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

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:

What is the typical velocity in the lake zurich ? http://www.rivermap.ch/ small cross section https://www.hydrodaten.admin.ch/fr/2099.html https://www.google.fr/maps/@47.3411488,8.5882158,12.79z?hl=fr Large cross section

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

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

An observation The water in the Limmat flows rapidly but in the lake it looks like it does not move ! small cross section https://www.youtube.com/watch?v=GYUZcLJLDZM Large cross section

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

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

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

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.

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.

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.

Blood flows

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

Can consider the blood as incompressible ?

Can consider the blood as incompressible ?

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

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

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