Stability of the flow over saturated porous medium containing dissolved admixture

K. B. Tsiberkin

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 25 (2015) no. 1, pp. 107-116

Résumé

A two-layer system consisting of a porous layer of finite thickness and a uniform fluid layer on top is considered. A rigid wall bounds the porous layer from below, while the upper fluid surface is assumed to be undeformable. We study the process of admixture extraction from the porous layer and its influence on the stability of the stationary plane-parallel flow above it. We describe a porous layer using a Brinkman model with interface boundary conditions by Ochoa–Tapia–Whitaker. We obtain an exact and an approximate solution for the concentration profile. The quasistationary velocity profile is obtained using “frozen” concentration distribution. We solve a linear stability problem for the plane-parallel stationary flow in a wide range of system parameters. Oscillatory instability evolved in the system at the sufficient flow velocity corresponds to traveling waves near the interface. We show that the convective and diffusion transport practically does not affect the structure of neutral stability curves and Reynolds numbers.

Détail

Keywords: flow over porous medium, two-layer system, bimodality, flow instability, Brinkman model.
Mots-clés : admixture transport

Bibliographie

[1] Thomson W. (Lord Kelvin), Mathematical and physical papers, v. 4, Hydrodynamics and general dynamics, Cambridge University Press, Cambridge, 1910, 563 pp. | Zbl

[2] Nield D. A., Bejan A., Convection in porous media, 4-th ed., Springer, New York, 2013, 778 pp. | Zbl

[3] Hill A. A., Straughan B., “Poiseuille flow in a fluid overlying a highly porous material”, Adv. Water Resour., 32:11 (2009), 1609–1614 | DOI | MR

[4] Berman A. S., “Laminar flow in channels with porous walls”, J. Appl. Phys., 24 (1953), 1232–1235 | DOI | MR | Zbl

[5] Tilton N., Cortelezzi L., “Linear stability analysis of pressure-driven flows in channels with porous walls”, J. Fluid Mech., 604 (2008), 411–445 | DOI | MR | Zbl

[6] Ochoa-Tapia J. A., Whitaker S., “Momentum transfer at the boundary between a porous medium and a homogeneous fluid – I. Theoretical development”, Int. J. Heat Mass Transfer, 38:14 (1995), 2635–2646 | DOI

[7] Ochoa-Tapia J. A., Whitaker S., “Momentum transfer at the boundary between a porous medium and a homogeneous fluid – II. Comparison with experiment”, Int. J. Heat Mass Transfer, 38:14 (1995), 2647–2655 | DOI

[8] Whitaker S., “The Forchheimer equation: a theoretical development”, Transport Porous Med., 25 (1996), 27–61 | DOI

[9] Beavers G. S., Joseph D. D., “Boundary conditions at a naturally permeable wall”, J. Fluid Mech., 30:1 (1967), 197–207 | DOI

[10] Kolchanova E. A., Lyubimov D. V., Lyubimova T. P., “The onset and nonlinear regimes of convection in a two-layer system of fluid and porous medium saturated by the fluid”, Transport Porous Med., 97:1 (2013), 25–42 | DOI | MR

[11] Govender S., “Stability of solutal convection in a rotating mushy layer solidifying from a vertical surface”, Transport Porous Med., 90:2 (2011), 393–402 | DOI | MR

[12] Lyubimov D. V., Muratov I. D., “On convective instability of fluid in layered system”, Hydrodynamics. Collection of papers, 10, Perm State Pedagogical Institute, Perm, 1977, 38–46 (in Russian)

[13] Chen F., Chen C. F., “Onset of finger convection in a horizontal porous layer underlying a fluid layer”, J. Heat Transfer, 110:2 (1988), 403–409 | DOI

[14] Gavrilov K., Accary G., Morvan D., Lyubimov D., Méradji S., Bessonov O., “Numerical simulation of coherent structures over plant canopy”, Flow Turbul. Combust., 86:1 (2011), 89–111 | DOI | Zbl

[15] Ghisalberti M., Nepf H., “The structure of the shear layer in flows over rigid and flexible canopies”, Environ. Fluid Mech., 6:3 (2006), 277–301 | DOI

[16] Aleshkova I. A., Sheremet M. A., “Mathematical simulation of conjugate natural convection in a porous medium”, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2010, no. 2, 49–56 (in Russian)

[17] le Bars M., Worster M. G., “Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification”, J. Fluid Mech., 550 (2006), 149–173 | DOI | MR | Zbl

[18] Conte S. D., “The numerical solution of linear boundary value problems”, SIAM Review, 8:3 (1966), 309–321 | DOI | MR | Zbl

[19] Godunov S. K., “Numerical solution of boundary-value problems for systems of linear ordinary differential equations”, Uspekhi Mat. Nauk, 16:3(99) (1961), 171–174 (in Russian) | MR | Zbl

[20] Gershuni G. Z., Zhukhovitskii E. M., Nepomnyashchii A. A., Stability of convective flows, Nauka, Moscow, 1989, 320 pp. | MR