Vocabulary: Viscosity Reynolds Number Laminar. Objectives:

Transcription

1 Fluid Mechanics for High School Author: Vishal Tandon Date Created: March 2010 Subject: Physics Level: High School Standards: New York State Physics Standards Standard 1 Analysis, Inquiry, and Design Standard 4 The Physical Setting Standard 7 Interdisciplinary Problem Solving Schedule: One 60-minute Lab Period Objectives: Students will learn about basic concepts in fluid mechanics, including turbulent vs. laminar flow, compressibility, and hydraulic resistance. Students will: Learn about viscosity by comparing glycerin to water Observe mixing at low and high Reynolds number by putting drops of dye into glycerin and water Do an experiment to see the (near) reversibility of low Re flow Determine the difference in compressibility between water and air Apply pressure by hand to actuate flow in 25 and 100 micron capillaries to observe the difference in hydraulic resistance in a tactile way Vocabulary: Viscosity Reynolds Number Laminar Materials: For Each Group (2-4): Water Food Coloring (or dyed water and dyed glycerin) Glycerin (honey or corn syrup can be used) Transfer Pipettes Stirring Sticks Gloves Safety: Compressibility Hydraulic Resistance Turbulent 5 ml Plastic Luer-Lok Syringes Luer-Lok fittings with plugs and PEEK fittings for capillaries (From LabSmith) Large Petri Dish Small Petri Dish 2 Beakers 1-25 µm Silica Capillary (polyimide coated), cut to about 5 cm in length µm Silica Capillary (polyimide coated), cut to about 5 cm in length ( Gloves should be worn at all times to prevent staining of hands. None of the chemicals involved are dangerous. The capillaries are sharp enough to prick skin (but not cut it).

2 Science Content for the Teacher: This lab is intended to give students a basic understanding of different ways in which fluids are characterized, and how flows can change dramatically depending on the nature of the fluid involved, the speed of flow, and the size of the system. Here, we assume that students have no knowledge of fluid mechanics. Fluid motion is described by utilizing continuum versions of conservation of momentum and conservation of mass, leading to the Cauchy Momentum and the continuity equations respectively. ρu x 0 Here represents the full stress tensor, is the fluid velocity vector, is the fluid density, and is the body force on the fluid per unit volume. These are the most general equations, but they simplify in certain situations (Newtonian fluid, incompressible flow, laminar flow, steady state, etc.) Formally, viscosity defines how a fluid responds to a shear stress. For Newtonian fluids, we assume that the deviatoric part of the stress tensor is linearly proportional to the velocity gradient tensor, and the viscosity defines the constant of proportionality. This is an excellent approximation for most common fluids (all of the fluids used in this lab). For Newtonian fluids, viscosity is associated with the thickness of the fluid, and the resistance to flow, making it somewhat analogous to friction for motion of solids. Students should gain this intuitive understanding of viscosity by going through the lab. The Cauchy momentum equation simplifies if the fluid is incompressible, i.e. its density is constant and uniform. Most common working liquids, such as water, are essentially incompressible, whereas gases, such as air, are compressible. For a steady, incompressible flow, the continuity equation becomes: x u 0 The momentum equation reduces to the steady-state, incompressible Navier- Stokes equations (for a Newtonian fluid). In non-dimensional form; - 2

