Sketch of a Hele-Shaw cell with an inviscid liquid, injected amid a viscous liquid. A close-up shows schematically the isocontours of the pressure field.
Flowfields for perturbation of wavenumber n = 8 and amplitude 0.05 for the Brinkman model and viscous potential flow, at , η = 1/2, and Ca = 1. Tangential velocities vary continuously only for the Brinkman solution. The interface is represented as a grey line.
Dispersion relations using different models: ( − · − ) potential flow, (- - -) viscous potential flow, and (—) Brinkman model. The aspect ratio is fixed to k = 200, while the capillary number changes between , and 1.
Most unstable modified wavenumber plotted against logarithm of capillary number. Plotted results are: (− · −) potential flow, (- - -) VPF, and (—) Brinkman model with aspect ratio k = 200. Maxworthy's experimental results are also shown, indicated by ○ symbols. The grey shaded region includes all wavenumbers of the Brinkman model, where the growth rate is within 1% of the maximal growth rate.
Most unstable finger number as a function of capillary number: (− · −) potential flow, (- - -) viscous potential flow, and (—) Brinkman model. Dotted lines show the experimental result of Maxworthy 7 for low surface tension and the theoretical result of Logvinov et al. 16 who used the Brinkman model for zero surface tension fingering of a flat interface with aspect ratio k = 200. For higher capillary number the number of fingers becomes inversely proportional to the gap width.
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