g(1 ρ 0 /ρ) (5.3) where ρ is the density of the material of the object and ρ 0 is the density of the fluid. As you can see, only the ratio of the two densities enters into the expression since the volumes of the object and of the fluid it displaces are the same. For example, for an Al sphere in water, ρ 0 = 1.00 10 3 kg/m 3 and ρ = 2.70 10 3 kg/m 3. 2 R.Clift, J.R.Grace, and M.E.Weber. Bubbles, Drops, and Particles, p.226. Academic Press, 1978
35 Apparatus The experimental apparatus consists of two different-size metal spheres, a long cylindrical container filled with water, a pulley system with a weight platform and variable weights, an ultrasonic position sensor, a digital weight scale, a micrometer. From the free-body diagram of Fig.5.2, drawn for the case of the sphere traveling downward as appropriate for the small pull mass of the counterweight, when v = v t = const, Denoting the pull mass of the counterweight as m, the mass of the weight platform as m p, the mass of the sphere as m s, and using the expressions of Eqs. 5.1 and 5.3 we obtain K f C d S ρ 0v 2 t 2 ( = m s g +T W counterweight W platform = 0, F d +F b +W counterweight +W platform W sphere = 0. 1 ρ 0 ρ Dividing through by g we obtain ) m p g mg m = m 0 K fc d Sρ 0 2g v 2 t, (5.4) where m 0 = m s (1 ρ 0 /ρ) m p. Procedure Figure 5.2: Drag force measurement apparatus In this experiment you will evaluate the drag coefficient for two different spheres using the setup in Fig.5.2. Using a digital scale, weigh separately both spheres and the weight platform. As you weigh different weights from the weight set, you may notice small differences in the measured weight of weights that have the same nominal value. W small =...±... W large =...±... W platform =...±... W 1 g =...±... W 2 g =...±... W 2 g =...±... W 5 g =...±... W 5 g =...±... W 10 g =...±... W 10 g =...±...
36 EXPERIMENT 5. VISCOSITY AND DRAG W 20 g =...±... W 20 g =...±...? Is the variation in the measured weight between weights of the same nominal value significant? How does it compare with the precision of the measuring instrument, the digital scale? Use a Vernier caliper or a micrometer to measure the diameter d of the two spheres: d small =...±... d large =...±...? Are these two spheres spherical within the limits of your measuring instrument? How would you check this experimentally? Measure and record the inner diameter D of the plastic cylinder filled with water. Verify that the cross-section is circular by measuring the diameter in a few different directions across the cylinder. D =...±... Part 1: Determination of C d for the small sphere Select the small sphere, adjust the length of the string so that the weight platform is at around 30 cm above the ultrasonic detector at its lowest position, and a few cm below the pulley at its highest position (i.e. when the sphere hits the bottom of the cylinder). Determine the range of masses from m 1 to m 2, in 1g resolution, for which the sphere does not move and hence v = 0. This result is related to the drag of the pulley system, typically spanning a few grams. Enter these two results (m 5,0) and (m 6,0) in Table 5.1. To complete Table 5.1, start with a value of m 1 = 0 and increase m to get four reasonably spaced data points in each direction of motion about the region where v = 0. Note that the sign of the terminal velocity differs depending on the direction of travel. trial, i 1 2 3 4 5 6 7 8 9 10 m i, pull mass v t,i, small sphere 0 0 Table 5.1: Terminal velocity data for small sphere? You may have noticed that there are small differences in the measured weight of weights that have the same nominal value. Do you need a precise value for the pull mass m? Do you need an estimate for (m), the uncertainty in the value of the pull mass? Close any open Physicalab windows, then start a new session by clicking on the Desktop icon. Login with your Brock email address. Your graphs are sent there for later inclusion in your online lab report. In Physicalab, check the Dig1 input channel to select the rangefinder as the data source, choose to collect 50 points at 0.025 s per point, then select scatter plot.
