Laboratory model for plastic fragmentation in the turbulent ocean

The fragmentation of solid objects in turbulence is of paramount importance in a large number of situations, especially for marine plastic pollution where small plastic debris are formed by the fragmentation of plastic litter under hydrodynamic forces. Up to now, investigations have focused on the fragmentation of particle aggregates in turbulent flows. Here we study the fragmentation of a single deformable object that behaves elastically up to breakage, in the inertial range of turbulence. Using laboratory experiments with glass fibers as a model system, complemented by numerical simulations and theoretical analyses, we exhibit a comprehensive fragmentation scenario, further modeled by an evolution equation. Our results demonstrate that the fragmentation process is limited at small scales by a physical cutoff length originating from the fluid-structure interactions between the objects and the turbulence, and therefore independent of the brittleness of the fibers. This scenario leads to the accumulation of fragments with a typical length slightly longer than the cutoff scale, as smaller fragments are too short to be deformed and broken by the turbulence.

Figure 1

Plastic fragmentation: from oceans to the laboratory. (a) Picture of the von Kármán turbulent flow used in the experimental setup and described in Sec. 2a1. The black arrows indicate the direction of rotation of the two motors. (b) Glass fiber observed under a microscope for diameter measurement. (c) 2D image of a glass fiber deformed by the experimental turbulent flow observed using a schlieren technique. (d) Breakup event obtained in the numerical simulations that are described in Sec. 2b. The event is illustrated using the superimposition of different time steps, with time going from left to right. The colors correspond to the local curvature $K$ and the location of the upcoming event on the fiber is shown by a dashed dotted line. More details on the fragmentation mechanism are given in Sec. 2c. (e) Random sets of fiber fragments collected in the experimental turbulent flow (see Sec. 2d), for different times and breaking parameters $K_B$. The fiber fragments are used to obtain fragment length distributions, shown in Fig. 2 and discussed in Sec. 3a. The scale bars in panels (c) and (e) all indicate 10 mm, while the diameter of the circles in panel (e) is equivalent to $20{\ell }_e$ for all cases.

Figure 2

Experimental (left) and numerical (right) fragment length distributions with $L\equiv \ell /{\ell }_e$. Top line shows the time evolution for a fixed breaking parameter $K_B=2.1$ in experiments (a) and $K_B=2$ in the simulations (c). Bottom line corresponds to the evolution as a function of breaking parameter for a fixed time $t/T_I=18\times {10}^3$ in experiments (b) and $t/T_I=100$ in the simulations (d). With corresponding colors, the thin dashed lines represent the model discussed further in the text (see Sec. 4c). The vertical dashed dotted lines indicate $L=1$ in all panels and the error bars represent $\pm 4$ fragments for the experiments.

Figure 3

Deformation statistics. (a) Local curvature $\langle K\rangle$ as a function of the curvilinear coordinate $s/L$ along the fiber (solid lines) and its approximation from Eq. (8) (thin dashed lines) for different fiber lengths. The notation $\langle \rangle$ denotes an average over time and realisations. (b) Maximum amplitude ${\langle K\rangle }^{★}$ of the curvature $\langle K\rangle$ along the fiber as a function of length $L$. The symbols represent the data from the numerical simulations while the dashed line correspond to the fit using Eq. (11). (c) Curvature statistics $C_{K^2}$ obtained from the numerical simulations (solid lines) and their fit (thin dashed lines) by Eq. (12) for different fiber lengths $L$. The gray region indicates the range of critical curvatures $K_B^2$ explored numerically.

Figure 4

Fragmentation statistics. (a) Breaking probability $p$ as a function of fiber length $L$ (solid lines) and its model from Eq. (19) (dashed lines) for different breaking parameters. The horizontal dashed dotted line indicates $p\equiv 1$ and the vertical thin solid line shows $L=L_c\equiv 2$. (b) Logarithm of the breaking location probability $\gamma$ as a function of fiber lengths $L$ and $L^{\prime }$ for $K_B=2$. The dashed line shows $L=L^{\prime }$ while the dashed dotted one indicates $L=L^{\prime }/2$.

