In section 2.2, I derived
a comparison that can be used to test whether the particles follow the flow.
The kinematic viscosity of the water-glycerol mixture, that is the bulk of the
can be found in table 3.3. If the
smallest particle radius, , is inserted as the particle radius in
equation 2.6 and the energy dissipation is
separated out we get an upper bound for the energy dissipation as the
condition for the particles to follow the flow of the soap film:
When the largest particle radius, , is inserted the condition is:
The energy dissipation can be found by comparing the mechanical energy
(potential plus kinetic) of the film at two locations along the channel. The
mechanical energy per unit mass is
where g denotes the gravitational acceleration, h denotes the height, and v denotes the velocity. The time it takes to flow a stretch l along the channel is
if the velocity does not change much. I let the subscripts 1 and 2 denote values at two locations along the channel separated by a distance l. The energy dissipation is the mechanical energy per unit mass per unit time that disappears from the system:
At two cross sections separated along the channel I find that the
mean square velocity is (upstream) and
(downstream). The inclination is
, and the gravitational acceleration is .
If we insert these values in expression
3.5 we find
This number is rather high compared to the magnitude of the energy dissipation you usually find in three-dimensional turbulence ( in wind tunnels), so I suspect that there is something wrong either in my derivation, or in my velocity measurements.
The energy dissipation in the soap film is (apparently) larger than what can be tolerated, if the largest particles I have poured on to the soap film should follow the flow of the soap film. Since the largest particles are those that are visible on the images, this means that the value of this collection of measurements is limited.
If we rewrite equation 2.6 so it expresses a limit
on the particle radius and insert the tabular value for the kinematic
viscosity and the value for the energy dissipation from the preceding
derivation we get
If we could separate the smallest particles from the mixture of particles of various sizes, and just use them as tracers, we would have some tracers that would follow the flow.