3 1 Where non-dimensionalized parameters are indicated by * s, the velocity has been normalized by a characteristic velocity in the system, and distances have been normalized by a characteristic length scale. Re, the Reynolds number, is a non-dimensional parameter that arises from this analysis. It is defined as Where is the fluid density, U is a characteristic (sometimes the maximum) velocity of the system, L is a characteristic length scale, and is the kinematic viscosity of the fluid. Solving the Navier-Stokes equations for pressure-driven flow in a pipe shows that the flow rate for a given applied pressure gradient is inversely proportional to the fourth power of the radius of the pipe. Students will see the powerful dependence of flow rate on channel size in a tactile way when they work with the microfluidic capillaries in this lab. The Reynolds number can be understood as the ratio of inertial forces to viscous forces; at high Reynolds number inertia dominates, fluids tend to coast (they continue moving even after actuation is ceased), fluids mix, and flow can be chaotic; at low Reynolds number viscosity dominates, flow creeps, and mixing is slow and mainly due to diffusion. The Reynolds number is also an indicator of whether a flow is laminar or turbulent. The flow transitions to turbulence at approximately Re>2000. In this lab, students will examine cases of relatively high and very low Re, by examining water as compared to a fluid with very high viscosity, glycerin. At very low Reynolds number (formally Re = 0), the Navier-Stokes equations are well-approximated by the Stokes equations. 0 1 Two important properties of Stokes flow are: 1) Instantaneity: The properties of the flow do not depend on the time-history of the flow, unless there are timedependent boundary conditions, and 2) Time-reversibility: A time-reversed version of a solution to the Stokes Equations also solves the Stokes Equations, i.e. a flow can be reversed and reverted back to the original configuration by exactly reversing the actuation. Students will explore time-reversibility in this lab by recreating a simplified version of one of a famous experiment from one of G.I. - 3

4 Taylor s lectures ( - see in particular, Low Reynolds Number Flow, about 15 min into the video). Preparation: 1. Photocopy print materials (Fluids_worksheet) 2. Distribute materials evenly to each lab group. 3. Pour chemicals into beakers and petri dishes as indicated in the lab handout. If more than one lab period is available, students can do this themselves (giving them a chance to see the differences in viscosity between the fluids). However, if only one lab period is available, all of the chemicals need to be in the appropriate containers for the experiment prior to the start of the lab period. - 4

5 Classroom Procedure: Engage (Time: 5 min) Allow students to play with the fluids using stirring sticks before they conduct any mixing experiments. Encourage them to discuss and write down their observations of differences between glycerin and water. Explore (Time: 25 min) Exploration and explanation are concurrent in this lab, owing to its many parts. Move students through each section of the lab, encouraging them to stop and observe flows before fully mixing fluids. When doing the annular shear flow experiment, encourage students to draw a design of their choosing, and to look at the effect of distance from the rotating petri dish. Explain (Time: 25 min) At each stage, ask directed questions about their observations (Are the flows the same? How are they different?) The students observations will immediately help them learn many fundamental aspects of fluid mechanics. Guide these observations with additional explanations; in particular point out fluid deformation, shear stress, increasing shear near the moving boundary, and turbulent/convective mixing vs. diffusive mixing. - 5

6 Assessment: The following rubric can be used to assess students during each part of the activity. The term expectations here refers to the content, process and attitudinal goals for this activity. Evidence for understanding may be in the form of oral as well as written communication, both with the teacher as well as observed communication with other students. Specifics are listed in the table below. 1= exceeds expectations 2= meets expectations consistently 3= meets expectations occasionally 4= not meeting expectations Engage Explore Explain 1 Shows leadership in discussion and engages others in making astute 2 Participates in working with fluids and makes 3 Participates in working with the fluids, but does not make any technical Completes experiments and writes down detailed observations of differences between different flows. Makes serious attempt at predicting outcomes. Completes experiments and writes down observations of flows. Completes experiments, but writes minimal Demonstrates understanding of differences between turbulent/laminar, and compressible/incompressible flows. Understands how predicted outcomes differed from the results (if at all). Demonstrates understanding of differences between flows, and documents outcomes. Wrote observations from experiments down, but does not necessarily understand how differences in experimental conditions led to different flows. 4 Does not participate at all. Does not participate at all. Does not participate at all. Supplemental Information: Fluids lectures at are useful. Safety: Gloves should be worn at all times to prevent staining from dye solutions. The capillaries are sharp enough to prick skin. Acknowledgments: Dr. Shivaun Archer, Dr. Chris Schaffer, Nevjinder Singhota - Cornell University. Walter Peck - Whitney Point High School, New York National Science Foundation Cornell GK-12 Program: DGE