Figure 5

Exponential tails of the length distributions. (a) Temporal evolution of the fragment length distribution obtained with the simplification of Eq. (18). The panel is in lin-log scales to emphasise on the exponential tail of the distribution. The vertical dashed dotted line shows $L=L_c\equiv 2$. The inset shows the temporal evolution of the mean fragment length $\langle L\rangle$. (b) Temporal evolution of the fragment length distribution with $K_B=2$, obtained in the numerical simulations (solid lines) and for the numerical solution of the full evolution equation (thin dashed lines, see Sec. 4c). The exponential fits of the tails are shown with dashed dotted thick lines. Note that this panel is also in lin-log scales to emphasise the tail of the distributions, contrary to Fig. 2 which is in log-log scales. (c) Correspondance between the theoretical timescale $\tau /p_L^0$ and the physical timescale $t/T_I$ for the numerical simulations and the numerical solution of the full evolution equation (see Sec. 4c).

Figure 6

Temporal evolution of mean fragment length $\langle L\rangle$. (a) Simulations (solid lines) and model (dashed lines). (b) Experiments (symbols) and model with a distribution of breaking parameters centered around the value indicated in the legend (dashed lines). The error bars on the experimental points are representative of the number of fragments collected for each experiment. In both panels, $L_0/2$ and $L=1$ are indicated by the solid and dashed dotted horizontal black lines.

Figure 7

Mechanical properties of the glass fibers used in this study. (a) Young modulus measurements: Vertical deflection $\delta$ from the horizontal of the tip of a fiber as a function of the length $\ell$. The dashed line shows the power law $\delta \propto {\ell }^4$ [see Eq. (A1)]. (b) Histograms of the critical curvature ${\kappa }_B$ for raw fibers (blue) and heat-treated fibers (red). The dashed lines show Gaussian fits of the two histograms.

Figure 8

Fragment length distributions with different initial fiber lengths $L_0$. (a) Experiments for $K_B=2.4$ at $t/T_I=18\times {10}^3$. The errorbars correspond to $\pm 4$ fragments. (b) Model for $K_B=2$ at $t={10}^3$. The inset represents the time evolution of the mean fragment length for three different initial fiber lengths $L_0$. The colors correspond to different initial fiber lengths $L_0$, given in the legends and represented by vertical dashed lines in both panels. In both panels, the vertical dashed dotted lines indicate $L=1$.

Figure 9

Results obtained with the model based on rare fragmentation events in highly turbulent regions. (a) Time evolution of the fragment length distribution for $L_0=10$. The vertical dashed line shows the initial length while the vertical dashed dotted line indicates $L=1$. (b) Time evolution of the mean fragment length $\langle L\rangle$, for three different initial fiber lengths $L_0$. The horizontal dashed lines show the initial length with corresponding colors and the horizontal dashed dotted line indicates $L=1$.

Figure 10

Comparison with the experiments. (a) Temporal evolution of the mean fragment length $\langle L\rangle$ for experiments, and for the two models: the one based on the rare fragmentation events (blue) and the statistical one (red). This last model is mainly discussed in the main body of the paper [see Sec. 4c and Fig. 6] and is here shown for a global comparison. For each model, the dashed line corresponds to the results with a single parameter ($T_0$ or $K_B$) while the solid line represents the result for a distribution of the parameter considered. The horizontal dashed dotted line indicates $L=1$. (b) Distribution of the mean timescale $T_0$ between two fragmentation events for long fibers that allows us to match the temporal evolution of the experimental mean fragment length in panel (a). (c) Time evolution of the laboratory fragment length distribution for a fixed breaking parameter $K_B=2.1$, with error bars indicating $\pm 4$ fragments. With corresponding colors, the thin dashed lines represent the model discussed in this section. The vertical dashed line indicates $L_0$ for the model.