Abstract.
In the present paper we consider slow filtration of a mixture through a close porous filter. The heavy solute penetrates slowly into the porous filter due to the external vertical filtration flow and diffusion. This process is accompanied by the formation of the domain with heavy fluid near the upper boundary of the filter. The developed stratification, at which the heavy fluid is located above the light fluid, is unstable. When the mass of the heavy fluid exceeds the critical value, one can observe the onset of the Rayleigh-Taylor instability. Due to the above peculiarities we can distinguish between two regimes of vertical filtration: 1) homogeneous seepage and 2) convective filtration. When considering the filtration process it is necessary to take into account the diffusion accompanied by the immobilization effect (or sorption) of the solute. The immobilization is described by the linear MIM (mobile/immobile media) model. It has been shown that the immobilization slows down the process of forming the unstable stratification. The purpose of the paper is to find the stability conditions for homogeneous vertical seepage of he solute into the close porous filter. The linear stability problem is solved using the quasi-static approach. The critical times of instability are estimated. The stability maps are plotted in the space of system parameters. The applicability of quasi-static approach is substantiated by direct numerical simulation of the full nonlinear equations.
Graphical abstract
Similar content being viewed by others
References
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, in Dover Books on Physics Series (Dover Publications, 1961)
F. Hattori, H. Takabe, K. Mima, Phys. Fluids 29, 1719 (1986)
H. Kull, Phys. Rep. 206, 197 (1991)
R.C. Sharma, n. Sunil, J. Plasma Phys. 55, 35 (1996)
H. Kalisch, D. Mitrovic, J.M. Nordbotten, Continuum Mech. Thermodyn. 28, 721 (2016)
A. Ivantsov, T. Lyubimova, J. Phys.: Conf. Ser. 681, 012040 (2016)
A. Riaz, M. Hesse, H.A. Tchelepi, F.M. Orr, J. Fluid Mech. 548, 87 (2006)
X. Xiaofeng, C. Shiyi, Z. Dongxiao, Adv. Water Res. 29, 397 (2006)
A.C. Slim, J. Fluid Mech. 741, 461 (2014)
H. Hassanzadeh, M. Pooladi-Darvish, D. Keith, Trans. Porous Media 65, 193 (2006)
D. Rees, A. Selim, J. Ennis-King, The Instability of Unsteady Boundary Layers in Porous Media (Springer, Netherlands, 2008) pp. 82--110
R. Wooding, J. Fluid Mech. 9, 183 (1960)
A. Noghrehabadi, D.A.S. Rees, A.P. Bassom, Trans. Porous Media 99, 493 (2013)
M.C. Kim, C.K. Choi, Phys. Fluids 24, 044102 (2012)
P.M. Patil, D.A.S. Rees, Trans. Porous Media 99, 207 (2013)
D. Lyubimov, T. Lyubimova, S. Amiroudine, D. Beysens, Eur. Phys. J. ST 192, 129 (2011)
P. Gresho, R. Sani, Int. J. Heat Mass Transf. 14, 207 (1971)
H. Deans et al., Soc. Petrol. Engin. J. 3, 49 (1963)
M.Th. Van Genuchten, P.J. Wierenga, Soil Sci. Soc. Am. J. 40, 473 (1976)
F.T. Lindstrom, R. Haque, V.H. Freed, L. Boersma, Environ. Sci. Technol. 1, 561 (1967)
H. Selim, M. Amacher, Reactivity and Transport of Heavy Metals in Soils, (Taylor & Francis, 1996)
M. Bromly, C. Hinz, Water Resour. Res. 40, W07402 (2004)
P. Gouze, T. Le Borgne, R. Leprovost, G. Lods, T. Poidras, P. Pezard, Water Resour. Res. 44, W06426 (2008)
R. Schumer, D.A. Benson, M.M. Meerschaert, B. Baeumer, Water Resour. Res. 39, WR002141 (2003)
B. Maryshev, M. Joelson, D. Lyubimov, T. Lyubimova, M.C. Néel, J. Phys. A: Math. Theor. 42, 115001 (2009)
B. Maryshev, A. Cartalade, C. Latrille, M. Joelson, M. Néel, Comp. Math. Appl. 66, 630 (2013)
B.S. Maryshev, T.P. Lyubimova, D.V. Lyubimov, Transp. Porous Media 65, 193 (2016)
R.D. Harter, D.E. Baker, Soil Sci. Soc. Am. J. 41, 1077 (1977)
H. Darcy, Les fontaines publiques de la ville de Dijon (Victor Dalmont, 1856)
D. Nield, A. Bejan, Convection in Porous Media (Springer, 2012)
C. Latrille, M. Néel, EPJ Web of Conferences 50, 04002 (2013)
M.A. Goldshtik, V.N. Stern, Hydrodynamic stability and turbulence (Nauka, Novosibirsk, 1977) (in Russian)
C. Van Duijn, G. Pieters, R. Wooding, A. van der Ploeg, Stability Criteria for the Vertical Boundary Layer Formed by Throughflow Near the Surface of a Porous Medium (American Geophysical Union, 2002) pp. 155--169
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Maryshev, B., Lyubimova, T. Stability of uniform vertical flow through a close porous filter in the presence of solute immobilization. Eur. Phys. J. E 39, 66 (2016). https://doi.org/10.1140/epje/i2016-16066-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epje/i2016-16066-9