Physics 10: 7-14 Fluid Mechanics and Bernoulli's Theorem

Homework

Reading Preparation
Key Equations
WebLecture
Study Activity
Chat Preparation Activities
Chapter Quiz
Lab Work

Reading Preparation

Text Reading: Giancoli, Physics - Principles with Applications, Chapter 10: Sections 8-14

Study Points

10.8: Because the particles are free to move in different directions, individual particles in flowing liquids and gases do not necessarily follow a nice trajectory, sliding easily past one another in layers. Any interference will cause one or more particles to vere off, creating a chain reaction that results in turbulence. Inherit friction or viscosity between the particles is always present, and becomes greater when turbulence occurs. The amount of fluid that can flow through a cross-sectional area A is determined by the density of the fluid and the rate of speed of the molecules: ρ * A * v. Flow of a incompressible fluid (ρ does not change) under constant pressure through a channel of changing diameter requires the speed to change: A₁v₁ = A₂v₂ [The equation of continuity.] As cross-section gets smaller, velocity increases, so that in any identical time interval, the same number of particles pass any point on the channel. If this weren't true, the particles would "back up" like cars in a traffic jam, increasing the density of the liquid -- which isn't allowed in an incompressible liquid.
10.9: Now consider an open system, where pressure is allowed to change. If we change the speed of the particles, density changes and pressure changes. Bernoulli's equation shows the relationship of pressure to altitude as air flows across a slanted surface: P₁ + ½ρv₁² + ρgy₁ = P₂ + ½ρv₂² + ρgy₂. We can now compare specific situations against a standard (unmoving air at sea level) or two different situations.
10.10: We can now apply Bernoulli's theorem to some common situations. One in particular is the flow of water through a small outlet from a large container (such as the flow of water from a reservoir through a dam spillway, or from a storage tank into the main pipe system). In this case, the only pressure is atmospheric pressure, which will be nearly the same at the top and bottom of the contain, so that P₁ = P₂. If the flow of water at the surface (point 2) is small compared to the flow of water through the spigot (point 1), we can assume v₂ ~ 0. Since P₁ and P₂ cancel, ρ cancels, and v₂ can be ignored, Bernoulli's equation simplifies to a form called Torricelli's theorem: ½v₁² + gy₁ = gy_2.We can solve for v₁ = sqrt(2g(y₂ - y₁)).
10.11-14: The remaining sections of this chapter discuss applications of these principles to the speed of flow (viscosity), flow within tubes where friction and drag occur, and blood flow in capillaries and through the heart.

Key Equations

Equation of continuity: $ρ_{1} A_{1} v_{1} = ρ_{2} A_{2} v_{2}$
If the fluid is incompressible (ρ₁ = ρ₂), rate flow is proportional to the volume of liquid passing through the cross-section A. $A_{1} v_{1} = A_{2} v_{2}$
Bernoulli's Principle: $P_{2} + ½ ρ {v_{2}}^{2} + ρ g y_{2} = P_{1} + ½ ρ {v_{1}}^{2} + ρ g y_{1}$
Poiseuille's equation: $Q = \frac{π R^{2} (P_{1} - P_{2})}{8 η l}$

Web Lecture

Read the following weblecture before chat: Inertia, Acceleration, and Force

Study Activity

Use the Fluid Pressure and Flow simulation at the PhET website. This is a browser compatible version of a Java Applet; it should run in most modern browsers.

Vary the water level by using the faucet to raise the level and the drain to lower it. Change the fluid density and gravity to note changes in behavior. What happens if there are objects in the tank that change how the water enters and leaves? What happens if you add weights to the third tank configuration?

Use the simulation on the flow tab to view what happens as you change the density of the fluid. What happens if you use the handles to change the shape of the pipe.

Use the water tower simulation to set up and "observe" the water tower problems in the text.

Physics simulation Java Applets are the product of the PHET Interactive Simulations project at the University of Colorado, Boulder.

Chat Preparation Activities

Forum question: The Moodle forum for the session will assign a specific study question for you to prepare for chat. You need to read this question and post your answer before chat starts for this session.
Mastery Exercise: The Moodle Mastery exercise for the chapter will contain sections related to our chat topic. Try to complete these before the chat starts, so that you can ask questions.

Chapter Quiz

Required: Complete the Mastery exercise with a passing score of 85% or better.
Go to the Moodle and take the quiz for this chat session to see how much you already know about astronomy!

Lab Work

If you want lab credit for this course, you must complete at least 18 labs; you may complete more if you are preparing for the AP exam.. One or more lab exercises are posted for each chapter as part of the homework assignment. We will be reviewing lab work at regular intervals, so do not get behind!

Lab Instructions: Bernoulli's Principle

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Physics

Chapter 10: 8-14