Motion through fluids

Transcription

1 Motion through fluids References: 1. J. P. Owen, W.S. Ryu: The effects of linear and quadratic drag on falling spheres: an undergraduate laboratory, Eur. J. Phys. 6 (005) P. Nelson: Biological Physics, Updated 1 st. ed., Freean (008), Ch. 5. Introduction In swiing bacteria or diffusing proteins, viscous rather than inertial forces doinate the dynaics of otion. A coon easure of the ratio of the inertial to viscous forces is known as the Reynolds nuber: l v R e (1) Where ρ is the fluid density, ν is the velocity of the object, l is a characteristic length of the object, and η is the fluid viscosity. Our everyday experience is ostly with high Reynolds nuber environents where inertial forces doinate. Swiing, for exaple, is a high Reynolds nuber activity. We propel ourselves through the water by accelerating the fluid behind us; the inertial force fro a single stroke lets us glide eters before we coe to a stop. Low Reynolds nuber activities are less coon, but stirring a jar of honey with a spoon is one exaple. It's the viscosity of the honey and not the ass of the honey that akes the stirring difficult. When you let go of the spoon, does it continue to swirl around the jar? No, the spoon stops oving fairly quickly. The viscous force doinates the inertial force. Swiing can be a low Reynolds nuber activity when the length scale of the swier is sall. Microorganiss fit this category. A bacteriu such as E. coli, is about one icron (10-6 eters) in diaeter and travels around 0 μ per second, so swiing bacteria have a Reynolds nuber uch less than one and the viscous forces doinate inertial forces. To us, this is a very alien hydrodynaic world. For you to swi at an equivalent Reynolds nuber,you would need to "swi" in soething viscous like honey, at speeds of about a foot a day, while cycling our ars at about 1 stroke per hour. Theoretical background Inertial forces (F = a) are failiar, but what are viscous forces? Iagine you have a fluid between two plates. Intuitively you know that as the viscosity of the fluid increases, it requires ore force to slide the plates apart (think water versus honey). Now assue that the botto plate is fixed while the top plate, at soe distance l, is free to ove parallel to the fixed plate (see Figure 1). 1

2 Figure 1: Fluid between oving plates: Force/area = viscosity*velocity/length If a force F is applied to the top plate, it will ove at soe velocity ν, foring a velocity gradient between the top and botto plates. As the viscosity increases, it will take a larger force to for the sae velocity gradient. It is this proportionality between the force per area (also known as shear stress) and the velocity per length (shear rate) that is known as viscosity. Fro Figure 1, we can define the viscous force (F v ): F v Av l Inertial forces are in the for F i = a and for a volue of fluid V with density ρ, this can be written as: () () F i V a l Av t (3) Taking the ratio gives the Reynolds nuber: Fi l v (4) Re Fv Equations of otion In this experient, you will be following the otion of objects falling in fluids. Check out the force diagra for a sphere falling in a fluid (Fig ). Figure : A falling sphere, where g is the weight of the object, g is the buoyant force, and F d is the drag force.

3 The equation of otion is: ( ' ) g F M ' g d F d (5) Where F d is the drag force and M' is the effective ass of the object corrected for buoyancy. When the object reaches its terinal velocity (a = dυ/ = 0) and: M ' g F d (6) The for of F d depends on the Reynolds nuber. At a high Reynolds nuber, the drag force is coonly written as: 1 Fd Cd Av (7) The drag coefficient C d is easured epirically, ρ is density of fluid and A is cross-sectional area of the object, perpendicular to the direction of otion. Terinal velocity is this regie can be obtained fro the equation of otion: 1 g C d Av (8) The equation of otion can be solved analytically by separation of variables. The general for of the equation is: g cv Solution takes the for: gt v( t) v ter tanh vter Where terinal velocity v ter is given by: (9) (10) v ter g Cd A 1/ (11) Integrate (9) and solve for υ(t). Verify that solution fro (10) is correct. To solve for the constant of integration, use the initial condition, υ(0) = 0. Terinal velocity in the high Reynolds nuber regie depends on the bead radius as: v ter g C d A 1/ r r 3 1/ r 1/ (1) 3

4 The derivation of drag force at a low Reynolds nuber is not as straightforward as the high Reynolds nuber case. However, fro the discussion of viscous forces above, one can see that F d is generally proportional to η, l, and ν. Soe specific cases have been solved exactly, such as the drag on a sphere of radius r oving at velocity ν in a fluid with viscosity η. This is known as Stokes drag: F d 6vr (13) In the following experients we will be working with objects of constant density but of different size. How should the terinal velocity vary as a function of linear diension? Fro the force equation at low Reynolds nuber, the terinal velocity of the sphere is: v ter M ' g 6 r (14) The assuption here is that the sphere's velocity is relatively constant throughout its fall. Let's check if this is expected. For an object falling at low Reynolds nuber, the equation of otion is (ignoring the buoyant force for now): g 6 rv (15) The general for of the equation of otion is: g bv (16) Separate variables and integrate. The solution is: g t / v( t) 1 e (17) 6 r Where 6 r Integrate and solve for ν(t). Verify that solution fro eq. (16) is correct. Qualitatively, we can see that the solution has the correct initial and asyptotic behaviour: At t = 0, ν = 0. At t =, ν = g/6πηr, the expected terinal velocity. How quickly the syste reaches the terinal velocity depends on the tie constant τ. Estiate τ for a 1 diaeter aluinu sphere in glycerine, r = 0.5 x 10-3, η = 1500cp, (1cp = 1centipoise = 10 - Poise), ρ =.7 g/c 3 (Aluinu). Terinal velocity in the low Reynolds nuber regie depends on the particle radius as: v ter M ' g 6 r r r 3 r (18) 4

