Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-24T07:28:07.484Z Has data issue: false hasContentIssue false

Linear stability analysis and direct numerical simulation of two-layer channel flow

Published online by Cambridge University Press:  13 June 2016

Kirti Chandra Sahu
Department of Chemical Engineering, Indian Institute of Technology Hyderabad, Sangareddy 502 285, Telangana, India
Rama Govindarajan*
TIFR Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Narsingi, Hyderabad 500 075, India
Email address for correspondence:


We study the stability of two-fluid flow through a plane channel at Reynolds numbers of 100–1000 in the linear and nonlinear regimes. The two fluids have the same density but different viscosities. The fluids, when miscible, are separated from each other by a mixed layer of small but finite thickness, across which the viscosity changes from that of one fluid to that of the other. When immiscible, the interface is sharp. Our study spans a range of Schmidt numbers, viscosity ratios and locations and thicknesses of the mixed layer. A region of instability distinct from that of the Tollmien–Schlichting mode is obtained at moderate Reynolds numbers. We show that the overlap of the layer of viscosity-stratification with the critical layer of the dominant disturbance provides a mechanism for this instability. At very low values of diffusivity, the miscible flow behaves exactly like the immiscible one in terms of stability characteristics. High levels of miscibility make the flow more stable. At intermediate levels of diffusivity however, in both linear and nonlinear regimes, miscible flow can be more unstable than the corresponding immiscible flow without surface tension. This difference is greater when the thickness of the mixed layer is decreased, since the thinner the layer of viscosity stratification, the more unstable the miscible flow. In direct numerical simulations, disturbance growth occurs at much earlier times in the miscible flow, and also the miscible flow breaks spanwise symmetry more readily to go into three-dimensionality. The following observations hold for both miscible and immiscible flows without surface tension. The stability of the flow is moderately sensitive to the location of the interface between the two fluids. The response is non-monotonic, with the least stable location of the layer being mid-way between the wall and the centreline. As expected, flow at higher Reynolds numbers is more unstable.

© 2016 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.)


Balasubramaniam, R., Rashidnia, N., Maxworthy, T. & Kuang, J. 2005 Instability of miscible interfaces in a cylindrical tube. Phys. Fluids 17, 052103.Google Scholar
Boomkamp, P. A. M. & Miesen, R. H. M. 1996 Classification of instabilities in parallel two-phase flow. Intl J. Multiphase Flow 22, 6788.Google Scholar
Chen, C.-Y. & Meiburg, E. 1996 Miscible displacement in capillary tubes. Part 2. Numerical simulations. J. Fluid Mech. 326, 5790.CrossRefGoogle Scholar
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.Google Scholar
Ern, P., Charru, F. & Luchini, P. 2003 Stability analysis of a shear flow with strongly stratified viscosity. J. Fluid Mech. 496, 295312.Google Scholar
Govindarajan, R. 2004 Effect of miscibility on the linear instability of two-fluid channel flow. Intl J. Multiphase Flow 30, 11771192.Google Scholar
Govindarajan, R. & Sahu, K. C. 2014 Instabilities in viscosity-stratified flows. Annu. Rev. Fluid Mech. 46, 331353.Google Scholar
Hinch, E. J. 1984 A note on the mechanism of the instability at the interface between two shearing fluids. J. Fluid Mech. 144, 463465.Google Scholar
Hooper, A. P. 1985 Long-wave instability at the interface between two viscous fluids: thin layer effects. Phys. Fluids 28 (6), 16131618.Google Scholar
Hooper, A. P. & Boyd, W. G. C. 1983 Shear flow instability at the interface between two fluids. J. Fluid Mech. 128, 507528.Google Scholar
John, M. O., Oliveira, R. M., Heussler, F. H. C. & Meiburg, E. 2013 Variable density and viscosity, miscible displacements in horizontal Hele–Shaw cells. Part 2. Nonlinear simulations. J. Fluid Mech. 721, 295323.Google Scholar
Joseph, D. D., Bai, R., Chen, K. P. & Renardy, Y. Y. 1997 Core-annular flows. Annu. Rev. Fluid Mech. 29, 6590.Google Scholar
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.Google Scholar
Malik, S. V. & Hooper, A. P. 2005 Linear stability and energy growth of viscosity stratified flows. Phys. Fluids 17, 024101.Google Scholar
Naraigh, L. O., Valluri, P., Scott, D. M., Bethune, I. & Spelt, P. D. M. 2014 Linear instability, nonlinear instability and ligament dynamics in three-dimensional laminar two-layer liquid–liquid flows. J. Fluid Mech. 750, 464506.Google Scholar
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.Google Scholar
Petitjeans, P. & Maxworthy, P. 1996 Miscible displacements in capillary tubes. Part 1. Experiments. J. Fluid Mech. 326, 3756.Google Scholar
Preziosi, L., Chen, K. & Joseph, D. D. 1989 Lubricated pipelining: stability of core-annular flow. J. Fluid Mech. 201, 323.Google Scholar
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.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Selvam, B., Merk, S., Govindarajan, R. & Meiburg, E. 2007 Stability of miscible core-annular flows with viscosity stratification. J. Fluid Mech. 592, 2349.CrossRefGoogle Scholar
Selvam, B., Talon, L., Lesshafft, L. & Meiburg, E. 2009 Convective/absolute instability in miscible core-annular flow. Part 2. Numerical simulations and nonlinear global modes. J. Fluid Mech. 618, 323348.Google Scholar
Squire, H. B. 1933 On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls. Proc. R. Soc. Lond. A 142, 621628.Google Scholar
Talon, L., Goyal, N. & Meiburg, E. 2013 Variable density and viscosity, miscible displacements in horizontal Hele–Shaw cells. Part 1. Linear stability analysis. J. Fluid. Mech. 721, 268294.Google Scholar
Talon, L. & Meiburg, E. 2011 Plane Poiseuille flow of miscible layers with different viscosities: instabilities in the Stokes flow regime. J. Fluid Mech. 686, 484506.Google Scholar
Valluri, P., Naraigh, L. O., Ding, H. & Spelt, P. D. M. 2010 Linear and nonlinear spatio-temporal instability in laminar two-layer flows. J. Fluid Mech. 656, 458480.Google Scholar
Yih, C. S. 1967 Instability due to viscous stratification. J. Fluid Mech. 27, 337352.Google Scholar