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Transport in Porous Media

, Volume 104, Issue 2, pp 407–433 | Cite as

Linear and Nonlinear Analyses of the Onset of Buoyancy-Induced Instability in an Unbounded Porous Medium Saturated by Miscible Fluids

  • Min Chan KimEmail author
  • Dhananjay Yadav
Article

Abstract

This study analyzes the stability of an initially sharp interface between two miscible fluids in a porous medium. Linear stability equations are first derived using the similarity variable of the basic state, and then transformed into a system of ordinary differential equations using a spectral expansion with and without quasi-steady-state approximation (QSSA). These transformed equations are solved using the eigenanalysis and initial value problem approach. The initial growth rate analysis shows that initially the system is unconditionally stable. The stability characteristics obtained under the present QSSA are quantitatively same as those obtained without the QSSA. To support these theoretical results, numerical simulations are conducted using the Fourier-spectral method. The results of theoretical linear stability analyses and the numerical simulations validate to each other.

Keywords

Gravitational fingering Porous media Linear stability analysis  Nonlinear numerical simulation 

List of Symbols

Variables

\(a\)

Dimensionless wavenumber, \(\sqrt{a_x^2 +a_y^2 }\)

\(a^{*}\)

Modified dimensionless wave number, \(a\sqrt{\tau }\)

\(C\)

Concentration (M)

\(c\)

Dimensionless concentration, \({\left( {C-C_+ } \right) }/{\left( {C_{-} -C_+ } \right) }\)

\(\mathcal{D}\)

Effective diffusion coefficient \(\left( {{\hbox {m}^{2}}/\hbox {s}} \right) \)

g

Gravitational acceleration vector \(\left( {\hbox {m}/{\hbox {s}^{2}}} \right) \)

\(K\)

Permeability \(\left( {\hbox {m}^{2}} \right) \)

\(\mathcal{L}\)

Diffusional operator in \(\left( {\tau ,\zeta } \right) \)-domain, \({\partial ^{2}}/{\partial \zeta ^{2}}+\left( {\zeta /2} \right) \left( {\partial /{\partial \zeta }} \right) \)

\(P\)

Pressure \(\left( {\hbox {Pa}} \right) \)

\(t\)

Time \(\left( \hbox {s} \right) \)

U

Velocity vector \(\left( {\hbox {m}/\hbox {s}} \right) \)

\(w\)

Dimensionless vertical velocity component

\(\left( {x,y,z} \right) \)

Dimensionless Cartesian coordinates

Greek Symbols

\(\mu \)

Viscosity \(\left( {\hbox {Pa}\, \hbox {s}} \right) \)

\(\rho \)

Density \(\left( {{\hbox {kg}}/{\hbox {m}^{3}}} \right) \)

\(\sigma \)

Dimensionless growth rate

\(\tau \)

Dimensionless time \(\left( {{\mathcal{D}t}/K} \right) \)

\(\zeta \)

similarity Variable \(\left( {z/{\sqrt{\tau }}} \right) \)

Subscripts

c

Critical state

0

Basic quantity

1

Perturbed quantity

Notes

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2038983). The authors are grateful to all the reviewers for their lucid comments which have served to greatly improve the status of the present article.

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Chemical EngineeringJeju National UniversityJeju Republic of Korea

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