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Side branched patterns, coalescence and stable interfaces during radial displacement of a viscoelastic fluid

We explore the interfacial instability that results when a Newtonian fluid (a glycerol-water mixture, inner fluid) displaces a viscoelastic fluid (a dense cornstarch suspension, outer fluid) in a radial Hele-Shaw cell. As the ratio of viscosities of the inner and outer fluids is increased, side branched interfacial patterns are replaced by more stable interfaces that display proportionate growth and finger coalescence. We correlate the average finger spacing with the most dominant wavelength of interfacial instability, computed using a mathematical model that accounts for viscous fingering in miscible Hele-Shaw displacements. The model predictions on the role of viscosity ratio on finger spacing are in close agreement with the experimental observations. Our study lends insight into the significant contribution of the viscoelasticity of the outer fluid on the morphology and growth of interfacial patterns.


Introduction
The displacement of a more viscous fluid by a less viscous fluid renders the fluid-fluid interface unstable and leads to intricate patterns called viscous fingers [1][2][3][4][5][6][7]. A fundamental understanding of these instability mechanisms is key to the design and enhancement of processes like oil recovery, sugar refining and carbon sequestration [8][9][10][11]. The radial Hele-Shaw cell is often used to track and obtain information about viscous fingering instabilities in a quasi two-dimensional flow [12,13]. A few studies have focused on the vital role that viscosity mismatch plays in the emergence of interfacial patterns [14,15]. In experiments by Lajeunesse et al. [14], the onset of instability at the interface between two miscible Newtonian fluids was studied. In spite of the low interfacial tension of the fluids, a complete suppression of instability was reported above a critical value of viscosity ratio, η in /η out , where η in and η out are, respectively, the viscosities of the inner and outer fluids. Bischofberger et al. [15] studied the pattern evolution at the interface between a pair of miscible Newtonian fluids over a wide range of viscosity ratios. They reported a transition from a fractal growth regime (similar to that reported by Saffman and Taylor [1]) to a stable regime (as reported by Lajeunesse et al. [14]) as η in /η out is increased.
Owing to the complex flow behaviour of non-Newtonian fluids at finite shear rates, a detailed understanding of instabilities at the interface between a Newtonian and a non-Newtonian fluid remains elusive. Interfaces involving non-Newtonian fluids such as polymer solutions [16,17] and colloidal suspensions [18,19] have been investigated in Hele-Shaw cells. A range of distinct pattern morphologies in the gas-driven displacement of cornstarch suspensions exhibiting discontinuous shear-thickening has been reported [20]. In a significant advance to earlier studies, we report here the emergence and growth of large scale patterns of distinct morphologies at a Newtonian/non-Newtonian liquid interface over a wide range of viscosity ratios. We observe and model the displacement of dense aqueous suspensions of cornstarch particles (outer fluid) by glycerol-water mixtures (inner fluid) in a radial Hele-Shaw cell. Transitions from side branched patterns to increasingly stable interfaces exhibiting proportionate growth and a new double coalescence phenomenon are reported. The concentration-dependent elasticity of the cornstarch suspension is seen to reduce the finger length, while resulting in the breakup of the interface into multiple small protrusions. The pattern morphologies at the initial stages are explained by employing a linear stability analysis of the interface between a Newtonian and a shear-thinning fluid [21]. The instability wavelength computed numerically shows close agreement with the average finger spacing estimated from experiments over three decades of viscosity ratios. Figure 1a shows a schematic of the Hele-Shaw cell (radius L = 30 cm, gap between plates b = 170 µm), used for studying interfacial patterns. Figure 1b shows the non-monotonic viscosity profile of 35 wt.% aqueous cornstarch suspensions with increasing shear rates. For shear rates upto 30 s −1 , shear-thinning behaviour is observed which fits a simplified Carreau model [22] as noted in the figure caption. The irregular-shaped cornstarch particles have a characteristic size of approximately 15 µm (inset of Fig. 1b). It has been proposed that shear-thinning cannot be attributed to particle rearrangement alone, but also to a constant hydrodynamic viscosity contribution arising from viscous stresses and an entropic contribution triggered by random particle collisions [23,24]. Beyond a critical shear rate, inter-particle frictional forces lead to the formation of a fragile soft solid which exhibits shear-thickening [25][26][27]. In this work, shear-thinning cornstarch suspensions are displaced by glycerol-water mixtures in the radial Hele-Shaw geometry and the formation of interfacial patterns is recorded at different viscosity ratios of the two fluids. The viscosity ratio, η in /η out , is controlled by changing only η in by carefully varying the glycerol concentration in the glycerol-water mixture. The viscosity of the cornstarch suspension that corresponds to the shear rate (γ = 0.08 s −1 ) imposed by the glycerol-water mixture at the time of injection is η out . Representative interfacial patterns are shown in Figure 2. At low η in /η out , side branched patterns (Supplementary Movie 1) are observed. In contrast to the thick finger-like patterns reported for low η in /η out in a pair of miscible Newtonian fluids [14,15], we observe long fractal patterns with side branches under similar conditions of flow rate and viscosity ratios. As η in /η out is increased, a dark inner circular region, devoid of fingers and comprising only the inner fluid, appears and grows larger ( Fig. 2b-f). At the highest η in /η out , this inner circular region covers almost the entire pattern (Fig. 2f). Bischofberger et al. [15] had reported the complete suppression of instabilities for a pair of Newtonian fluids at η in /η out ≈ 3.3 × 10 −1 . We note the formation of small protrusions at the periphery of the pattern even at η in /η out = 5.0 × 10 −1 .

