2.1 Experimental set-up

We generated extensional flows experimentally by discharging a thin layer of viscous fluid into a bath filled with low-viscosity but denser fluid. This way traction on the viscous fluid was reduced substantially, allowing extensional stresses to dominate. The experimental apparatus (figure 1) consisted of a transparent bath of $1\times 1~\text{m}^{2}$ filled with low-viscosity, transparent salt solution to a depth of 15 mm. At the centre of the bath and attached to its bottom we placed a conical section made of transparent polydimethylsiloxane (PDMS), which had a trapezoidal cross-section (figure 1 b) with dimensions 15 mm height, 10 cm top base diameter and 18 cm bottom base diameter. The displacing complex fluid was driven from a container by a peristaltic pump at constant flux ( $1\lesssim Q\lesssim 14$ ) $\text{g}~\text{s}^{-1}$ , which was measured with an accuracy of $\pm 0.05~\text{g}~\text{s}^{-1}$ using a computer connected to a digital balance that supported the container. From the pump the fluid was discharged axisymmetrically through a tube, 10 mm in diameter, over the centre of the top base of the conical section. The fluid continued to propagate under gravity over the conical section and beyond, displacing the ambient salt solution. The propagation of the interface was imaged using a digital SLR camera in plan view from the bottom of the bath through a 45 $^{\circ }$ mirror at a rate of one frame every 2–6 s, depending on the source flux.

Figure 1. (a) Schematic diagram of the experimental apparatus to generate extensional flows. The region where the flow evolves (dotted rectangular) is expanded in (b). (b, top) Schematic side view of the flow region, showing the polymer solution (blue) propagating radially over the conic plate and into a bath of salt solution (cyan), resulting in a no-stress interface at the surface and the base of the displacing fluid. (b, bottom) An image showing the camera plan view of the flow at a stage where the front of the polymer solution developed four fingers. The vertical dashed lines mark the radius $r_{G}$ , which is where the fluid–fluid interface intersects with the conical section.

To keep the level of the ambient fluid in the tank fixed as the complex fluid was intruding, we used a secondary pump controlled by an optical feedback circuit. Specifically, a red laser beam was emitted through the ambient fluid and underwent total internal reflection from the ambient-fluid free surface towards an optical sensor. A rise in the level of the ambient fluid shifted the beam trajectory from the sensor and consequently operated the pump that drained excess fluid, until the beam resumed intersection with the sensor, thereby turning the pump off. This way we managed to keep the level of the ambient fluid fixed to within a 0.1 mm accuracy.

The complex fluid consisted of Xanthan gum (Jungbunzlauer) dissolved in distilled water (1 % in weight), having density ${\approx}1.02~\text{g}~\text{cm}^{-3}$ . Xanthan is a semirigid and high-molecular-weight polysaccharide produced by the Xanthomonas campestris bacterium (Lapasin Reference Lapasin1995). To prepare the solution we used a laboratory mixer to form a smooth vortex in the centre of a 3 l beaker containing the distilled water. The Xanthan powder (from a single batch) was then spread into the centre of the vortex through a sieve to disperse powder aggregates, thereby allowing faster polymer hydration. Next, we added a couple of millilitres of blue dye and maintained the mixing for one hour at the maximum speed possible so long as air bubbles did not intrude into the fluid. Then the solution was left for the next 24 h in a closed container to allow full hydration of the polymer. This way we obtained an agglomerate-free and bubble-free Xanthan solution.

