Fig. 8: ηinv–Q phase diagrams.
From: Frictional fluid instabilities shaped by viscous forces
a Experiments and (b) simulations of the invasion of water-glycerol mixtures into dry hydrophobic grains. The viscous deformability at which the pattern transitions from single to multiple fingers (\({D}_{{{{{{{{\rm{visc}}}}}}}}}^{*}\)) and from multiple fingers to radial spokes (\({D}_{{{{{{{{\rm{visc}}}}}}}}}^{*\ast }\)) are plotted in blue and orange, respectively (Eqs. (8) and (9)). c Definitions of front length Sfront (red curve) and displaced area Adis (invaded area plus compaction front, blue region), with \({r}_{\max }\) the reach of the most advanced finger. d Pattern compactness c = Adis/Areach, (e) front instability number s = Sfront/Sfinger, and (f) finger width 2R normalised by the characteristic rate-independent finger width 2Rf (Eq. (1)) as functions of Dvisc. \({D}_{{{{{{{{\rm{visc}}}}}}}}}^{*}\) and \({D}_{{{{{{{{\rm{visc}}}}}}}}}^{*\ast }\) in blue and orange lines, and theoretical lower \({c}^{{\prime} }\) indicated in (d). In all panels, φ = 0.49, b = 0.9 mm, and the outer radius is rout = 13.4 cm.
