Destabilization of a Saffman-Taylor Fingerlike Pattern in a Granular Suspension

We study the Saffman-Taylor instability in a granular suspension formed by micrometric beads immersed in a viscous liquid. When using an effective viscosity for the flow of the suspension in the Hele-Shaw cell to define the control parameter of the system, the results for the finger width of stable fingers are found to be close to the classical results of Saffman-Taylor. One observes, however, an early destabilization of the fingers that can be attributed to the discrete nature of the individual grains. Classically, the threshold of destabilization is linked to the noise in the cell and is thus difficult to quantify. We show that the grains represent a “controlled noise” and produce an initial perturbation of the interface with an amplitude proportional to the grain size. The finite amplitude instability mechanism proposed by Bensimon et al. allows us to link this perturbation to the value of the threshold observed.

Figure 1

Schematic drawing of the experimental setup. Evolution of a typical finger with increasing finger velocity (a)–(f).

Figure 2

(a) Relative viscosity of the suspensions ($D=80\mu \mathrm{m}$) as a function of the grain fraction $\varphi$ obtained from rheological measurements ${\eta }_{\mathrm{Rh}}(\varphi )/{\eta }_0$ (□) and from flow in the Hele-Shaw cell ${\eta }_C(\varphi )/{\eta }_0$ (●). The solid line represents the prediction of Zarraga et al. [8]. (b) and (c) Relative finger width $\lambda$ as a function of $1/B=12(W/b{)}^2({\eta }_{C}U/\gamma )$ for the same suspensions. Cell width $W=4\mathrm{cm}$ and cell thickness $b=1.43\mathrm{mm}$ (b) and $b=0.75\mathrm{mm}$ (c). The solid line represents the results for pure fluid.

Figure 3

(a) Image processing and measurement of the fluctuation of the relative finger width. (b) $\delta \lambda$ as a function of $1/B$ for pure fluid (■) and a suspension (○) ($\varphi =10\%$, $D=80\mu \mathrm{m}$) in a cell geometry of $W=4\mathrm{cm}$ $b=1.43\mathrm{mm}$. The arrow indicates the stability threshold $1/B_c$.

Figure 4

Control parameter $1/B_c$ at the stability threshold as a function of grain fraction $\varphi$. $D=80\mu \mathrm{m}$, $W=4\mathrm{cm}$, $b=0.75\mathrm{mm}$ (○) and $b=1.45\mathrm{mm}$ (■) (not shown here: $1/B_c>15000$ for pure fluid). Inset: Localization of the perturbation (○) during a typical experimental finger destabilization.

Figure 5

Experimental values (□) and theoretical prediction from Bensimon et al. (solid line). Experiments performed by changing the grain diameter $D=20$, 40, 80, and $140\mu \mathrm{m}$, the cell thickness $b=0.75–1.43\mathrm{mm}$ and the cell width $W=2–4\mathrm{cm}$. Additional experiments (●) using a different silicon oil (Dow Corning 704). Inset: Schematic drawing of a particle approaching a free interface in a stagnation point flow.