The rheology of Xanthan gum solutions of ${\sim}1\,\%$ concentration has been studied extensively in various flow configurations. Under steady-shear flow, such solutions behave as a power-law, shear-thinning fluid with exponent ${\approx}6$ (Sayag & Worster Reference Sayag and Worster2013) for the solution we used, although other measurements find that it can vary between ${\approx}5$ (Martin-Alfonso et al. Reference Martin-Alfonso, Cuadri, Berta and Stading2018) and ${\approx}7$ (Song, Kim & Chang Reference Song, Kim and Chang2006a ) (we define shear thinning with exponents larger than 1). At low shear rates there are indications of bounded shear viscosity at more dilute polymer concentrations than 1 % (Wyatt & Liberatore Reference Wyatt and Liberatore2009; Jaishankar & McKinley Reference Jaishankar and McKinley2014; Allouche et al. Reference Allouche, Botton, Henry, Millet, Usha and Hadid2015) with a zero strain-rate viscosity of over $10^{6}$ times that of water at $25\,^{\circ }\text{C}$ at shear rates lower than $10^{-3}~\text{s}^{-1}$ (Jaishankar & McKinley Reference Jaishankar and McKinley2014), and possibly the presence of yield stress (e.g. Herschel-Bulkley fluid, Song et al. Reference Song, Kim and Chang2006a ). These solutions also exhibit elastic properties such as primary normal-stress difference over a wide range of strain rates (Jaishankar & McKinley Reference Jaishankar and McKinley2014), and a relaxation time of $\unicode[STIX]{x1D706}\approx 0.1~\text{s}^{-1}$ (Stokes et al. Reference Stokes, Macakova, Chojnicka-Paszun, de Kruif and de Jongh2011). In oscillatory-shear flow, the storage modulus of such suspensions is 2–4 times larger than the loss modulus in frequencies ranging between 0.01 and $100~\text{rad}~\text{s}^{-1}$ (Song, Kuk & Chang Reference Song, Kuk and Chang2006b ). In uniaxial extensional flows, Xanthan solutions exhibit strain hardening (Stelter & Brenn Reference Stelter and Brenn2002; Stelter et al. Reference Stelter, Brenn, Yarin, Singh and Durst2002), as well as strain-rate softening of the extensional viscosity (extensional thinning) with an exponent of ${\approx}3$ (Jones, Walters & Williams Reference Jones, Walters and Williams1987; Ferguson, Walters & Wolff Reference Ferguson, Walters and Wolff1990; Martin-Alfonso et al. Reference Martin-Alfonso, Cuadri, Berta and Stading2018), while the behaviour at extensional rates lower than approximately $0.5~\text{s}^{-1}$ is not known. The ratio of extensional to shear viscosity (Trouton ratio) of such solutions grows with the strain rate, reaching approximately 30 at strain rate of $1~\text{s}^{-1}$ (Jones et al. Reference Jones, Walters and Williams1987; Martin-Alfonso et al. Reference Martin-Alfonso, Cuadri, Berta and Stading2018), but at lower strain rates it drops and is expected to converge to the Newtonian value of 3, as typical of polymeric solutions (Bird, Armstrong & Hassager Reference Bird, Armstrong and Hassager1987). Therefore, at the higher end of the strain rate, such Xanthan solutions have a more significant elastic deformation, while viscous deformation becomes more significant at the lower end.

The ambient fluid consisted of sodium chloride solution with density ${\approx}1.181\,\text{g}\,\text{cm}^{-3}$ , and therefore denser than the Xanthan gum solution. This combination of fluids resulted in the formation of a region in the flow where the Xanthan solution was propagating over the denser ambient fluid, which was significantly less viscous and consequently applied negligible traction.

2.2 Evolution of the interface

We performed a range of experiments in which we varied the source flux $Q$ of the polymer solution, while keeping fixed the level and density of the ambient salt solution, and the concentration of the polymer solution (table 1).

Table 1. The source flux $Q$ ( $\pm 0.05~\text{g}~\text{s}^{-1}$ uncertainty) that was specified in each experiment and the resulting measured grounding-line position $r_{G}$ and the fluid thickness $h_{G}$ at the grounding line. In all the experiments the polymer solution consisted of 1 % Xanthan gum and the density of the salt solution was $\unicode[STIX]{x1D70C}=1.181~\text{g}~\text{cm}^{-3}$ .

