Skip to main content Accessibility help

Double diffusive effects on pressure-driven miscible displacement flows in a channel

  • Manoranjan Mishra (a1), A. De Wit (a2) and Kirti Chandra Sahu (a3)

The pressure-driven miscible displacement of a less viscous fluid by a more viscous one in a horizontal channel is studied. This is a classically stable system if the more viscous solution is the displacing one. However, we show by numerical simulations based on the finite-volume approach that, in this system, double diffusive effects can be destabilizing. Such effects can appear if the fluid consists of a solvent containing two solutes both influencing the viscosity of the solution and diffusing at different rates. The continuity and Navier–Stokes equations coupled to two convection–diffusion equations for the evolution of the solute concentrations are solved. The viscosity is assumed to depend on the concentrations of both solutes, while density contrast is neglected. The results demonstrate the development of various instability patterns of the miscible ‘interface’ separating the fluids provided the two solutes diffuse at different rates. The intensity of the instability increases when increasing the diffusivity ratio between the faster-diffusing and the slower-diffusing solutes. This brings about fluid mixing and accelerates the displacement of the fluid originally filling the channel. The effects of varying dimensionless parameters, such as the Reynolds number and Schmidt number, on the development of the ‘interfacial’ instability pattern are also studied. The double diffusive instability appears after the moment when the invading fluid penetrates inside the channel. This is attributed to the presence of inertia in the problem.

