Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-17T21:36:48.434Z Has data issue: false hasContentIssue false
  • English
  • Français

Influence of viscosity contrast on buoyantly unstable miscible fluids in porous media

Published online by Cambridge University Press:  04 September 2015

Satyajit Pramanik ,
Tapan Kumar Hota  and
Manoranjan Mishra
Satyajit Pramanik
Affiliation:
Department of Mathematics, Indian Institute of Technology Ropar, 140001 Rupnagar, India
Tapan Kumar Hota
Affiliation:
Department of Mathematics, Indian Institute of Technology Ropar, 140001 Rupnagar, India
Manoranjan Mishra*
Affiliation:
Department of Mathematics, Indian Institute of Technology Ropar, 140001 Rupnagar, India
*
†Email address for correspondence: manoranjan.mishra@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

The influence of viscosity contrast on buoyantly unstable miscible fluids in a porous medium is investigated through a linear stability analysis (LSA) as well as direct numerical simulations (DNS). The linear stability method implemented in this paper is based on an initial value approach, which helps to capture the onset of instability more accurately than the quasi-steady-state analysis. In the absence of displacement, we show that viscosity contrast delays the onset of instability in buoyantly unstable miscible fluids. Further, it is observed that by suitably choosing the viscosity contrast and injection velocity a gravitationally unstable miscible interface can be stabilized completely. Through LSA we draw a phase diagram, which shows three distinct stability regions in a parameter space spanned by the displacement velocity and the viscosity contrast. DNS are performed corresponding to parameters from each regime and the results obtained are in accordance with the linear stability results. Moreover, the conversion from one dimensionless formulation to another and its importance when comparing the different type of flow problem associated with each dimensionless formulation are discussed. We also calculate ${\it\epsilon}$-pseudo-spectra of the time dependent linearized operator to investigate the response to external excitation.

JFM classification

Instability: Absolute/convective instability Interfacial Flows (free surface): Fingering instability Low-Reynolds-number flows: Porous media
Type
Papers
Information
Journal of Fluid Mechanics , Volume 780 , 10 October 2015 , pp. 388 - 406
Check for updates
Copyright
© 2015 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Backhaus, K., Turitsyn, K & Ecke, R. E. 2011 Convective instability and mass transport of diffusion layers in a Hele–Shaw geometry. Phys. Rev. Lett. 106, 104501.CrossRefGoogle Scholar
Berkowitz, B., Dror, I. & Yaron, B. 2008 Contaminations Geochemistry: Interactions and Transport in the Subsurface Environment. Springer.Google Scholar
Daniel, D. & Riaz, A. 2014 Effect of viscosity contrast on gravitationally unstable diffusive layers in porous media. Phys. Fluids 26, 116601.Google Scholar
Daniel, D., Tilton, N. & Riaz, A. 2013 Optimal perturbations of gravitationally unstable, transient boundary layers in porous media. J. Fluid Mech. 727, 456–487.Google Scholar
Doumenc, F., Boeck, T., Guerrier, B. & Rossi, M. 2010 Transient Rayleigh–Bénard–Marangoni convection due to evaporation: a linear non-normal stability analysis. J. Fluid Mech. 648, 512–539.Google Scholar
Golub, G. H. & van Loan, C. F. 2007 Matrix Computation, 3rd edn. Hindustan Book Agency.Google Scholar
Guiochon, G., Felinger, A., Shirazi, D. G. & Katti, A. M. 2008 Fundamentals of Preparative and Nonlinear Chromatography, 2nd edn. Academic.Google Scholar
Homsy, G. M. 1987 Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19, 271–311.CrossRefGoogle Scholar
Hota, T. K., Pramanik, S. & Mishra, M.2015a Non-modal linear stability analysis of miscible viscous fingering in a Hele–Shaw cell. arXiv:1504.03734.CrossRefGoogle Scholar
Hota, T. K., Pramanik, S. & Mishra, M. 2015b Onset of fingering instability in a finite slice of adsorbed solute. Phys. Rev. E 92, 023013.Google Scholar
Huppert, H. E. & Neufeld, J. A. 2014 The fluid mechanics of carbon dioxide sequestration. Annu. Rev. Fluid Mech. 46, 255–272.Google Scholar
Kumar, S. & Homsy, G. M. 1999 Direct numerical simulation of hydrodynamic instabilities in two- and three-dimensional viscoelastic free shear layers. J. Non-Newtonian Fluid Mech. 83, 249–276.Google Scholar
Loodts, V., Thomas, C., Rongy, L. & De Wit, A. 2014 Control of convective dissolution by chemical reactions: general classification and application to $\text{CO}_{2}$ dissolution in reactive aqueous solutions. Phys. Rev. Lett. 114, 114501.Google Scholar
Manickam, O. & Homsy, G. M. 1995 Fingering instabilities in vertical miscible displacement flow in porous media. J. Fluid Mech. 288, 75–102.Google Scholar
Mishra, M., Martin, M. & De Wit, A. 2009 Influence of miscible viscous fingering of finite slices on an adsorbed solute dynamics. Phys. Fluids 21, 083101.Google Scholar
Nield, D. A. & Bejan, A. 1992 Convection in Porous Media, 2nd edn. p. 15. Springer.Google Scholar
Pramanik, S. & Mishra, M. 2015 Viscosity scaling of fingering instability in finite slices with Korteweg stress. Europhys. Lett. 109, 64001.CrossRefGoogle Scholar
Rana, C. & Mishra, M. 2014 Fingering dynamics on the adsorbed solute with influence of less viscous and strong sample solvent. J. Chem. Phys. 141, 214701.CrossRefGoogle ScholarPubMed
Rapaka, S., Chen, S., Pawar, R. J., Stauffer, P. H. & Zhang, D. 2008 Non-modal growth of perturbations in density-driven convection in porous media. J. Fluid Mech. 609, 285–303.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.Google Scholar
Tan, C. T. & Homsy, G. M. 1988 Simulation of non-linear viscous fingering in miscible displacement. Phys. Fluids 31, 1330–1338.Google Scholar
Trefethen, L. N. & Embree, M. 2005 Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press.CrossRefGoogle Scholar
Trefethen, L. N., Trefethen, A. E., Redddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 285–303.CrossRefGoogle ScholarPubMed
Wright, T. G.2002 EigTool. http://www.comlab.ox.ac.uk/pseudospectra/eigtool/.Google Scholar