Once discharged from the tube, the gravity-driven polymer solution propagated axisymmetrically over the truncated cone surrounded by a deep layer of denser and low-viscosity salt solution. When the circular fluid front intruded into the ambient salt solution, as it advanced along the sloping section of the cone, buoyancy forces due to the surrounding denser fluid caused it to detach from the substrate and float (figure 1 b, top). This way, a circular contact line (grounding line) was formed at radius $r_{G}>5~\text{cm}$ over the slope of the cone, separating between an inner region ( $r<r_{G}$ ) where no-slip basal conditions were applied to the polymer solution, and an outer region ( $r>r_{G}$ ) where no-stress basal conditions were applied to the polymer solution. Within a short transient the grounding line advanced down the slope of the substrate and reached a steady position that was maintained for all $t>0$ . We discovered that the leading front of the displacing polymer solution split into a number of tongues in the vicinity of $r_{G}$ , which moved as solid blocks similar to the movement of floating ice tongues (e.g. Holdsworth Reference Holdsworth1983) or of foam under wall slip (Lindner et al. Reference Lindner, Coussot and Bonn2000b ). The region between the tongues resembled fractures with sharp tips near $r_{G}$ , reminiscent of fracture tips (figure 1 b, bottom). As the tongues grew longer, some of those tips were advected with the flow and, as they did, the space between the tongues closed by the joining of adjacent tongues into wider ones. Consequently, the number of tongues declined with time (figure 2 and supplementary movies are available at https://doi.org/10.1017/jfm.2019.777). Such an inverse cascade of the number of tongues also appears to characterise patterns in squeezed pastes (Mascia et al. Reference Mascia, Patel, Rough, Martin and Wilson2006), in fractured thin elastic plates (Vermorel, Vandenberghe & Villermaux Reference Vermorel, Vandenberghe and Villermaux2010; Vandenberghe, Vermorel & Villermaux Reference Vandenberghe, Vermorel and Villermaux2013) and in melt transport (Spiegelman, Kelemen & Aharonov Reference Spiegelman, Kelemen and Aharonov2001). However, the breakdown of the circular symmetry that we report is in sharp contrast with similar flows that involved Newtonian displacing fluids (Pegler & Worster Reference Pegler and Worster2012), in which the fluid–fluid interface remained circular.

Figure 2. Snapshots (plan view) from two experiments with source flux (a–d) $Q=3.90\pm 0.05~\text{g}~\text{s}^{-1}$ and (e–h) $Q=2.64\pm 0.05~\text{g}~\text{s}^{-1}$ , showing the evolution of the fluid–fluid interface. The injected complex fluid is coloured blue, and the ambient less viscous, denser fluid as well as the conical substrate are transparent. Time is measured with respect to the instant when the interface reached the grounding-line position. The Fourier modes of each interface in (e–h) are shown in figure 5(b).

Figure 3. Snapshots (plan view) from the experiment with source flux $Q=2.64\pm 0.05~\text{g}~\text{s}^{-1}$ tracing the evolution of the interface as it transits the steady grounding-line position at $t=0$ . The injected polymer solution is coloured blue, and the ambient denser fluid as well as the conical substrate are transparent. The origin is the centre of the tube outlet (

). The lower base of the conical substrate (18 cm in diameter) is marked in (– – –, cyan) in (a). The steady position of the grounding line is marked at all snapshots (– – –, grey). The inset in panel (e) shows the region in the flow (dashed square) that is viewed in the five snapshots (a–e). Panel (f) shows the intensity of the light (green channel) transmitted through the flow along the radius ( $\cdots \cdots$ , green) in each of the previous panels, normalised by the intensity of the background (ambient fluid). The grounding-line position at each time is the local minimum of the intensity between 5 cm (the top edge of the substrate slope, $\cdots \cdots$ ) and ${\sim}6~\text{cm}$ (▴), where (- - -, red) marks the grounding-line steady position.

Figure 4. Fourier decomposition of the fluid–fluid interface: examples from three different experiments ( $Q=2.64,1.36,1.37\pm 0.05~\text{g}~\text{s}^{-1}$ ), showing snapshots of the traced interface (——) and the results of the regression (——, red) of part of the interface ( $r\leqslant 2.5\,r_{G}$ ) to the Fourier series (2.1).

2.3 Analysis of the results

Having image sequences of the experimental results, we first determine the grounding-line steady position $r_{G}$ and the instant when the front crossed that position for each experiment. We define the origin $r=0$ at the centre of the feeding tube using snapshots taken before the fluid was discharged. We also identify the rim of the larger base of the PDMS substrate (figure 3 a) and use its known diameter (18 cm) to convert from pixels to dimensional length. Then we identify the position of the grounding line using the intensity of the light transmitted through the polymer solution (figure 3 f). Specifically, the part of the fluid that spreads along the higher base of the conical section from the nozzle is relatively thick and appears as a dark disk that becomes radially brighter towards the edge of that top base. Near that edge and just before the fluid intrusion into the salt solution, the fluid thickness is minimal and appears as a thin ring of brighter colour (figure 3 e), with a corresponding local maximum in the light intensity at radius 5 cm (figure 3 f). Across that ring, the polymer solution intrudes into the ambient fluid along the substrate slope (figure 1 b), so it becomes thicker and consequently darker with radius, as also implied from the corresponding decline in the light intensity. Further down along the substrate slope, another sharp transition occurs into a brighter region, which implies that the fluid layer has gone through a sudden thinning. Such abrupt thinning that coincides with a local minimum in the light intensity (figure 3 f) is due to the detachment of the fluid layer from the slope of the substrate (figure 1 b, top). Therefore, we identified that local minimum in intensity, which varies with azimuth by less than $\pm 2$ mm, with the position of the grounding line. We follow the propagation of the grounding line along the slope from the time it is formed very close to the top base and until it reaches a steady position within a short transient (figure 3 f). Upon detachment of the polymer solution, the basal friction drops abruptly and substantially and the fluid layer flows as a plug (no vertical shear), becoming a pure extensional flow with principal axes along the radial and azimuthal directions resulting in horizontal stretching, and perpendicular to the direction of flow resulting in vertical thinning (e.g. Robison, Huppert & Worster Reference Robison, Huppert and Worster2010; Pegler & Worster Reference Pegler and Worster2013). Having the steady grounding-line position and knowing the substrate geometry, we calculate the fluid thickness at that position, $h_{G}$ , assuming that the fluid is floating (table 1). In addition we define $t=0$ at the instant when the fluid front reaches the steady position of the grounding line (figure 3).

The sharp colour contrast between the polymer solution and the salt solution allowed a straightforward tracing of the fluid–fluid interface $r_{N}(\unicode[STIX]{x1D703},t)$ as a function of time (figure 4) using image-processing tools. Applying a fast Fourier transform to the raw interface leads to inconsistent results, particularly when the interface roughness is substantial, having sections with multiple fracture-like elements that enhance the interface length in these parts disproportionally (e.g. figure 4 c). To avoid such complications we first truncate at each instant the part of the interface having $r>2.5r_{G}$ and then fit to the residual traced interface a Fourier series of the form

(2.1) $$\begin{eqnarray}f(t)=a_{0}(t)+\displaystyle \mathop{\sum }_{k=1}^{N}a_{k}(t)\sin (k\unicode[STIX]{x1D703}+\unicode[STIX]{x1D719}_{k}(t)),\end{eqnarray}$$

where the amplitudes $a_{0\ldots N}$ and the phases $\unicode[STIX]{x1D719}_{1\ldots N}$ are the fitted parameters (figure 4). We chose $N=30$ noticing that a larger value does not affect the amplitude distribution $a_{k}(t)$ in a measurable way. In addition, the value of the truncation radius was robust and was used for all experiments. We applied this routine to each experimental snapshot and got a time series of the wavenumber amplitude distribution (figure 5 a,b). Then, we normalised the amplitude distribution at each instant with the maximal amplitude at that instant, obtaining the normalised amplitude distribution of the different Fourier modes for the whole duration of the experiment (figure 5 c). For clarity, that distribution is a smoothed (interpolated) version of the original discrete distribution (appendix A). This wavenumber–amplitude distribution, which is typical for a wide range of experiments, shows that the initial circular front ( $t<0$ ) first develops a high-wavenumber pattern ( $0<t<20~\text{s}$ ) that then declines progressively and monotonically through a descending set of dominating wavenumbers to a finite $k=3$ dominating wavenumber. This reduction of the dominating wavenumber is reflected in the experiment through the repeated merging of adjacent fingers into wider ones (figure 2).