Corresponding author
Email address for correspondence:
Hide All
Balasubramaniam, R., Rashidnia, N., Maxworthy, T. & Kuang, J. 2005 Instability of miscible interfaces in a cylindrical tube. Phys. Fluids 17, 052103.
Cao, Q., Ventresca, L., Sreenivas, K. R. & Prasad, A. K. 2003 Instability due to viscosity stratification downstream of a centreline injector. Can. J. Chem. Engng 81, 913.
Chen, C.-Y. & Meiburg, E. 1996 Miscible displacement in capillary tubes. Part 2. Numerical simulations. J. Fluid Mech. 326, 5790.
Chouke, R. L., Van Meurs, P. & Van Der Pol, C. 1959 The instability of slow, immiscible, viscous liquid–liquid displacements in permeable media. Trans. AIME 216, 188.
Cox, B. G. 1962 On driving a viscous fluid out of a tube. J. Fluid Mech. 14, 81.
Curtiss, C. F. & Hirschfelder, J. O. 1949 Transport properties of multicomponent gas mixtures. J. Chem. Phys. 17, 550555.
Ding, H., Spelt, P. D. M. & Shu, C. 2007 Diffuse interface model for incompressible two-phase flows with large density ratios. J. Comput. Phys. 226, 20782095.
d’Olce, M., Martin, J., Rakotomalala, N., Salin, D. & Talon, L. 2008 Pearl and mushroom instability patterns in two miscible fluids’ core annular flows. Phys. Fluids 20, 024104.
Gabard, C. & Hulin, J.-P. 2003 Miscible displacement of non-Newtonian fluids in a vertical tube. Eur. Phys. J. E 11, 231.
Goyal, N., Pichler, H. & Meiburg, E. 2007 Variable-density miscible displacements in a vertical Hele-Shaw cell: linear stability. J. Fluid Mech. 584, 357372.
Hickox, C. E. 1971 Instability due to viscosity and density stratification in axisymmetric pipe flow. Phys. Fluids 14, 251.
Homsy, G. M. 1987 Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19, 271311.
Hu, H. H. & Joseph, D. D. 1989 Lubricated pipelining: stability of core–annular flows. Part 2. J. Fluid Mech. 205, 395.
Iglesias-Silva, G. A. & Hall, K. R. 2010 Equivalence of the McAllister and Heric equations for correlating the liquid viscosity of multicomponent mixtures. Ind. Engng Chem. Res. 49, 62506254.
Joseph, D. D., Bai, R., Chen, K. P. & Renardy, Y. Y. 1997 Core–annular flows. Annu. Rev. Fluid Mech. 29, 65.
Joseph, D. D. & Renardy, Y. Y. 1992 Fundamentals of Two-Fluid Dynamics. Part II: Lubricated Transport, Drops and Miscible Liquids. Springer.
Kalidas, R. & Laddha, S. 1964 Viscosity of ternary liquid mixtures. J. Chem. Engng Data 9, 142145.
Kuang, J., Maxworthy, T. & Petitjeans, P. 2003 Miscible displacements between silicone oils in capillary tubes. Eur. J. Mech. 22, 271.
Lajeunesse, E., Martin, J., Rakotomalala, N. & Salin, D. 1997 3D instability of miscible displacements in a Hele-Shaw cell. Phys. Rev. Lett. 79, 5254.
Lajeunesse, E., Martin, J., Rakotomalala, N., Salin, D. & Yortsos, Y. C. 1999 Miscible displacement in a Hele-Shaw cell at high rates. J. Fluid Mech. 398, 299.
Manickam, O. & Homsy, G. M. 1993 Stability of miscible displacements in porous media with nonmonotonic viscosity profiles. Phys. Fluids 5, 13561367.
Mishra, M., Trevelyan, P. M. J., Almarcha, C. & De Wit, A. 2010 Influence of double diffusive effects on miscible viscous fingering. Phys. Rev. Lett 105, 204501.
Petitjeans, P. & Maxworthy, T. 1996 Miscible displacements in capillary tubes. Part 1. Experiments. J. Fluid Mech. 326, 37.
Pritchard, D. 2009 The linear stability of double-diffusive miscible rectilinear displacements in a Hele-Shaw cell. Eur. J. Mech. (B/Fluids) 28 (4), 564577.
Probstein, R. F. 1994 Physicochemical Hydrodynamics. Wiley.
Ranganathan, B. T. & Govindarajan, R. 2001 Stabilisation and destabilisation of channel flow by location of viscosity-stratified fluid layer. Phys. Fluids 13 (1), 13.
Rashidnia, N., Balasubramaniam, R. & Schroer, R. T. 2004 The formation of spikes in the displacement of miscible fluids. Ann. N.Y. Acad. Sci. 1027, 311316.
Saffman, P. G. & Taylor, G. I. 1958 The penetration of a finger into a porous medium in a Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245, 312329.
Sahu, K. C., Ding, H. & Matar, O. K. 2010 Numerical simulation of non-isothermal pressure-driven miscible channel flow with viscous heating. Chem. Engng Sci. 65, 32603267.
Sahu, K. C., Ding, H., Valluri, P. & Matar, O. K. 2009a Linear stability analysis and numerical simulation of miscible channel flows. Phys. Fluids 21, 042104.
Sahu, K. C., Ding, H., Valluri, P. & Matar, O. K. 2009b Pressure-driven miscible two-fluid channel flow with density gradients. Phys. Fluids 21, 043603.
Sahu, K. C. & Govindarajan, R. 2011 Linear stability of double-diffusive two-fluid channel flow. J. Fluid Mech. 687, 529539.
Sahu, K. C. & Govindarajan, R. 2012 Spatio-temporal linear stability of double-diffusive two-fluid channel flow. Phys. Fluids 24, 054103.
Sahu, K. C. & Matar, O. K. 2010 Three-dimensional linear instability in pressure-driven two-layer channel flow of a Newtonian and a Herschel–Bulkley fluid. Phys. Fluids 22, 112103.
Sahu, K. C. & Matar, O. K. 2011 Three-dimensional convective and absolute instabilities in pressure-driven two-layer channel flow. Intl J. Multiphase Flow 37, 987993.
Scoffoni, J., Lajeunesse, E. & Homsy, G. M. 2001 Interface instabilities during displacement of two miscible fluids in a vertical pipe. Phys. Fluids 13, 553.
Selvam, B., Merk, S., Govindarajan, R. & Meiburg, E. 2007 Stability of miscible core–annular flows with viscosity stratification. J. Fluid Mech. 592, 2349.
Tan, C. T. & Homsy, G. M. 1986 Stability of miscible displacements: rectilinear flow. Phys. Fluids 29, 73549.
Taylor, G. I. 1961 Deposition of viscous fluid on the wall of a tube. J. Fluid Mech. 10, 161.
Yang, Z. & Yortsos, Y. C. 1997 Asymptotic solutions of miscible displacements in geometries of large aspect ratio. Phys. Fluids 9, 286298.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed