Saffman–Taylor instability
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, described mathematically by Philip Saffman and G. I. Taylor in a paper of 1958.[1][2] This situation is most often encountered during drainage processes through media such as soils.[3] 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 instability is seen. 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, whilst in the radial configuration the pattern grows forming fingers by successive tip-splitting.[4]
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 centre of the cell. Instabilities analogous to viscous fingering can also be self-generated in biological systems.[5]
Derivation for a planar interface[edit]
The simplest case of the instability arises at a planar interface within a porous medium or Hele-Shaw cell, and was treated by Saffman and Taylor[1] but also earlier by other authors.[6] A fluid of viscosity is driven in the -direction into another fluid of viscosity at some velocity . Denoting the permeability of the porous medium as a constant, isotropic, , Darcy's law gives the unperturbed pressure fields in the two fluids to be
with surface tension and the mean curvature. This suppresses small-wavelength (high-wavenumber) disturbances, and we would expect to see instabilities with wavenumber close to the value of which results in the maximal value of ; in this case with surface tension, there is a unique maximal value.
In radial geometry[edit]
The Saffman–Taylor instability is usually seen in an axisymmetric context as opposed to the simple planar case derived above.[8][9] The mechanisms for the instability remain the same in this case, and the selection of the most unstable wavenumber in this case corresponds to a given number of fingers (an integer).
See also[edit]
References[edit]
- ^ a b Saffman, Philip Geoffrey; Taylor, Geoffrey Ingram (1958-06-24). "The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 245 (1242): 312–329. Bibcode:1958RSPSA.245..312S. doi:10.1098/rspa.1958.0085. S2CID 95750900.
- ^ Homsy, G M (1987-01-01). "Viscous Fingering in Porous Media". Annual Review of Fluid Mechanics. 19 (1): 271–311. Bibcode:1987AnRFM..19..271H. doi:10.1146/annurev.fl.19.010187.001415. ISSN 0066-4189.
- ^ 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. S2CID 53323967.
- ^ Lajeunesse, E.; Couder, Y. (2000-09-01). "On the tip-splitting instability of viscous fingers". Journal of Fluid Mechanics. 419 (1): 125–149. Bibcode:2000JFM...419..125L. doi:10.1017/S0022112000001324. ISSN 1469-7645. S2CID 121812154.
- ^ 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.
- ^ Hill, S. (1952). "Channeling in packed columns". Chemical Engineering Science. 1 (6): 247–253. doi:10.1016/0009-2509(52)87017-4. ISSN 0009-2509.
- ^ Chuoke, R. L.; van Meurs, P.; van der Poel, C. (1959-12-01). "The Instability of Slow, Immiscible, Viscous Liquid-Liquid Displacements in Permeable Media". Transactions of the AIME. 216 (1): 188–194. doi:10.2118/1141-G. ISSN 0081-1696.
- ^ Wilson, S. D. R (1975-06-01). "A note on the measurement of dynamic contact angles". Journal of Colloid and Interface Science. 51 (3): 532–534. Bibcode:1975JCIS...51..532W. doi:10.1016/0021-9797(75)90151-4. ISSN 0021-9797.
- ^ Paterson, Lincoln (1981-12-01). "Radial fingering in a Hele Shaw cell". Journal of Fluid Mechanics. 113: 513–529. Bibcode:1981JFM...113..513P. doi:10.1017/S0022112081003613. ISSN 1469-7645. S2CID 122222282.