Figure 5. (a) The traced fluid–fluid interface at times $t=20$ , 50, 80, 160 s (thick curves) in the experiment with $Q=2.64\pm 0.05~\text{g}~\text{s}^{-1}$ (figure 2 e–h), with the grounding line marked in (——, green). (b) Amplitude distribution (lines) of the dominating Fourier wavenumbers that compose each of the interfaces in (a) (colours match) obtained by fitting a Fourier series of the front $r_{N}(\unicode[STIX]{x1D703},t)$ (♢ marking the maximum amplitude in each case). (c) Time evolution of the amplitude distributions (interpolated) normalised by the maximum amplitude at each time, where darker colour represents modes with higher amplitudes (discrete version is shown in appendix A). Vertical gridlines mark the distributions at times that match (by colour) those shown in (b).

Figure 6. Time evolution of the normalised amplitude of the Fourier modes (dark, smoothed) that comprise the fluid–fluid interfaces for experiments with source flux in the range $1.3\lesssim Q\lesssim 13.4\pm 0.05~\text{g}~\text{s}^{-1}$ . The maximal amplitudes at each instant are highlighted with colour patches. In each panel, the horizontal gridline marks the average wavelength after the initial cascade of wavenumbers.

We repeated the same analysis procedure for all of the experiments with varying source flux $Q$ (table 1), and generated the corresponding wavenumber–amplitude distributions (figure 6, 7). The evolution of the interface is reflected in the evolution of the dominant wavenumbers – those modes that have maximum amplitude at each instant (figure 8). In these experiments we find that the wavenumber evolution at early time has a different pattern compared to late time. At early time, the patterns that emerged from the initial circular front had high dominating wavenumber, which gradually cascaded into a lower dominating wavenumber at increasingly growing transition times. This behaviour characterised all experiments, and quantitatively the rate of the cascade (transition time to lower wavenumbers) and the lowest wavenumber that the pattern converged to was strongly dependent on the source flux $Q$ . At later times, after the initial cascade to the lower wavenumber, the interface evolved in two distinguished patterns depending on the input flux. In the higher-flux regime ( $Q\gtrsim 1.37~\text{g}~\text{s}^{-1}$ ), the asymptotic pattern remained in the lowest wavenumber of the initial cascade. That lowest wavenumber declined with the source flux (figure 8 a). In the lower-flux regime ( $Q\lesssim 1.37~\text{g}~\text{s}^{-1}$ ), the dominating wavenumber did not settle on a well-defined fixed value, rather, it changed stochastically within a finite band of wavenumbers (figure 8 b). This appears to imply that, while instantaneously the pattern has an integer dominant wavenumber, on average the system can asymptotically settle on a fractional wavenumber.

Figure 7. Time evolution of the normalised amplitude of the Fourier modes (dark, smoothed) that comprise the experimental fronts for experiments with source flux in the range $Q=0.73,1.37,1.38\pm 0.05~\text{g}~\text{s}^{-1}$ , in which the dominant wavenumber varies stochastically after the initial wavenumber cascade. The maximal amplitudes at each instant are highlighted with colour patches. In each panel the ○, green marks the instant when the pattern had a maximum dominant wavenumber. The ○, cyan marks the instant when the pattern had the lowest wavenumber at the termination of the wavenumber cascade, beyond which the dominant wavenumber evolved stochastically. In each panel, the horizontal gridline marks the average wavenumber after the initial cascade of wavenumbers.