1. conservation and continuity
Fluid Flow
conservation and continuity
§ 15.6

2. Volume Flow Rate
Volume per time through an imaginary surface perpendicular to the velocity
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. Volume Flow Continuity
Constant volume flow if incompressible
If r1 = r2, DV Dt = 1 2

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

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

8. Poll 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
§ 15.7

10. Incompressible Fluid 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?
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!

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?
Here.
Same for both.
Can’t tell.

15. Work-Energy Theorem Wnet = DK K1 + Wnet = K2
K1 + U1 + Wother = K2 + U2
Wother = DK + DU
dW = dK + dU 16

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
Total work done on fluid : W = 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 = rV
DK = 1/2 rV (v22–v12)

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

19. Put It All Together Wother = DK + DU
(p1 – p2)V = 1/2 rV (v22–v12) + rgV (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?