5 The experient Notes about the procedure By easuring the vertical speed of different sized spherical particles in glycerol or water, you will deterine how the terinal velocity scales with the radius of the sphere and the viscosity of the fluid. In order to find out how fast the terinal velocity is setup, estiate τ. You will use a video tracking ethod. Open the LabView application Motion through Fluids (shortcut on the coputer desktop). Confir the default caera. The progra asks you to select the nuber of fraes to be recorded. Do the trial experients with 10 fraes; adjust the nuber according to your experiental conditions later on. The frae rate eans how any fraes per second. Try 0 (this will cover 6 seconds when cobined with 10 fraes), adjust it later. You have to provide a location for saving the *.avi file (ovie of the falling particle). Be ready to drop the bead: suberge the tweezers + bead in glycerol in order to avoid surface tension probles. Click on Start the avi ovie capture. You ll hear two short warning beeps followed by a long one: at the long one release the bead. The application will output a text file with tie (in seconds) and position (in ) of the falling bead. Correction for the wall effect* We have already discussed the forces acting on a sphere falling in a liquid. There is a gravity force, a buoyancy force, and a drag force. For the calculation of the drag force, we used Stokes law, which assues the sphere to be oving in an unbound or infinite fluid. Objects falling near a boundary (like the wall of a container) fall ore slowly than an object falling far fro a wall. You can see this for yourself by siultaneously dropping two identical spheres at the sae tie: one near the wall and one at the center of the container (try it!). To account for the effect of the container wall on the otion of the falling sphere, a correction factor has been deterined. The velocity correction for a sphere falling in the center of a fluidfilled cylinder is: v corr v d d D D (19) Where ν is the experientally easured ean velocity, d is the diaeter of the sphere, D is the diension of the container, perpendicular to the fall direction, and ν corr is the expected velocity of the sphere if it were falling in an unbounded fluid. *This is derived fro experiental easureents and taken fro Loatzsch, T., et al. Conceptual study of an absolute falling-ball viscoeter, Metrologia, 001, 38 ( ). Estiate the correction for all sizes. 5

6 Exercise 1: Low Reynolds nuber You ll receive a box arked Glycerine with five different sizes of spherical beads ade of Aluinu or Teflon. Beads range fro 1.94 to 6.35 in diaeter. A rectangular container (9.5c 9.5c 31.0c) filled with glycerol will be provided. The container is placed in front of a dark chaber. The video caera is located at the rear of the chaber. Test your setup by dropping beads and tiing the fall using a stopwatch. This would give you a hint about the nuber of fraes and fraes/second. Do -3 easureents for each bead size. Measure the diaeter of each size, using a caliper. Calculate the ean velocity for each bead size (this is the ean terinal velocity). Correct the data for the wall effect. Calculate the Reynolds nuber. Does the value atch the low R e condition of your proble? Using Python, plot the ean terinal velocity of the spheres in glycerine as a function of radius. Fit the data using eq. (18). The slope of the plot should be close to 1. Did you notice any discrepancy? Exercise : High Reynolds Nuber You will also receive a box with Nylon beads arked Water. Measure the diaeter of each bead. Replace the large container arked Pure Glycerine with the container arked Water. Please take care not to spill glycerine (it is essy and slippery). Make sure you reove all the air bubbles fro the container walls. Test your setup again by dropping beads without the tracking software to estiate the fall tie. As you will notice, the fall of larger particles is not in a straight line. Soeties, they wobble due to water turbulence. Practice until you get the best trajectory. Take 5 easureents for each bead size using the tracking progra. Calculate the ean velocity for each bead size. Can this be interpreted as terinal velocity? Correct the data for the wall effect. Calculate the Reynolds nuber in each case. Is there a critical velocity (or a critical bead size) that confirs the high R e nuber (R e >00)? Using Python, plot ean velocity of the spheres in water, as a function of the radius. Fit the data using eq. (1). Do you notice any discrepancy with theory? Appendix (Physical Reference Data): Glycerol viscosity = 934 centipoises (cp) or 9.34 g/(c s) at 5 C Glycerol density = 1.6 (g/c 3 ) Viscosity of water 1 5 C Density of water 1 g/c 5 C Aluinu density =.7 g/c 3, Teflon density =. g/c 3, Nylon density 1.1 g/c 3. The experiental setup was built by Larry Avraidis. Larry also designed the LabView tracking progra. This guide was written by Ruxandra Serbanescu in 013. In includes aterial fro Reference 1, which was part of an experient designed by W. Ryu at Princeton University in