37 Place a mass m on the platform and move the sphere to the starting point of the travel regime. Press the Get data button, then release the string to record a trace of the sphere position in the tube as a function of time. Your graph should display a upwardly curved region when the sphere is accelerating, followed by a linear region of constant slope near the end of the motion when the sphere is moving at a constant speed. The trajectory ends with a roughly horizontal line that represents the motion of the ball after it is suddenly stopped when it reaches an end of the tube. If the sphereflutters or wanders erratically as it moves along the tube, repeat the run. Do not include any trials where the sphere fails to reach terminal velocity; repeat the trial using a different pull mass. Note that the sign of the slope, and thus of the terminal velocity v t that you record, changes as the sphere switches directions of travel. Click Send to: to email yourself a copy of the graph. You should save a copy of all the trials for later analysis, but you only need to include one sample graph in your report. For each trial, identify the region of terminal velocity from x = X min to x = X max at the end of the sphere trajectory where the position-time graph is a straight line. A series of 8-10 points is sufficient, if properly chosen.? Did you select only the region of your graph just before the end of the trajectory where the data should be changing at a constant rate and thus fit to a straight line? Use A*(x-B)*(x>X min )*(x<x max ) to fit a straight line segment to your data in the limited range X min <x<x max. To get a correct error value, you must also enter (x>x min )*(x<x max ) in the constraint box.? How well was the fitted straight line segment placed over your data points? What would cause a mismatch between our data and the fitted line segment? If the fitted line segment did not overlap the data points completely, redo the fit using more appropriate limits. Following a good result, record the terminal velocity v t as a function of m in Table 5.1. Enter the values of (v t,m) for the small sphere into Physicalab and click Draw to make a scatter plot of the data set. The relationship between m and v t is given by Equation 5.4. Fit your data to the equivalent quadratic expression, A+B*sign(x*x,x), where the function sign() here accounts for a change in the direction of the drag force as v t changes sign. Save a properly labelled graph of your resulting fit of the small sphere data.? How does your fit look, especially in the region where v t = 0? How might you explain this mismatch between the velocity data and the fitting equation? As you previously noted, there is a region over several grams of mass where v = 0. To account for the effect of this drag on the system, repeat the previous fit, and use the following equation: (A+B*x**2)*(x<0)+((A-C)-B*x**2)*(x>0) with constraint ((x<0)+(x>0). The added fit parameter C compensates for the offset in the data set at v = 0 and hence this value represents the approximate drag of the pulley system. Save the graph and record the fit parameters below: A =...±... B =...±... C =...±... Determine values for the cross-sectional area S, the sphere to pipe diameter ratio x, the finite-size correction coefficient K f and their associated errors (S), (x) and (K f ):
38 EXPERIMENT 5. VISCOSITY AND DRAG S =... =... =... (S) =... =... =... x =... =... =... (x) =... =... =... K f =... =... =... (K f ) = α2 x α 1 x (1 αx α ) 2 =... =... Extract from Equation 5.4 the term corresponding to the fit parameter B, then rearrange this term to solve for C d and finally calculate a value for the drag coefficient C d and error (C d ) for the small sphere: B =... C d =... =... =... (C d ) =... =... =... Part 2: Determination of C d for the large sphere trial, i 1 2 3 4 5 6 7 8 9 10 m i, pull mass v t,i, large sphere 0 0 Table 5.2: Terminal velocity data for large sphere Repeat the procedure of Part 1, using the large sphere. Save all the graphs, then summarize the fit results from your graph that included the drag term C, A =...±... B =...±... C =...±...
39 S =... =... =... (S) =... =... =... x =... =... =... (x) =... =... =... K f =... =... =... (K f ) =... =... =... C d =... =... =... (C d ) =... =... =... Part 3: Summary of results Summarize in Table 5.3 the correctly rounded results from Parts 1 and 2: with the correct units: Table 5.3: Drag coefficient results small sphere large sphere S ± S x± x K F ± K f C d ± C d C± C Finally, calculate an average value for the drag coefficient C d and the pulley drag given by the fit parameter C: C d =...=...=... C =...=...=...
40 EXPERIMENT 5. VISCOSITY AND DRAG IMPORTANT: BEFORE LEAVING THE LAB, HAVE A T.A. INITIAL YOUR WORKBOOK! Lab report Go to your course homepage on Sakai (Resources, Lab templates) to access the online lab report worksheet for this experiment. The worksheet has to be completed as instructed and sent to Turnitin before the lab report submission deadline, at 11:00pm six days following your scheduled lab session. Turnitin will not accept submissions after the due date. Unsubmitted lab reports are assigned a grade of zero.
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