Breakup dynamics of toroidal droplets in shear-thinning fluids

We use toroidal droplets to study the breakup dynamics of a Newtonian liquid jet in the presence of rheologically nonlinear materials. We find that the droplets resist breakup for times that are longer than those in the presence of Newtonian liquids. More importantly, we show that our experiments can be explained by incorporating the nonlinearities into the linear treatment of the problem through the strain-rate-dependent viscosity. Finally, we show that the scaling factor required to relate the viscosity to the growth rate associated to unstable modes is given by the elastic modulus of the outside material, illustrating that both the viscoelastic and shear-thinning nature of the outside material play a crucial role in the dynamics of the problem.

Figure 1

(a) Growth rate, $\nu$, of an unstable mode as a function of its dimensionless wave number, $2\pi a_0/\lambda$, with $\lambda$ the wavelength, in the case of a viscous jet immersed in another viscous liquid [4]. The values used for the inner and outer viscosities are ${\mu }_i=1\mathrm{mPa}\mathrm{s}$ and ${\mu }_o={10}^3\mathrm{mPa}\mathrm{s}$ (solid line), and ${\mu }_i={10}^3\mathrm{mPa}\mathrm{s}$ and ${\mu }_o=1\mathrm{mPa}\mathrm{s}$ (dashed line). The value of the interfacial tension is $\sigma =30$ mN/m and the jet radius is $a_0=1$ mm. (b) Dimensionless wavelength of the fastest unstable mode as a function of the viscosity contrast ${\mu }_i/{\mu }_o$ according to Tomotika [4]. The vertical line corresponds to the case of ${\mu }_i={\mu }_o$. Note the lack of symmetry around ${\mu }_i={\mu }_o$.

Figure 2

(a) Viscosity of mixture 1 (squares) and mixture 2 (circles) as a function of the strain rate, $\dot{\gamma }$, as measured using steady-state rheology. The viscosity can be modeled as power law with an exponent of $-0.78\pm 0.05$. (b) Schematic of the experimental setup. The symbols $R$ and $a_0$ correspond to the inner and tube radii of the torus, respectively. The value of the typical angular speed is $\sim 10$ rad/s and of the typical flow rate is 40 ml/h. (c)–(e) Evolution of a 10 cSt silicone oil toroidal drop in mixture 1. The neck radius, $a_n$, defined in (d), decreases very slowly in time until it abruptly changes near breakup. The torus has an initial aspect ratio of $R/a_0=3.8$ and $a_n(t=0)=0.56$ mm.

Figure 3

(a) Time evolution of the local strain, $\gamma$, for a torus made out of 1000 cSt silicone oil in mixture 2. The torus has an initial aspect ratio of $R/a_0=2.8$ and $a_n(t=0)=1.23$ mm. (b) $ln\gamma$ as a function of time for the same torus showing that the behavior is highly nonlinear; this is due to the effects of the outside shear-thinning fluid. (c) Evolution of the inner radius, $R$, for the same torus, illustrating that in our experiments, $R$ does not change in time; this confirms that the time dependence of $\gamma$ is not due to shrinking. (d) Comparison between the time evolution of $ln\gamma$ for a torus in mixture 2 (squares) and a torus made out of water with 10 mM SDS in $30000$ cSt silicone oil (circles). The breakup time, $t_b$, is 144.7 s in the presence of the Carbopol mixture and 18.3 s in the presence of the Newtonian liquid. We also emphasize that the values of $\gamma$ are limited by the resolution of the camera. Hence, we only include points where the radius $a_n$ has changed at least a pixel size; this is why the time intervals are not constant.

Figure 4

Inverse growth rate, obtained from the local slope of $ln\gamma$ versus time, as a function of the strain rate, obtained from the local slope of $\gamma$ versus time. Closed symbols correspond to data from 15 tori in mixture 1, with $2.4<R/a_0<14.1$ and $0.51<a_0<1.2$ mm. Open symbols correspond to data from 10 tori in mixture 2, with $2.8<R/a_0<9.3$ and $0.55<a_0<1.3$ mm. The data is consistent with a power-law behavior with exponent $\beta \approx 0.8$. The two lines correspond to $\mu /G^{\prime }$ for mixture 1 (dashed) and mixture 2 (solid), where $\mu$ is the viscosity measured using steady-state rheology and shown in Fig. 2, and $G^{\prime }$ is the elastic modulus obtained using oscillatory rheology and shown in Fig. 5.

Figure 5

Shear elastic, $G^{\prime }$, and viscous, $G^{″}$, moduli as a function of frequency, $\omega$, for mixtures 1 (squares) and 2 (circles). We use a cone-plate configuration, with angle $2^{\circ }$ and diameter 25 mm. The strain used to perform the measurements is 0.01, which is within the linear regime.