Results
Previous numerical and experimental studies with a pair of Newtonian and non-Newtonian fluids have reported a reduction in the width of the fingers and a high dominant wavenumber at the onset of instability [16,28,29]. A breakdown of the interface into multiple protrusions has also been noted [30].
In order to study the impact of elasticity on instabilities, we have performed experiments with cornstarch suspensions of different concentrations at a fixed η in /η out (Supplementary Fig. 1). We observe a suppression in the growth of fingers at the periphery of the patterns and an increase in the pattern density as the concentration of the cornstarch suspensions is increased (supplementary information section S1). While the instability at the interface between a pair of Newtonian fluids depends on η in /η out [15], our results suggest that the morphological evolution at the interface between a Newtonian and a non-Newtonian fluid is extremely sensitive to the elasticity of the outer fluid even when η in /η out is constant.
We follow the protocol adopted by Bischofberger et al. [15] and quantify the patterns using different length scales (inset of Fig. 2g). R o is the radius of the smallest circle enclosing the entire pattern, R i is when a dyed glycerol-water mixture invades a 35 wt.% cornstarch suspension in a radial Hele-Shaw cell. The scale bar is 5 cm. g, The size ratio R f /R i vs. η in /η out for R o = 13.5 cm. Inset shows the three characteristic length scales, R o , R i and R f . h, Sweep efficiency vs. time (t) for different viscosity ratios: the radius of the largest circle enclosing only the dark region comprising the inner dyed fluid and R f is Figure 2g shows the variation of the size ratios, R f /R i , vs.
η in /η out for fixed R o = 13.5 cm. Our analysis shows that R f /R i decreases monotonically with an increase in viscosity ratio (Fig. 2g) and resembles the observation for a pair of Newtonian fluids [15]. As η in /η out is increased, patterns with small fingers and dark inner regions ( Fig. 2b-d) that are followed by more stable interfaces with small protrusions at the periphery (Fig. 2e,f), are observed. This leads to a flatter R f /R i curve at high η in /η out . Partly, the stable regime can be attributed to the stabilising effect of non-vanishing normal stress differences when either the displacing or the displaced fluid is viscoelastic.
The sweep efficiency measures the significant displacement of the outer fluid by the inner fluid. Here, following the work of Shokri et al. [28], sweep efficiency is defined as the ratio of the area contacted by the inner fluid (gray area in Fig. 1a) and the total area occupied by both inner and outer fluids (sum of the black and gray areas in the same figure). Figure 2h shows the variation of sweep efficiency with time for various η in /η out . At early times, the absence of finger formation results in sweep efficiency close to one. As the fingers grow, sweep efficiency gradually decreases with time. While numerical simulations   Figure 3b shows magnified images of the circled regions. These patterns appear indistinguishable and indicate proportionate growth [31,32], reminiscent of the observations for Newtonian fluids [15]. They are distinct from the growing structures observed in diffusion-limited aggregation [33,34] and viscous fingering [4,5] in which growth occurs at the outer surface while inner parts, once formed, remain preserved. An expression for the growth rate of the interfacial perturbation in terms of the perturbation wavelength (or wavenumber) and other liquid parameters is obtained from the equations governing the flow: Darcy's law [13], continuity equation and convection-diffusion equation [35]. Details of derivation of the growth rate are provided in the supplementary information section S3. The wavenumber (i.e., 2π/wavelength) above which the interface is stable and perturbations do not grow with time is termed the cut-off wavenumber, k c , and is obtained by setting the growth rate to 0. The expression for the dimensionless form of k c is [21]: A comparison of λ m , obtained from numerical analysis, and λ, estimated from experimental data, for different η in /η out shows good correlation (Fig. 4). This suggests that the most dominant wavelength of the interfacial perturbation is a reliable estimate of the average spacing between experimentally obtained finger patterns that subsequently develop. The successful application of the Carreau model-based linear stability provides an additional source for the production of vorticity disturbance leading to a destabilising effect on the flow. The interfacial instability is then determined by a competition between this destabilising source and momentum dissipation that increases with increasing viscosity ratios [21].

Hele-Shaw Cell:
The interfacial patterns are studied in a radial Hele-Shaw cell having two circular glass plates of radius L = 30 cm with a gap b = 170 µm. Teflon spacers are used to ensure a constant gap between the plates.   Fig. 7). The viscosity of the glycerol-water mixture, η in , does not change with shear rates, which is indicative of Newtonian behaviour.