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. 2011:2:288.
doi: 10.1038/ncomms1289.

Patterns and flow in frictional fluid dynamics

B Sandnes  1 , E G Flekkøy, H A Knudsen, K J Måløy, H See
Affiliations
Free PMC article

Patterns and flow in frictional fluid dynamics

B Sandnes et al. Nat Commun. 2011.
Free PMC article

Abstract

Pattern-forming processes in simple fluids and suspensions have been studied extensively, and the basic displacement structures, similar to viscous fingers and fractals in capillary dominated flows, have been identified. However, the fundamental displacement morphologies in frictional fluids and granular mixtures have not been mapped out. Here we consider Coulomb friction and compressibility in the fluid dynamics, and discover surprising responses including highly intermittent flow and a transition to quasi-continuous dynamics. Moreover, by varying the injection rate over several orders of magnitude, we characterize new dynamic modes ranging from stick-slip bubbles at low rate to destabilized viscous fingers at high rate. We classify the fluid dynamics into frictional and viscous regimes, and present a unified description of emerging morphologies in granular mixtures in the form of extended phase diagrams.

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Figures

Figure 1
Figure 1. Setup and displacement process.
(a) Air is slowly injected into a linear Hele-Shaw cell loaded with polydisperse glass beads (∼100 μm diameter) submersed in a water/glycerol solution. The gap is Δz=0.5 mm and the cell forms a channel 20 cm wide and 30 cm long. The granular material settles after loading. (b) The invading air/fluid interface accumulates a front of close-packed grains in the gap between the plates.
Figure 2
Figure 2. Friction fingers and stick-slip bubbles.
(a) At low granular filling fraction ϕ, air slowly displaces the granular-fluid mixture in a creeping fingering process. (b) A transition to highly intermittent dynamics occurs at high ϕ. A single bubble of air suddenly inflates after a long static period. Both experiments: Vair=30 ml, q=0.02 ml min−1, scale bar, 5 cm. (c) Gas pressure, (d) invaded volume V and (e) the flow rate [Image: see text] as a function of time for the bubbles labelled 1–7 in b (Supplementary Movie 1).
Figure 3
Figure 3. Frictional dynamics transition.
(a) Displacement structures obtained by varying ϕ and Vair, keeping q constant (0.03 ml min−1). Frictional fingers are stable at low filling fraction and elasticity, whereas stick-slip bubbles prevail at high ϕ and Vair. The theoretical model for the phase boundary is superimposed on the images (black line), including the effect of a ±10% variation in the friction coefficient μ (red lines). Scale bar, 10 cm. (b) The finger width decreases with increasing ϕ (squares, experiments; solid line, theory). Inset: Bubble size increases with Vair, but varies considerably as evident from this box plot showing median, quartiles and extremes (whiskers). ϕ=0.58 for this data set. (c) Interface to area ratio S/A. Error bars on two data points denote standard deviations in replicate experiments. (d) Ratio of adjacent maximum to minimum width plotted as a function of ϕ. Mean and standard deviation obtained by 20 random measurements. Red line: theoretical result for the phase boundary.
Figure 4
Figure 4. Rate dependency.
Displacement structures as a function of injection rate q. Frictional stick-slip flow at low rates gives way to continuous motion as viscous forces fluidize the accumulated front, and, at yet higher rates, suspends the granular material (ϕ=0.58, Vair=30 ml, scale bar, 10 cm). (Supplementary Movie 2).
Figure 5
Figure 5. Phase space.
(a) Tentative phase diagram of morphologies in the ϕ−1−q plane extended to the ϕ=1 limit corresponding to a close-packed porous medium. Various displacement morphologies in the frictional fluid give way to fracturing at ϕ>0.9, followed by a transition to capillary/viscous fingering in porous media (ϕ≈1).The phase boundaries are 'guides to the eye'. The images of capillary and viscous fingering in porous media have been obtained using different experimental setups by other authors. (b) A framework for displacement dynamics in frictional fluids illustrating the transitions between intermittent frictional, quasi-continuous frictional and continuous viscous fluid dynamics.
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References

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