This question might seem relatively trivial, but if the particles do not follow the flow, what are we looking at?
The question can be reformulated: Do the particles adjust to the flow
faster than the flow changes? I will present some rough estimates that
indicates that the particles do follow the flow. I have chosen to assume that
the particles behave as if they were inside the fluid even though they are on
the surface.
We can use Stokes formula for drag on a sphere of radius flowing with a
velocity
through a liquid with a dynamic viscosity
(see for example [11]),
to estimate the typical time, , it takes a particle to adjust to the
velocity of the surrounding fluid. Solving Stokes formula with respect to the
particle velocity we get
The time constant in the exponent tells us how fast the drag brakes the
particles. Using the fact that the density of the particles is close to that
of the liquid, we can change from dynamic viscosity to kinematic viscosity.
If we use Kolmogorov's assumptions about turbulent velocity fields, wanting a
better description of the small scale features of the flow, we have a relation
between the average energy dissipation,
, the average velocity
differences,
, on a length scale, R, and the length scale. This
relation can be used to estimate a typical time,
, between changes in
the flow velocity the particle will have to adjust to:
Now that we have expressions for the two typical times, we simply have to
compare them. Using the particle radius as the length scale in
2.5 the question we are asking can be written
as:
The particle diameter is . The kinematic viscosity of water is
close to
at room temperature. We have not been able to
measure the energy dissipation in the fluid, but our measurements show that it
is less than
.
When we insert these numbers in 2.6, we see that
the particles adjust to the flow ten times faster than the flow changes,
so I am inclined to say that the particles are following the flow (QED).