Figure 1

Viscous fingering patterns of air (black) penetrating into oil (white) in rectangular and circular Hele-Shaw cells for different forcing levels. A fjord is the region of oil between two adjacent fingers. Rectangular cell: $\mathrm{Ca}=\left(\mathrm{a}\right)$ 0.0056 and (b) 0.046; the rectangles are $25\mathrm{cm}$ wide and $190\mathrm{cm}$ long, while the entire cell is $254\mathrm{cm}$ long. Circular cell: $\mathrm{Ca}=\left(\mathrm{c}\right)$ 0.000 88 and (d) 0.27; the radial patterns are approximately $20\mathrm{cm}$ in diameter, grown in a cell of diameter $28.8\mathrm{cm}$. The Ca values were computed using the average velocity of the farthest tip.Reuse & Permissions

Figure 2

(a) Time development of a fjord grown in the rectangular cell at $\mathrm{Ca}=0.0076$ (cf. Fig. 1 caption). Adjacent interfaces are separated by $1\mathrm{s}$. (b) Curvature $\kappa$ and (c) velocity $V$ of the point on the finger $(\kappa >0)$ that evolves into a fjord $(\kappa <0)$ and stops moving; $t=0$ is the time at which $\kappa =0$. The velocity $V_0$ used in computing the instability wavelength ${\lambda }_c$ [Eq. (2)] corresponds to $\kappa =0$; here $V_0=0.25\pm 0.01\mathrm{cm}∕\mathrm{s}$. The arrow points to the minimum of curvature in time, which defines the fjord centerline arc length $\ell =0$. (d) Fjord width $W$ as a function of the fjord centerline arc length $\ell$. The base width of the fjord, $W_0$, is given by the fjord width extrapolated to $\ell =0$; here $W_0=1.08\pm 0.06\mathrm{cm}$. (e) The selected opening angle of the fjord, given by the plateau in $\psi \left(\ell \right)$, is ${\psi }_0=$7.1°±0.5°.Reuse & Permissions

Figure 3

The measured fjord base width $W_0$ [Fig. 2d] versus the predicted width ${\lambda }_c∕2$, where ${\lambda }_c$ is the instability wavelength, Eq. (2). The inset illustrates how an initial infinitesimal perturbation of a front sets the wavelength for a large-amplitude fjord. Here, the front is moving to the right and is observed in a comoving frame. The lengths of the error bars are the rms deviations from the line.Reuse & Permissions

Figure 4

(a) The fjord opening angle ${\psi }_0$ in the rectangular cell increases slightly with forcing parameter (the modified capillary number $1∕B$). (b) Probability distributions $p\left({\psi }_0\right)$ measured for the opening angle of all fjords observed in the circular cell (●) and of fjords in the high-fjord-density regime $(1∕B>25000)$ of the rectangular cell (▵). The data for the rectangular and circular cells yield, respectively, ${\psi }_0=7.9°\pm 0.8°$ and ${\psi }_0=8.2°\pm 1.1°$, where the uncertainties are one standard deviation.Reuse & Permissions