The Saffman–Taylor instability, also known as viscous fingering, is the formation of patterns in a morphologically unstable interface between two fluids in a porous medium. This situation is most often encountered during drainage processes through media such as soils.[1] It occurs when a less viscous fluid is injected, displacing a more viscous fluid. (In the inverse situation, with the more viscous displacing the other, the interface is stable and no patterns form.) Essentially the same effect occurs driven by gravity (without injection) if the interface is horizontal and separates two fluids of different densities, the heavier one being above the other: this is known as the Rayleigh-Taylor instability. In the rectangular configuration the system evolves until a single finger (the Saffman–Taylor finger) forms. In the radial configuration the pattern grows forming fingers by successive tip-splitting.
Most experimental research on viscous fingering has been performed on Hele-Shaw cells, which consist of two closely spaced, parallel sheets of glass containing a viscous fluid. The two most common set-ups are the channel configuration, in which the less viscous fluid is injected at one end of the channel, and the radial configuration, in which the less viscous fluid is injected at the center of the cell. Instabilities analogous to viscous fingering can also be self-generated in biological systems.[2]
The mathematical description of viscous fingering is Darcy's law for the flow in the bulk of each fluid, with a boundary condition at the interface accounting for surface tension.
Simulation methods for viscous fingering problems include boundary integral methods, phase field models, etc. A solution to the Saffman-Taylor finger problem was proposed by Roland Combescot, Thierry Dombre, Vincent Hakim, Yves Pomeau and Alain Pumir [3][4]
See also
Fractal canopy
References
Li, S; et al. (2018). "Dynamics of Viscous Entrapped Saturated Zones in Partially Wetted Porous Media". Transport in Porous Media. 125 (2): 193–210. arXiv:1802.07387. doi:10.1007/s11242-018-1113-3.
Mather, W.; Mondragón-Palomino, O.; Danino, T.; Hasty, J.; Tsimring, L. S. (2010). "Streaming Instability in Growing Cell Populations". Physical Review Letters. 104 (20): 208101. Bibcode:2010PhRvL.104t8101M. doi:10.1103/PhysRevLett.104.208101. PMC 2947335. PMID 20867071.
"The Boltzmann Medal for 2016".
Combescot, R., Dombre, T., Hakim, V. V., Pomeau, Y., & Pumir, A. (1986). Shape selection of Saffman-Taylor fingers. Physical review letters, 56(19), 2036-2039.
Viscous Fingering at Center for Nonlinear Dynamics
Saffman, P. G.; Taylor, G. (1958). "The Penetration of a Fluid into a Medium or Hele-Shaw Cell Containing a more Viscous Liquid". Proceedings of the Royal Society of London, Series A. 245 (1242): 312–329. Bibcode:1958RSPSA.245..312S. doi:10.1098/rspa.1958.0085.
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