1. CE 3305 Engineering FLUID MECHANICS
Lecture 6: bernoulli’s equation for a fluid

2. Outline
Bernoulli Equation
Application to some practical cases

3. Bernoulli’s equation Sort of a derivation
Textbook derives along a streamline (which saves a step).
Start with Euler’s equation

4. Bernoulli’s equation Select a useful coordinate system
Incompressible fluid

5. Bernoulli’s equation Write in differential form
Only showing x-component to the right; similar structure for y and z.
Z-component will have a weight term.
Require irrotational flow (vorticity vanishes)
<watch vorticity video>

6. Bernoulli’s equation
Y and Z acceleration terms

7. Bernoulli’s equation
Irrotational (zero vorticity) lets us refactor the cross-terms in the acceleration vector
Euler’s equation after substitutions (still ugly calculus)
Use chain rule

8. words

9. Bernoulli’s equation Rearrange the component equations
Group terms within the partial differential operation
Recall definition of the length of a vector in 3-space

10. Bernoulli’s equation Recall what a constant does when differentiated
Three derivatives, all equal to each other and all equal to zero and all with respect to a different variable
They must be the same function!

11. Bernoulli’s equation
The textbook derives along a streamline (which by definition means flow is irrotational)
Typically the equation is memorized as total head between two locations on the same streamline
Bernoulli’s equation is a special case of Euler’s equation of motion
It can be applied to compressible flow with minor modifications

12. Example using bernoulli’s equation
Problem Statement

13. Example using bernoulli’s equation
Known
Total head in system
Free surface and outlet pressure
Water is working fluid

14. Example using bernoulli’s equation
Unknown
Velocity at outlet

15. Example using bernoulli’s equation
Governing Equations
Bernoulli’s equation

16. example
Solution

17. example Discussion Water/oil same (specific weight cancels)
Steady flow requires important assumption about relative “areas”
Assumed pressure across “jet” is zero
No frictional losses (yet – that’s coming soon!)

18. Example bernoulli’s equation
Problem Statement

19. Example bernoulli’s equation
Known:
Working head
Outlet velocity
Working fluid (water)

20. Example bernoulli’s equation
Unknown:
Outlet pressure

21. Example bernoulli’s equation
Governing equation:
Bernoulli’s equation

22. example
Solution

23. example
Discussion
Almost same as prior example; but to find pressure, we need to know the working fluid
Outlet velocity specified, don’t know if it is a jet, so no assumption about pressure
No frictional losses (yet!)

24. Bernoulli example
Problem Statement

25. Bernoulli example Known: Speed of jet at fountain nozzle
Vertical speed of water at apogee (high point) in fountain
Working fluid (water)

26. Bernoulli example
Unknown:
Height of the jet (fountain)

27. Bernoulli example
Governing Equations:
Bernoulli’s equation

28. example
Solution

29. example
Solution

30. example
Discussion
Working fluid was irrelevant (it does matter when we introduce friction later on)
Height controlled by exit speed
Imagine the speed at the Bellagio Fountains in Las Vegas, Nevada (find a video)

31. Next Time
Reynold’s Transport Theorem