1. Part II: Application of the mass conservation principle in an incompressible fluid.

2. The mass conservation implies that the same amount of water flows through the narrow and large tubes. In particular the same mass of water enters and leaves the system.
If the fluid is incompressible and all thermal fluctuations are small, the density is uniform in the system, it follows:

3. What is the typical velocity in the lake zurich ?
small cross section
Large cross section

4. 48m 1.94km Calculate the typical
Average flow rate: 101 m3/s
48m
Let’s assume 5m depth.
small cross section
You can get the measurement from googlemaps
Average depth: 50m
Calculate the typical
lake current velocity in the Zurichsee
Large cross section
1.94km

5. 1.94km We apply the mass conservation between the river and the lake:
S1=240m2
50m
S2=97’000m2
48m
1940m
The velocity in the Limitat is
Thus the velocity in the lake is typically:
Or even quicker:
1.94km

6. An observation
The water in the Limmat flows rapidly but in the lake it looks like it does not move !
small cross section
Large cross section

7. A laboratory equivalent of the system
small cross section
9.5cm
1.5mm
Large cross section

8. The streamlines are curved tangent to the velocity at all points.

9. Streamlines
The streamlines are curves tangent to the velocity at all points.
The velocity component normal to a streamline is null
If two streamlines crosses, the velocity at the intersection is zero

10. Streamlines
Seeding the fluid with reflecting particles and using a laser to produce a light sheet, one can visualize 2D flows by tacking a picture with a shutter time, dt, long enough so that particles leaves a streak that represents a small displacement, dl, parallel to the local velocity.
The exposure time has to be short enough so that the velocity remained constant both in direction and amplitude.

11. Streamtubes
Streamtubes are volumes enclosed by the streamlines originating from an arbitrary surface.
Since the walls of a streamtube are formed by streamlines, no particles can cross the wall.

12. Assuming we know Ua, Ub can be deduced from the mass conservation:
Streamtubes
Let’s choose a small dt over which the flow is quasi-steady, all the mass dm entering the tube during the time dt must exit the tube at the other end.
It follows that for an incompressible fluid, the volume entering and living the tube must be the same.
Assuming we know Ua, Ub can be deduced from the mass conservation:
For an incompressible fluid, the velocity increases inversely proportional to the cross-section of a stream tube.

13. Blood flows

14. What is the flow rate of blood in the various blood vessels ?
Vein system
deoxygenated blood cells
Artery system
Oxygenised blood cells.
Wikipedia
Wikipedia

16. Can consider the blood as incompressible ?

17. Can consider the blood as incompressible ?

18. Very Large total cross section,
small velocities
Low velocity allow for more efficient osmosis processes, oxygen transfer!

19. Low velocity allow for more efficient osmosis processes, oxygen transfer!

20. Take away message:
In an incompressible fluid the mass conservation is equivalent to volume conservation laws: