×

Selection mechanism in non-Newtonian Saffman-Taylor fingers. (English) Zbl 07681414

Summary: We present an analytical approach to the problem of predicting the finger width of a simple fluid driving a non-Newtonian (power-law) fluid. Our analysis is based on the Wentzel-Kramers-Brillouin approximation, by representing the deviation from the Newtonian viscosity as a singular perturbation in a parameter, leading to a solvability condition at the finger tip, which selects a unique finger width from the family of solutions. We find that the relation between the dimensionless finger width, \(\Lambda\), and the dimensionless group of parameters containing the viscosity and surface tension, \(\nu\), has the form \(\Lambda \sim \frac{1}{2} - \mathcal{O}(\nu^{-1/2})\) for the shear thinning case and \(\Lambda \sim \frac{1}{2} + \mathcal{O}(\nu^{2/(4-n)})\) for the shear thickening case, in the limit of small \(\nu\). This theoretical estimate is compared with the existing experimental, finger width data as well as the one computed with the linearized model, and a good agreement is found near the power-law exponent, \(n=1\).

MSC:

76E17 Interfacial stability and instability in hydrodynamic stability
76A10 Viscoelastic fluids
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
76D45 Capillarity (surface tension) for incompressible viscous fluids
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bansal, D., Chauhan, T., and Sircar, S., Spatiotemporal linear stability of viscoelastic Saffman-Taylor flows, Phys. Fluids, 34 (2022), 104105.
[2] Bansal, D., Ghosh, D., and Sircar, S., Spatiotemporal linear stability of viscoelastic free shear flows: Nonaffine response regime, Phys. Fluids, 33 (2021), 054106.
[3] Bird, R. B., Armstrong, R. C., and Hassager, O., Dynamics of Polymeric Liquids, Volume 1: Fluid Mechanics, Wiley Interscience, New York, 1987.
[4] Bonn, D. and Meunier, J., Viscoelastic free-boundary problems: Non-Newtonian viscosity vs normal stress effects, Phys. Rev. Lett., 79 (1997), pp. 2662-2665.
[5] Combescot, R., Dombre, T., Hakim, V., and Pomeau, Y., Shape selection of Saffman-Taylor fingers, Phys. Rev. Lett., 56 (1986), pp. 2036-2039.
[6] Fast, P., Kondic, L., Palffy-Muhoray, P., and Shelley, M. J., Pattern formation in non-Newtonian Hele-Shaw flow, Phys. Fluids, 13 (2001), pp. 1191-1212. · Zbl 1184.76156
[7] Hakim, V., Asymptotics beyond all orders, in Proceedings of the NATO ARW, , 1991.
[8] Hong, D. C. and Langer, J. S., Analytic theory of the selection mechanism in the Saffman-Taylor problem, Phys. Rev. Lett., 56 (1986), pp. 2032-2035.
[9] Huerre, P. and Monkewitz, P. A., Absolute and convective instabilities in free shear layers, J. Fluid Mech., 159 (1985), pp. 151-168. · Zbl 0588.76067
[10] Kondic, L., Palffy-Muhoray, P., and Shelley, M. J., Models of non-Newtonian Hele-Shaw flow, Phys. Rev. E, 54 (1996), pp. R4536-R4539.
[11] Kondic, L., Palffy-Muhoray, P., and Shelley, M. J., Non-Newtonian Hele-Shaw flow and the Saffman-Taylor instability, Phys. Rev. Lett., 80 (1998), pp. 1433-1436.
[12] Lindner, A., Bonn, D., and Meunier, J., Viscous fingering in a shear-thinning fluid, Phys. Fluids, 12 (2000), pp. 256-261. · Zbl 1149.76459
[13] Lindner, A., Bonn, D., Poire, E. C., Amar, M. B., and Meunier, J., Viscous fingering in non-Newtonian fluids, J. Fluid Mech., 469 (2002), pp. 237-256. · Zbl 1152.76309
[14] Rabaud, Y. C. M. and Gerard, N., Dynamics and stability of anomalous Saffman-Taylor fingers, Phys. Rev. A, 37 (1988), pp. 935-947.
[15] McLean, J. W., Fingering Problem in Flow Through Porous Media, Ph.D. thesis, CalTech, Pasadena, 1980.
[16] Ostwald, W., Ueber die geschwindigkeitsfunktion der viskosität disperser systeme. I, Kolloid-Z., 36 (1925), pp. 99-117.
[17] Chuoke, R. L., van Meurs, P., and van der Pol, C., The instability of slow immiscible viscous liquid-liquid displacements in permeable media, Trans. AIME, 216 (1959), pp. 188-194.
[18] Saffman, P. and Taylor, G., The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid, Proc. A, 245 (1958), pp. 312-329. · Zbl 0086.41603
[19] Schröder, M., Kassner, K., Rehberg, I., Claret, J., and Sagués, F., Experimental investigation of the initial regime in fingering electrodeposition: Dispersion relation and velocity measurements, Phys. Rev. E, 65 (2002), 041607.
[20] Shraiman, B. I., Velocity selection in the Saffman-Taylor problem, Phys. Rev. Lett., 56 (1986), pp. 2028-2031.
[21] Sircar, S. and Bansal, D., Spatiotemporal linear stability of viscoelastic free shear flows: Dilute regime, Phys. Fluids, 31 (2019), 084104.
[22] Sircar, S., Li, J., and Wang, Q., Biaxial phases of bent-core liquid crystal polymers in shear flows, Commun. Math. Sci., 8 (2010), pp. 697-720. · Zbl 1213.82091
[23] Sircar, S. and Wang, Q., Transient rheological responses in sheared biaxial liquid crystals, Rheol. Acta, 49 (2010), pp. 699-717.
[24] Tabeling, P. and Libchaber, A., Film draining and the Saffman-Taylor problem, Phys. Rev. A, 33 (1985), pp. 794-796.
[25] Waele, A. D., Viscometry and plastometry, Oil Color Chem. Assoc. J., 6 (1923), pp. 33-88.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.