1. conservation and continuity
Fluid Flow
conservation and continuity
§ 11.7–11.8

2. Volume Flow Rate
Volume per time through an imaginary surface perpendicular to the path
DV/Dt
units: m3/s

3. Volume Flow Rate
DV/Dt = v·A if v is constant over A

4. Mass Flow Continuity Constant mass flow for a closed system Dm = Dt1 2
r1A1v1 = r2A2v2

5. Flow Continuity For an incompressible fluid: constant r Dm = Dt1 2
r1A1v1 = r2A2v2
Because r1 = r2,
A1v1 = A2v2 DV Dt = 1 2

6. Question
Where is the speed greatest in this stream of incompressible fluid?
Here.
Same for both.
Can’t tell.

7. Question
Where is the density greatest in this stream of incompressible fluid?
Here.
Same for both.
Can’t tell.

8. Question
Where is the volume flow rate greatest in this stream of incompressible fluid?
Here.
Same for both.
Can’t tell.

9. Bernoulli’s Equation
Energy in fluid flow
§ 11.9–11.10

10. Incompressible Fluid Recall:
Continuity condition: constant volume flow rate
DV1 = DV2
v1A1 = v2A2

11. Poll Question
Where is the kinetic energy of a parcel greatest in this stream of incompressible fluid?
Up here.
Down here.
Same for both.
Can’t tell.

12. Changing Cross-Section
Fluid speed varies
Faster where narrow, slower where wide
Kinetic energy changes
Work is done (somehow).

13. Ideal Fluid
No internal friction (viscosity)
No non-conservative work

14. Poll Question
Where would the pressure be greatest if the fluid were stationary?
Up here.
Down here.
Same for both.
Can’t tell.

15. Conservation of Energy
K1 + Ug1 + Wnon-g = K2 + Ug2
Wnon-g = K2 + Ug2 – K1 – Ug1
Wnon-g = K2 – K1+ Ug2 – Ug1
Wnon-g = DK + DUg
What is this “Wnon-g”? 15

16. Work done by Pressure W = F·Ds
Work done on fluid at bottom: W1 = p1A1·Ds1
Work done on fluid at top: W2 = –p2A2·Ds2
Non-g work done on fluid : Wnon-g = p1A1·Ds1–p2A2·Ds2
= (p1 – p2)DV

17. Kinetic Energy Change Steady between “end caps” Lower cap: K1 = ½ mv12
Upper cap: K2 = ½ mv22
m = rDV
DK = 1/2 rDV (v22–v12)

18. Potential Energy Change
Steady between “end caps”
Lower cap: U1 = mgy1
Upper cap: U2 = mgy2
m = rDV
DU = rgDV (y2–y1)

19. Put It All Together Wnon-g = DK + DU
(p1 – p2)DV = 1/2 rDV (v22–v12) + rgDV (y2–y1)
(p1 – p2) = 1/2 r (v22–v12) + r g(y2–y1)
p1 + 1/2 rv12 + rgy1 = p2 + 1/2 rv22 + rgy2
This is a conservation equation
Strictly valid only for incompressible, inviscid fluid

20. What Does It Mean? Faster flow  lower pressure
Maximum pressure when static
pV is energy

21. Example problem
A bullet punctures an open water tank, creating a hole that is a distance h below the water level. How fast does water emerge from the hole?

22. Torricelli’s Law p1 + 1/2 rv12 + rgy1 = p2 + 1/2 rv22 + rgy2 v22 = 2gh
1/2 rv22 = rg(y2–y1) + (p2–p1) – 1/2 rv12
1/2 rv22 = rgh
v22 = 2gh
v2 =    2gh
look familiar?

23. Exercise
P. 333 Question 65
A Venturi meter is a device for measuring the speed of a fluid within a pipe. A gas with density 1.30 kg/m3 flows at speed v through a pipe with cross section A2 = m2. The meter has cross section A1 = m2 and is substituted for a section of the pipe. The pressure difference is 120 Pa.
What is the speed v?
What is the volume flow rate?