1. Reminder: HW #10 due Thursday, Dec 2, 11:59 p.m.
(last HW that contributes to the final grade)
Recitation Quiz #11 tomorrow
(last Recitation Quiz)
Formula Sheet for Final Exam posted on Bb
Last Time: Pressure vs. Depth, Buoyant Forces and Archimedes’ Principle
Today: Fluid Motion, Bernoulli’s Equation

2. Colorado River, Grand Canyon
Fluids in Motion
When a fluid is in motion, its flow can be either …
Streamline (Laminar) : Every particle that passes a particular
point moves along exactly the same smooth path
Turbulent : In contrast, the flow of a fluid becomes irregular
above a certain velocity or under conditions that cause abrupt
changes in velocity
Colorado River, Grand Canyon

3. Viscosity
In fluid flow, the term viscosity is used to describe the degree of internal friction in the fluid.
This internal friction is associated with the resistance between
two “adjacent layers” of the fluid moving relative to each other !
vs.

4. Definition of an “Ideal Fluid”
“Ideal fluids” satisfy the following conditions :
(1) The fluid is non-viscous (no internal friction between
adjacent layers)
(2) The fluid is incompressible (density is constant)
(3) The fluid motion is steady (velocity, density, and pressure
each point in the fluid do not change with time)
(4) The fluid moves without turbulence. Each element has
no angular velocity about its center (no “eddy currents”)

5. Continuity Equation
Consider a fluid flowing through a pipe. Enters the pipe at (1) and exits at (2).
Let’s apply conservation of mass to find a relation between the density, velocity, and cross-sectional area
at (1) and (2) …
Result is the “continuity equation” :
ideal fluid

6. Continuity Equation Implications of the continuity equation :
(1) Product of the cross-sectional area and velocity is a constant
(2) Speed is high when tube is constricted, low when larger
(3) The continuity equation is equivalent to the statement that the
volume of fluid entering the tube in some time interval Δt equals
the volume of fluid exiting the tube in the same time interval

7. Continuity Equation
The continuity equation is also a measure of a “volume flow rate”
Units of (A x v) are :
(A x v) tells you how many m3 of the fluid flows past a particular point every second.

8. Everyday Example: Garden Hose
Q: What happens when you squeeze a garden hose ?
A: Water sprays out with greater speed (and, thus, travels farther).
This is just because of the continuity equation !!
If A2 decreases, v2 must increase !!

9. Example: 9.44
Water flowing through a garden hose of diameter 2.74 cm fills a 25-liter bucket in 1.5 minutes.
(a) What is the speed of the water leaving the end of the hose ?
(b) A nozzle is now attached to the end of the hose. If the nozzle diameter is 1/3 the diameter of the hose, what is the speed of the water leaving the hose ?

10. Bernoulli’s Equation
As a fluid flows through a pipe of varying cross-sectional area and elevation (height), the pressure changes along the pipe.
Bernoulli’s Equation relates the pressure in the pipe to the kinetic and potential energy of the fluid along the pipe :

11. Bernoulli’s Equation pressure
kinetic energy per unit volume = KE/Volume
potential energy per unit volume = PE/Volume
The sum of the pressure, kinetic energy per unit volume, and the potential energy per unit volume has the same value at all points along a streamline.

12. Venturi Tube Can be used to measure the speed of fluid flow.
Horizontal  Heights are equal so y1 = y2
From continuity equation:
v2 > v1
So this means: P1 > P2
Swiftly moving fluids exert less pressure than do more slowly moving fluids.

13. Conceptual Question
An ideal fluid flows through a horizontal pipe having a diameter that varies along its length. Does the sum of the pressure and kinetic energy per unit volume at different sections of the pipe …
(a) decrease as the pipe diameter increases
(b) increase as the pipe diameter increases
(c) increase as the pipe diameter decreases
(d) decrease as the pipe diameter decreases
(e) remain the same as the pipe diameter changes

14. Example: 9.52 Water flows through a constricted pipe.
At the lower point, the pressure is 1.75 x 105 Pa, and the radius is 3.0 cm.
At a point 2.5 m higher, the pressure is 1.20 x 105 Pa, and the radius is 1.5 cm.
Find the speed of flow …
(a) in the lower section, and
(b) in the upper section.
(c) What is the volume flow rate through the pipe ?

15. Application: Airplane Wings
higher air speed
Lift on an airplane wing can be partly explained with the concepts of Bernoulli’s Equation.
Air molecules striking the bottom of the wing are deflected downward. Wing exerts downward force on air molecules (action force). Reaction force is upward force on wing !!
slight upward tilt
lower air speed
[Newton’s 3rd Law !!]

16. Example
An airplane has a mass M, and the two wings have a total area A. During level flight, the pressure on the lower wing surfaces is P1.
Determine the pressure P2 on the upper wing surfaces.

17. Next Class
13.1 – 13.4 : Hooke’s Law, Simple Harmonic Motion