Extensional viscosity and thinning of a fiber suspension thread

The extensional effective viscosity, ${\eta }_{\mathrm{e}}$, of a suspension of solid cylindrical fibers in a Newtonian liquid is measured from the gravitational stretching of a quasisteady jet. We vary the length-to-diameter aspect ratio of the fibers, $a$, from $\sim 10$ to $\sim {10}^2$, and the particle volume fraction, $\varphi$, in the range $a\varphi \sim 0.1–2$. For low values of $\varphi$, the extensional viscosity is found to agree with Batchelor's model for the dilute regime, which assumes noninteracting fibers aligned with the stretching direction, but for higher concentrations, ${\eta }_{\mathrm{e}}$ is found to increase much more strongly with increasing $\varphi$ than predicted by available models assuming purely hydrodynamics interactions between the fibers. Additional experiments are performed on the breakup of an unstable capillary bridge for $a=11$. Although the variability in the bridge shape and the deviation from a viscous Newtonian dynamics in the late stage of the breakup increase significantly with increasing $\varphi$, it is found that the mean total duration of the breakup is in good agreement with a Newtonian effective dynamics limited by the extensional viscosity, ${\eta }_{\mathrm{e}}$.

Figure 1

Setups and fibers. (a) Constant flow rate jet stretched by gravity used for the extensional rheometry. (b) Shear cell used for the shear rheometry. (c) Capillary bridge breakup configuration. (d) Cylindrical polyamide fibers used for the experiments: (i) $d=28.9$ $\mu \mathrm{m}$, $a=10.8$. (ii) $d=28.3$ $\mu \mathrm{m}$, $a=27.6$. (iii) $d=28.1$ $\mu \mathrm{m}$, $a=106$. (iv) $d=48.1$ $\mu \mathrm{m}$, $a=11.5$.

Figure 2

(a) Typical jets observed for $a\approx 11$ and $d\approx 28$ $\mu \mathrm{m}$, $\varphi =0.01$ (left, ${\eta }_0=3.1$ Pa s) and $\varphi =0.20$ (right, ${\eta }_0=0.66$ Pa s), and $u_0=1.36$ cm/s. (b) Mean velocity versus distance to the nozzle for different volume fractions [same fibers as in (a), ${\eta }_0=3.1$ Pa s for $\varphi <0.10$ and ${\eta }_0=0.66$ Pa s for $\varphi \ge 0.10$]. The solid line shows the velocity for a Newtonian liquid, Eq. (4), and $z_0$ determines the velocity at the nozzle $u_0$, i. e., $u\left(z_0\right)=u_0$. The shaded envelops indicate the standard deviation of $u_0{h}_0^2/U{h}^2$. Inset: Same data in linear scales. (c) Relative effective viscosity of the suspensions as measured from the jet velocity profile (${\eta }_{\mathrm{e}}/{\eta }_0$) and with the shear cell (${\eta }_{\mathrm{s}}/{\eta }_0$).

Figure 3

(a) Specific extensional effective viscosity as a function of the diluteness parameter, $\varepsilon Q\left(\varepsilon \right)a^2\varphi$. The circles represent the values obtained with the jet setup for different aspect ratios but a similar particle diameter ($d\approx 28$ $\mu \mathrm{m}$). The triangles represent the experimental data by Weinberger [7], Mewis and Metzner [6], Pittman and Bayram [8], and the numerical data by Mackaplow and Shaqfeh [20] (the value of $a$ is given by the symbol darkness as indicated in the scale bar). The solid lines show the predictions for the dilute regime with aligned, Eq. (1), and isotropic fibers (see text in Sec. 1). (b) Same viscosity as in (a) normalized by $\varepsilon Q\left(\varepsilon \right)a^2$ as a function of $\varphi$. The dash-dotted and dotted lines are the predictions for the semidilute regime, Eqs. (2a) and (2b), at the same aspect ratios as in the experiments.

Figure 4

(a)–(c) Typical breakup dynamics of the capillary bridge. From top to bottom, $\varphi =0$, 0.05, and 0.24 ($a=10.8\pm 1.8,d=28.9\pm 1.2\mu \mathrm{m},{\eta }_0=3.05$ Pa s [for (a) and (b)] and 0.13 Pa s [for (c)]). The tags indicate the time, $t_{\mathrm{break}}-t$, remaining before breakup (from left to right, $h_{\min }=2.5$, 1, 0.5, and 0.1 mm, respectively). (d) Variability in the bridge shape for four different breakup events with $\varphi =0.24$ ($h_{\min }\approx 0.5\mathrm{mm}$). (e) Time evolution of the minimal diameter for 16 repetitions of the experiment with $\varphi =0.24$ (the time origin is taken when $h_{\min }=h_{\mathrm{ref}}=2.5$ mm). The large filled symbols and the envelope show, respectively, the average and the standard deviation of $t$ at a given value of $h_{\min }$. The small filled symbols correspond to the images shown in (c) and (d). (f) Time evolution of the bridge minimal diameter for different particle volume fractions [average and standard deviation over 16 repetitions of the breakup, same value of $a$ and $d$ as in (a)–(d)]. Inset: Same data as a function of the time made dimensionless with the initial thinning rate $|{\dot{h}}_{\min }{|}_{h_{\min }=h_{\mathrm{ref}}}/h_{\mathrm{ref}}$ (the rate is measured between $h_{\min }=h_{\mathrm{ref}}$ and $h_{\min }=0.9h_{\mathrm{ref}}$). (g) Relative thinning rate as inferred from the mean breakup time $t_{\mathrm{break}}$ and from the mean initial thinning rate $|{\dot{h}}_{\min }{|}_{h_{\min }=h_{\mathrm{ref}}}/h_{\mathrm{ref}}$ [same data for ${\eta }_{\mathrm{e}}/{\eta }_0$ and ${\eta }_{\mathrm{s}}/{\eta }_0$ as in Fig. 2].