Breakup of a particulate suspension jet

As viscosity is increased, a liquid capillary jet accelerated by gravity stretches over increasingly large distances before eventually breaking up. This Newtonian behavior is profoundly altered for particulate suspensions. Adding solid particles to a liquid, which increases the effective viscosity, can paradoxically shorten the jet considerably [as first reported by Furbank and Morris, Phys. Fluids 16, 1777 (2004)]. This apparent contradiction is rationalized by considering finite-size effects occurring at the scale of a few particles. A model is presented which captures the breakup length of suspension jets observed experimentally for a broad range of liquid viscosities, particle sizes, and extrusion velocities of the jet and recovers the Newtonian case for vanishing particle sizes. These results can be readily extended to any stretched jet configuration and potentially to other fluid media having a granularity.

Figure 1

(a) A particulate suspension jet stretched by gravity $\left(d=550\mu \mathrm{m},u_0=16.8\mathrm{cm}/\mathrm{s}\right)$. (b) From left to right: Newtonian liquid jet and suspension jets with a particle volume fraction of 50% and increasing particle diameter $(d=40$, 135, and $550\mu \mathrm{m})$ having the same effective shear viscosity $\left(\sim 5\mathrm{Pa}\mathrm{s}\right)$ and flow rate $\left(u_0=4.2\mathrm{cm}/\mathrm{s}\right)$. The liquid jet is five times longer than shown. (c) Breakup length vs extrusion velocity for suspensions $(•$ PEGPG, $▴45$ wt% PEGPG, $▾30.5$ wt% Ucon oil), and pure liquids $(\circ$ PEGPG, $▵30.5$ wt% Ucon oil).

Figure 2

(a) Localized pinching hastening the breakup of a suspension jet $(d=135\mu \mathrm{m},{\eta }_0=0.13\mathrm{Pa}\mathrm{s},u_0=4.2\mathrm{cm}/\mathrm{s},h_0=2.90\mathrm{mm}$, times to breakup are indicated at the bottom). See movies in the Supplemental Material [19]. [(b), (c)] Jet thickness and velocity profiles from breakup to breakup for the sequence shown in panel (a) (the profiles are sampled every 5.6 ms; later profiles are darker). (d) Time evolution of the local minimal diameter until pinchoff for 17 uncorrelated breakup events [same conditions as in panels (a)–(c)]. (e) Steady velocity profile upstream of the breakup for two different particles sizes and a large nozzle, $h_0=11.6\mathrm{mm}\left({\eta }_0=0.13\mathrm{Pa}\mathrm{s}\right)$. The solid line is the Newtonian profile (1). $z_0$ embeds the extrusion velocity $u_0$. The envelop indicates the standard deviation. Inset: Same data in logarithmic scales.

Figure 3

(a) Dimensionless breakup length vs extrusion velocity for the suspensions [same data as in Fig. 1]. The dashed and dotted lines are, respectively, the predictions (3) and (4) for the viscous and inertial stretching regimes, with $n=10$. The solid line is the numerical solution (obtained with a relative precision smaller than ${10}^{-3}$ using a standard integration routine of mathematica) of Eqs. (1) and (2) connecting both asymptotes. The gray domain encloses the cases developing a capillary modulation (see text). (b) Jet with small particles $\left(d=10\mu \mathrm{m}\right)$ developing a Plateau-Rayleigh modulation prior to breaking up (at the arrow). (c) Breakup length in the presence of a capillary modulation of the jet vs $\tilde{d}={(2U^2{L}_0/u_0^2)}^{1/4}nd/h_0$. The data are those inside the gray domain in panel (a). The solid line is Eq. (6) with $n=10$. The value of $\alpha \left(=30\right)$ is calibrated with a pure liquid $\left({\eta }_0=2.4\mathrm{Pa}\mathrm{s}\right)$.