Fragmentation of magnetic particle aggregates in turbulence

Particle aggregates are frequently encountered in many natural and industrial environments. We describe here the stationary state of the fragmentation process of inertial scale particle aggregates in turbulence, i.e., when the particles and the aggregates are larger than the Kolmogorov dissipative scale $\eta$. For this purpose, we place at the initial time a large aggregate of millimetric, nearly neutrally buoyant magnetic particles in a high-Reynolds-number turbulent von Kármán flow. Turbulent fluctuations impose external stresses that tend to fragment the initial and the subsequent aggregates, contrary to the magnetic dipoles that impose torques and forces on the magnets responsible for cohesion. Using video image analyses, we perform the three-dimensional reconstruction of the aggregates and measure their characteristic sizes. The average number of particles inside each aggregate can then be deduced as a function of the intensity of turbulence. Assuming a Kolmogorov inertial scaling law for the turbulent velocity increments, we predict theoretically an aggregate mean size which is in agreement with our experimental results.

Figure 1

Sketch of the turbulent von Kármán flow setup.

Figure 2

(a) Normalized longitudinal second-order velocity structure function for the different impeller rotation frequencies $f$. (b) Energy dissipation rate $\epsilon$ as a function of the impellers rotation frequency $f$.

Figure 3

(a) Magnetic force $F_B$ between the magnets as a function of their separation distance $r$. (b) Photograph of some beads coated with wax. (c) Main axis length probability distribution function (PDF) and (d) density PDF, with their Gaussian fits.

Figure 4

Observation of the three different regimes as a function of the turbulence intensity: (a) $f=8.5$ Hz, (b) $f=14.5$ Hz, and (c) $f=19$ Hz.

Figure 5

(a) Experimental results for fresh water are shown by red closed circles ($\langle N_{\text{expt}}\rangle$) and blue closed squares ($\beta$) and the salt water results are shown by red open squares ($\langle N_{\text{expt}}\rangle$) and blue open triangles ($\beta$). Also shown are the theoretical predictions (solid lines) of the average number of particles $N$ per aggregate versus the dissipation rate $\epsilon$ of turbulence. A smooth sigmoid evolution for $\beta$ is fitted through the fresh water data and is used together with the measured threshold $f_c=25$ and fitting parameters $a=0.9$ and $6c_3/\pi =1$ to predict $\langle N_{\text{expt}}\rangle$. (b) Measure of the average number $\langle N_T\rangle$ of particles per aggregate taking into account individual particles in the statistics.

Figure 6

(a) Probability density functions of the size of the aggregates for different rotation rates of the impellers in the case of the fresh water experiments. In agreement with a hierarchical breaking mechanism, log-normal distributions that have the same means and standard deviations as the experimental distributions have been plotted through each distribution. (b) Evolution of the variance $\sigma$ of the PDFs (fresh water) as a function of the dissipation rate $\epsilon$ of turbulence.