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Data analysis

 

I set out to study relative diffusion in the two experiments I have presented in chapter 2 and 3. The quantity I extract from my measurements is the time derivative of the squared separation of pairs of particles as a function of their separation (or rather an approximation thereof).

The data I collect from the experiments are tracks of particles. Each particle track consists of positions of the particle sampled at a fixed frequency. For the Faraday experiment, the sampling frequency was tex2html_wrap_inline4685, and several tracks consisted of more than 1000 positions. In the soap film experiment, I worked with a sampling frequency of tex2html_wrap_inline4687, and all tracks consisted of two positions (the two ends of an image of a particle track).

  figure1069
Figure 5.1: Particle separation. The sketch shows the tracks of two particles P and Q, and the positions of the particles at two times. The time between successive known positions is tex2html_wrap_inline4693 in the data from the Faraday experiment and tex2html_wrap_inline4695 in the data from the soap film experiment.

Since I do not have continous data for the locations, I have to approximate the time derivative of the squared particle separation with a difference
eqnarray1077

Richardson's collection of experimental data for the atmosphere and Kolmogorov's derivations gives a scaling relation
eqnarray1083

Instead of just splitting the R-axis in small intervals, and calculating the average of tex2html_wrap_inline4389 in each interval, as we did for our articles on the Faraday experiment, I sorted all the tex2html_wrap_inline4389 by the particle separation and calculated an approximation to the integral of tex2html_wrap_inline4389 with respect to R:
eqnarray1095
In the following, an integral with respect to R of some quantity measured on the experiments, denotes such an approximation.

The integral with respect to R of a quantity which scales as a power of R is proportional to the average of the quantity times R.
equation1102



next up previous contents index
Next: Faraday experiment Up: Measurements of relative diffusion Previous: Uncertainties in the velocity

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