Momentum Shell Balances

  • Set up axes and draw picture
  • Write balance (make sure units of each term is force), considering:
  • Viscous transport terms – what variables do we multiply?
  • Continuity terms (usually cancel) – what variables do we multiply?
  • Body forces – what are the kinds?
  • Pressure forces – when do we consider pressure?
  • Divide by repeated terms and by anything with a delta.
  • Take the limit as delta (any dimension) goes to zero – this will create a derivative of shear stress in terms of a dimension of the system.
  • Be careful—if you’re using a radial system, you may have to have a radius factored into this derivative! Write what this would look like before and after taking the limit in a radial system:
  • Separate variables and integrate.
  • Don’t forget a constant!
  • Apply boundary conditions, if they are relevant at this stage. What is one kind of boundary condition that would apply at this point?
  • Substitute Newton’s law of viscosity. Write it below:
  • Separate variables and integrate again
  • Don’t forget a constant!
  • Apply boundary conditions to solve for constants (explain and give example for each):
  • No-slip condition
  • Interfacial condition
  • Symmetry
  • Need for physically meaningful velocity profile
  • Simplify to write velocity profile.
  • Other things you can find with velocity profile:
  • Average velocity, <v>
  • Forces on plates or surfaces (shear stress at that point * area)
  • Volumetric flow rate (<v> * A, avg velocity * cross sectional area)

Practice Problems (assume Newtonian fluids): Find velocity profile, average velocity, volumetric flowrate, and the force exerted by the fluid on any flat surfaces. Draw a diagram of the velocity profile.

1)A flat plate tilted at an angle with a fluid flowing down the surface. (lecture)

2)A flat plate tilted at an angle with a fluid. On top of the fluid is a flat plate moving with constant velocity U uphill. (lecture)

3)An upright cylinder with a pressure gradient po – pL has fluid flowing down the middle. (lecture)

4)A fluid fills the gap between two concentric cylinders. The inner cylinder is moving with a constant velocity U uphill.

5)Two parallel plates each with length L and width W are separated by a distance d. A fluid fills the space between the plates. The plates are situated horizontally and the bottom plate is fixed. The top plate moves with constant velocityU.

6)Consider the same geometry as (5), but now let both plates be fixed. The pressure at one end of the plates is po and the pressure at the other end of the plates is pL.

7)Homework problem 2B.3.

Fluid Statics

  • What is the main difference between a fluid and a solid?
  • Define, and give the symbol and units for:
  • Specific gravity
  1. Specific weight
  1. Viscosity
  1. Shear stress
  1. Pressure
  • Write an equation that relates Pgauge, Patm, and Pabs.
  • When doing fluid statics problems,
  1. What do we know about the pressure in a continuous fluid?
  1. What happens to pressure as you go down through a fluid?
  1. What happens to pressure as you go up through a fluid?
  • What is the buoyant force? What is the equation? Which density, volume, etc. do we use?
  • Briefly describe the way we derived buoyant force.
  • Do you know how vapor pressure was used in the spinning u-tube homework problem? If you are unsure, review vapor pressure in your 210 book.
  • A common theme among examples and homework problems (mercury sphere, upside-down T-shaped container) has been the idea of a uniform pressure at every height in a continuous fluid, despite the fact that only part of the container is open to atmospheric pressure. Can you explain how to use this ideal in solving problems? Do you know why, in the example in class, the same pressure was exerted on a scale in both cases?
  • Another common theme among examples and homework problems has been slanted walls. Do you know how to treat a slanted wall? If not, review the numerous examples from the notes and learn how to use the equation.

Practice Problems:

1)All lecture and homework problems. If you know how to do them and understand them, you will likely do well on the exam.

2)In the figure below, the pressure at point A is 25 lbf/in2. All fluids are at 20 degrees C. What is the air pressure in the closed chamber B, in Pa? (from Dr. Hebert’s homework)

3)Determine the elevation difference, delta h, between the water levels in the two open tanks shown below: (Dr. Olson’s first exam, Fall ’07)

4)A teapot with a brewer at the top is used to brew tea, as shown below. The brewer blocks the vapor from escaping the teapot, causing the pressure in the teapot to rise and the water to rise up the spout. If the volume of the water in the spout is negligible compared to the volume of water in the teapot, determine the maximum cold-water height (i.e., before any heat is added), H, that would not cause an overflow at the spout when enough heat is added to produce a gage pressure of 0.32 kPa for the vapor. (Dr. Olson’s first exam, Spring ’06)

5)Approximate a fishing boat as a box measuring 20 m long by 5 m wide by 5 m high with a mass of 100,000 kg.

a)What is the draft, d, of the boat? (the draft is the depth the boat sits in water).

b)The fishing net accidentally catches a 1m x 1m x 1m block of aluminum (SG = 2.7), and the fishermen slowly begin to reel the net in. What is the draft of the boat now (assume the block of aluminum is completely submerged in the water)?