Statistics of a two-dimensional immersed granular gas magnetically forced in volume

We present an experimental study of the dynamics of a set of magnets within a fluid in which a remote torque applied by a vertical oscillating magnetic field transfers angular momentum to individual magnets. This system differs from previous experimental studies of granular gas where the energy is injected by vibrating the boundaries. Here, we do not observe any cluster formation, orientational correlation and equipartition of the energy. The magnets' linear velocity distributions are stretched exponentials, similar to three-dimensional boundary-forced dry granular gas systems, but the exponent does not depend on the number of magnets. The value of the exponent of the stretched exponential distributions is close to the value of 3/2 previously derived theoretically. Our results also show that the conversion rate of angular momentum into linear momentum during the collisions controls the dynamics of this homogenously forced granular gas. We report the differences among this homogeneously forced granular gas, ideal gas, and nonequilibrium boundary-forced dissipative granular gas.

Figure 1

[(a), (b)] Schematic of the experimental setup for the 2D granular gas of magnets immersed in water and remotely energized by a vertical oscillating magnetic. The vertical magnetic field is generated by a pair of Helmholtz coils. The magnets are illuminated from the rear by a light-emitting diode (LED) panel, and a high-speed camera records time series of images. (c) Trajectories of $N=13$ magnets ($\varphi =0.06$) in the fluid reservoir (red). The black cores are neodymium magnets surrounded by transparent Plexiglas shells.

Figure 2

(a) Probability density functions (PDFs) of the rescaled angular velocity $\omega /{\sigma }_{\omega }$ for different frequencies $f_B$ of the oscillating magnetic field: 7, 20, 30, and 50 Hz (cold to hot colors). The number of magnets is equal to $N=25$ ($\varphi =0.12$) and the intensity of the magnetic field to $B=0.0162$ T. The equation of the solid line is $y=a_{\omega }{e}^{-b_{\omega }x}$, with $a_{\omega }=0.58$ and $b_{\omega }=1.1$. (b) Temporal signal of the angular velocity $\omega /2\pi$ of a magnet for $f_B=7$ Hz. (c) Temporal signal of the angular velocity $\omega /2\pi$ of a magnet for $f_B=50$ Hz. In both (b) and (c), the solid lines represent the standard deviation ${\sigma }_{\omega }$ of the signal and the dashed lines represent the frequency of the magnetic field $f_B$.

Figure 3

(a) Standard deviation of the rescaled angular frequency ${\sigma }_{\omega }/\left(2\pi \right)$ as a function of the frequency of the oscillating magnetic field $f_B$ for $B=0.0162$ T and $N=25$ ($\varphi =0.12$). The dashed line represents a power law fit of ${\sigma }_{\omega }/\left(2\pi \right)$. (b) Standard deviation of the angular velocity ${\sigma }_{\omega }/\left(2\pi \right)$ as a function of the intensity of the oscillating magnetic field $B$ for $f_B=50$ Hz and $N=25$ ($\varphi =0.12$). The dashed line represents a power law fit of ${\sigma }_{\omega }/\left(2\pi \right)$. (c) Standard deviation of the angular velocity ${\sigma }_{\omega }/\left(2\pi \right)$ as a function of the number of magnets $N$ or the volume fraction $\varphi$. (d) Diffusion constant $D$ of the magnets as a function of the number of magnets $N$ or the volume fraction $\varphi$. In both (c) and (d), the values of the control parameters are $B=0.0162$ T and $f_B=50$ Hz.

Figure 4

(a) Probability density functions (PDFs) of the rescaled horizontal velocity $v_x/{\sigma }_{v_x}$ for different frequencies of the magnetic field $f_B$: 7, 10, 40, and 50 Hz (cold to hot colors). The equation of the dashed line is $y=a_{v_x}{e}^{-b_{v_x}{x}^{1.6}}$, with $a_{v_x}=0.45$ and $b_{v_x}=0.7$ and the dotted line represents a Gaussian distribution. (b) PDFs of the rescaled modulus of the velocity $v/{\sigma }_v$ for different frequencies of the magnetic field $f_B$. The colors are the same as in panel (a). The equation of the dashed line is $y=a_v{x}^{1.3}{e}^{-b_v{x}^{1.35}}$, with $a_v=0.9$ and $b_v=0.75$, and the dotted line represents a Rayleigh distribution. In both (a) and (b), the control parameters are $B=0.0162$ T and $N=25$. (c) PDFs of the rescaled horizontal velocity $v_x/{\sigma }_{v_x}$ for a different number of magnets $N$: 10, 20, 30, and 40 (cold to hot colors). The corresponding volume fractions $\varphi$ are equal to 0.045, 0.09, 0.135, and 0.18. The equations of the dashed and dotted lines are the same as in panel (a).

Figure 5

(a) Probability density functions of the angle difference $\mathrm{\Theta }$ for different frequencies of the magnetic field $f_B$: 7, 10, 40, and 50 Hz (cold to hot colors) for $B=0.0162$ T and $N=25$. (b) Main figure: Standard deviation of the linear velocity ${\sigma }_v$ as a function of the control parameters $(f_B,B,N)$. Top inset: ${\sigma }_v$ as a function of the number of magnets $N$. The dashed line represents a power law fit of ${\sigma }_v$ as a function of $N$. Bottom inset: ${\sigma }_v$ as a function of the frequency of the oscillating magnetic field $f_B$. The dashed line represents a power law fit of ${\sigma }_v$ as a function of $f_B$. Red squares correspond to $B=0.0162$ T and $N=25$, green circles correspond to $B=0.0162$ T and $f_B=50$ Hz, and blue diamonds correspond to $f_B=50$ Hz and $N=25$.

Figure 6

(a) Collision frequency $f_{\mathrm{col}}$ as a function of the number of magnets $N$. The equation of the dashed line is $y=0.87x$. Top inset: Collision frequency $f_{\mathrm{col}}$ as a function of the intensity of the magnetic field $B$. Bottom inset: Collision frequency $f_{\mathrm{col}}$ as a function of the frequency of the oscillating magnetic field $f_B$. (b) Mean free path $l$ as a function of the number of magnets $N$. The equation of the dashed line is $y=22/x$. Top inset: mean free path $l$ as a function of the intensity of the magnetic field $B$. Bottom inset: mean free path $l$ as a function of the frequency of the oscillating magnetic field $f_B$. In both figures, red squares correspond to $B=0.0162$ T and $N=25$, green circles correspond to $B=0.0162$ T and $f_B=50$ Hz, and blue diamonds correspond to $f_B=50$ Hz and $N=25$.

Figure 7

Probability density functions (PDFs) of the rescaled angular velocity $\omega /{\sigma }_{\omega }$ for different number of magnets $N$: 11, 20, 30, and 45 (cold to hot colors). The frequency of the magnetic field is equal to $f=50$ Hz and the intensity of the magnetic field to $B=0.0162$ T.