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# Do the particles follow the flow?

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).

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