1. Physics 151: Lecture 30 Today’s Agenda
Today’s Topic:
Fluids in Motion
Bernoulli’s Equation and applications

2. Example / Fluid Statics
A beaker of mass mbeaker containing oil of mass moil (density = roil) rests on a scale. A block of iron of mass miron is suspended from a spring scale and completely submerged in the oil as in Figure on the right.
Determine the equilibrium readings of both scales.

3. Fluids in Motion See text: 14.5
Up to now we have described fluids in terms of their static properties:
density r
pressure p
To describe fluid motion, we need something that can describe flow:
velocity v
There are different kinds of fluid flow of varying complexity
non-steady / steady
compressible / incompressible
rotational / irrotational
viscous / ideal

4. See text: 14.5
Ideal Fluids
Fluid dynamics is very complicated in general (turbulence, vortices, etc.)
Consider the simplest case first: the Ideal Fluid
no “viscosity” - no flow resistance (no internal friction)
incompressible - density constant in space and time
Simplest situation: consider ideal fluid moving with steady flow - velocity at each point in the flow is constant in time
In this case, fluid moves on streamlines
streamline

5. follows from mass conservation if flow is incompressible.
See text: 14.6
Ideal Fluids
streamlines do not meet or cross
velocity vector is tangent to streamline
volume of fluid follows a tube of flow bounded by streamlines
streamline
Flow obeys continuity equation
volume flow rate Q = A·v is constant along flow tube.
A1v1 = A2v2
follows from mass conservation if flow is incompressible.

6. Conservation of Energy for Ideal Fluid
See text: 14.7
Conservation of Energy for Ideal Fluid
Recall the standard work-energy relation
Apply the principle to a section of flowing fluid with volume dV and mass dm = r dV (here W is work done on fluid) dV
Doesn’t matter whether these particles are distinguishable or not - we are only asking how many are on the left and right.
Bernoulli Equation

7. Lecture 30 Act 1 Continuity
A housing contractor saves some money by reducing the size of a pipe from 1” diameter to 1/2” diameter at some point in your house. v1
v1/2
1) Assuming the water moving in the pipe is an ideal fluid, relative to its speed in the 1” diameter pipe, how fast is the water going in the 1/2” pipe?
a) 2 v1
b) 4 v1
c) 1/2 v1
c) 1/4 v1

8. Lecture 30 Act 2 Bernoulli’s Principle
A housing contractor saves some money by reducing the size of a pipe from 1” diameter to 1/2” diameter at some point in your house. v1
v1/2
2) What is the pressure in the 1/2” pipe relative to the 1” pipe?
a) smaller
b) same
c) larger

9. DEMO SLIDE
The smaller the diameter the lower is the pressure

10. Example
A Pitot tube (see Fig. below) can be used to determine the velocity of air flow by measuring the difference between the total pressure and the static pressure. If the fluid in the tube is mercury, density rHg = kg/m3, and h = 5.00 cm, find the speed of air flow. (Assume that the air is stagnant at point A and take rair = 1.25 kg/m3.)

11. Can we know what is v from what we can measure ?
Venturi Meter
v = ?
Can we know what is v from what we can measure ? h
rHg
rair

12. Example
A tank containing a liquid of density r has a hole in its side at a distance h below the surface of the liquid. The hole is open to the atmosphere and its diameter is much smaller than the diameter of the tank.
What is the speed with of the liquid as it leaves the tank. h r
v=?

13. Example
Figure on the right shows a stream of water in steady flow from a kitchen faucet. At the faucet the diameter of the stream is cm. The stream fills a 125-cm3 container in 16.3 s. Find the diameter of the stream 13.0 cm below the opening of the faucet.

14. Example
Water is forced out of a fire extinguisher by air pressure, as shown in Figure below. How much gauge air pressure in the tank (above atmospheric) is required for the water jet to have a speed of 30.0 m/s when the water level in the tank is m below the nozzle?

15. Recap of today’s lecture
14.4-7
Streamlines
Bernoulli’s Equation and applications