Dissipated power within a turbulent flow forced homogeneously by magnetic particles

We report measurements of global dissipated power within a turbulent flow homogeneously forced at small scale by an original forcing technique. The forcing is random in both time and space within the fluid by using magnetic particles in an alternating magnetic field. By measuring the growth rate of the fluid temperature, we show how the dissipated power is governed by the external control parameters (magnetic field and number $N$ of particles). We experimentally found that the mean dissipated power scales linearly with these parameters, as expected from the magnetic injected power scalings. These experimental results are well described by simple scaling arguments showing that the main origins of the energy dissipation are due to viscous turbulent friction of particles within the fluid and to the inelasticity of collisions. Finally, by measuring the particle collision statistics, we also show that the particle velocity is independent of $N$ and is only fixed by the magnetic “thermostat.”

Figure 1

Experimental setup. Top insets show pictures of a magnetic particle.

Figure 2

Temporal evolution of the fluid temperature for different numbers of particles $N=10$, 20, 30, 40, 50, and 60 (see arrow), with $B=225$ G and $h=13$ cm. Dashed lines are linear fits. The inset shows fit slopes $dT/dt$ versus $N$. The dashed line has a slope of $1.7\times {10}^{-5}$ K/s.

Figure 3

Temporal evolution of the fluid temperature for different magnetic fields $B=112.5$, 135, 157.5, 180, and 202.5 G (see the arrow), with $N=60$ and $h=13$ cm. Dashed lines are linear fits. The inset shows fit slopes $dT/dt$ versus $B$ for two fluid depths $h=6.5$ ($•$) and 13 ($\blacksquare$) cm. Here $N=60$. The dashed line has a slope of $5.4\times {10}^{-3} \mathrm{K}{\mathrm{s}}^{-1}{\mathrm{T}}^{-1}$.

Figure 4

Plot of $1/\overline{\tau }$ vs $N{B}^{1/3}/h^2$ for $1\le N\le 60$ ($B=180$ G and $h=4$ cm) ($•$), $2\le h\le 8$ cm ($N=30$ and $B=180$ G) ($\blacksquare$), and $112\le B\le 225$ G ($N=30$ and $h=4$ cm) ($⧫$). The bottom inset shows the temporal signal of the accelerometer showing 44 collisions in 2 s ($N=10, h=4$ cm, and $B=180$ G). The top inset shows $\mathrm{PDF}(A/\overline{A})$ for $1\le N\le 60$ ($B=180$ G and $h=4$ cm).

Figure 5

Mean acceleration (in volts) during particle collisions with the top wall vs $B$: $\overline{A}$ vs $B$ for $N=30$, with $h=4$ cm. The dashed line shows $B^{1/3}$ scaling. The top inset shows $\overline{A}$ vs $N$ for $B=180$ G, with $h=4$ cm. The bottom inset shows $\overline{A}$ vs $h$ for $B=180$ G, with $N=30$.