Spreading and fragmentation of particle-laden liquid sheets

The fragmentation of liquid sheets produces a collection of droplets. The size distribution of the droplets has a considerable impact on the coating efficiency of sprays and the transport of contaminants. Although many processes commonly used particulate suspensions, the influence of the particles on the spreading dynamics of the sheet and its subsequent fragmentation has so far been considered negligible. In this paper, we consider experimentally a transient suspension sheet that expands radially. We characterize the influence of the particles on the dynamics of the liquid sheet and the fragmentation process. We highlight that the presence of particles modifies the thickness and reduces the stability of the liquid sheet. Our study suggests that particles can significantly modify the dynamics of liquid films through capillary effects, even for volume fractions much smaller than the maximum packing.

Figure 1

(a) Schematic of the experimental setup. (b) Viscosity of a suspension of $2a=140\mu \mathrm{m}$ particles (black full circles) as a function of the volume fraction $\varphi$. The red curve is given by Eq. (1), which allows us to obtain water-glycerol mixtures with equivalent effective viscosities (brown empty diamonds). All viscosity values are measured at $T=23{}^{\mathrm{o}}\mathrm{C}$.

Figure 2

(a) Radial evolution of the thickness of the liquid film for different times after impact from 2 to 8 ms. These curves were obtained with the carrier fluid (without particles). The dashed lines show the particle sizes for comparison. (b) Typical spatial distributions of the thickness at maximum spreading ($t=\tau /3=4.2\mathrm{ms}$) for the carrier fluid. The correspondence between thickness and colors is indicated in the color bar. Scale bars are 2 mm.

Figure 3

Top view of the time evolution of the suspension sheet from the impact to the fragmentation in smaller droplets for (a) increasing volume fraction ($\varphi =0\%,15\%,25\%,35\%$, clockwise) of $140\mu \mathrm{m}$ particles and (b) increasing diameter of particles (equivalent liquid, $2a=40,80,140\mu \mathrm{m}$) at a volume fraction $\varphi =35\%$.

Figure 4

(a) Time evolution of the normalized mean diameter $d\left(t\right)/D$ for varying volume fraction $\varphi$ of $140\mu \mathrm{m}$ particles. (b) Rescaled maximum diameter of the liquid sheet $d_{\max }/D$ for suspensions of $140\mu \mathrm{m}$ particles (circles) and equivalent fluids (diamonds). The volume fraction $\varphi$ of the suspension is reported on the bottom $x$-axis, while the corresponding viscosities are shown on the top nonlinear $x$-axis. The dashed line shows the best fit using Eq. (2), with $\alpha =4.0$ and $\beta =4.6{\left(\mathrm{Pa}\mathrm{s}\right)}^{-1/2}$.

Figure 5

Mean velocity of the particles $u_p$ as a function of the time $t$ at which they leave the impactor, normalized by $\tau$. The velocity is normalized by the characteristic velocity $u_{\varphi }=u_0/[1+\beta \sqrt{\mu \left(\varphi \right)}]$ (see the main text), which is proportional to the ejection velocity of the equivalent fluid (the coefficient depends on the impactor geometry), and the dashed line corresponds to the theoretical prediction for the fluid velocity field $u=u_p/u_{\varphi }\propto {(t/\tau )}^{-1}$.

Figure 6

(a) Typical evolution of the diameter of the liquid sheet, for various particle diameter $2a$ ($\varphi =35\%$). Inset: Normalized maximum diameter of the liquid sheet as a function of the particle diameters for $\varphi =15\%,30\%$, and $35\%$. (b) Lifetime $t_{\mathrm{life}}$ and (c) expansion time $t_{\max }$ for varying particle diameters and volume fraction $\varphi$. In these figures, the yellow triangles, orange squares, and red circles correspond to particles of diameter $2a=40,80,140\mu \mathrm{m}$, respectively, and the equivalent fluid is shown in brown diamonds. The dashed line in (b) represents the average value for the equivalent fluid, and the continuous line in (c) is the theoretical prediction for a liquid [44].

Figure 7

Illustration of the retraction path: superposition of images during the retraction of the liquid sheet for (a) the carrier fluid, (b) the equivalent fluid (${\varphi }_{\text{eff}}=35\%$), and (c) a suspension of $140\mu \mathrm{m}$ particles at $\varphi =35\%$. The first image (in blue) is 3.5 ms after impact, and the interval between images is 0.375 ms. Scale bars correspond to 5 mm.

Figure 8

(a),(b) Typical spatial thickness distribution in the sheet at maximum spreading ($t=\tau /3$) for (a) the equivalent fluid ${\varphi }_{\mathrm{eff}}=35\%$ and (b) the suspension ($2a=140\mu \mathrm{m}$ and $\varphi =35\%$). (c) Droplet radius distribution resulting from the fragmentation of the liquid sheet for different volume fraction ($0\%<\varphi <35\%$, $2a=140\mu \mathrm{m}$) normalized by the mean droplet radius $\langle r\rangle$. Inset: Mean droplet radius $\langle r\rangle$ as a function of the volume fraction $\varphi$. The dashed line corresponds to Eq. (4) with a prefactor 0.82 and $\beta =4.6{\left(\mathrm{Pa}\mathrm{s}\right)}^{1/2}$. The color code is the same as in Fig. 4. (d) Probability distribution function (PDF) of the droplet radii for different particle sizes and a constant volume fraction $\varphi =35\%$. Inset: Mean droplet size for increasing particle sizes (filled symbols) at $\varphi =35\%$. For comparison, the brown line with diamonds and the black dotted line show the mean diameter obtained with the equivalent viscosity fluid and for the interstitial fluid, respectively. The black line is the mean value for the suspensions. In both figures, the black dashed line shows the PDF given by Eq. (3) with $n=10$.