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A weighted residual method for two-layer non-Newtonian channel flows: steady-state results and their stability

Published online by Cambridge University Press:  28 August 2013

K. Alba ,
S. M. Taghavi  and
I. A. Frigaard
K. Alba
Affiliation:
Department of Mechanical Engineering, University of British Columbia, 6250 Applied Science Lane, Vancouver, BC, V6T 1Z4, Canada
S. M. Taghavi
Affiliation:
Department of Chemical and Biological Engineering, University of British Columbia, 2360 East Mall, Vancouver, BC, V6T 1Z3, Canada
I. A. Frigaard*
Affiliation:
Department of Mechanical Engineering, University of British Columbia, 6250 Applied Science Lane, Vancouver, BC, V6T 1Z4, Canada Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada
*
†Email address for correspondence: [email protected]
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Abstract

We study buoyant displacement flows in a plane channel with two fluids in the long-wavelength limit in a stratified configuration. Weak inertial effects are accounted for by developing a weighted residual method. This gives a first-order approximation to the interface height and flux functions in each layer. As the fluids are shear-thinning and have a yield stress, to retain a formulation that can be resolved analytically requires the development of a system of special functions for the weight functions and various integrals related to the base flow. For displacement flows, the addition of inertia can either slightly increase or decrease the speed of the leading displacement front, which governs the displacement efficiency. A more subtle effect is that a wider range of interface heights are stretched between advancing fronts than without inertia. We study stability of these systems via both a linear temporal analysis and a numerical spatiotemporal method. To start with, the Orr–Sommerfeld equations are first derived for two generalized non-Newtonian fluids satisfying the Herschel–Bulkley model, and analytical expressions for growth rate and wave speed are obtained for the long-wavelength limit. The predictions of linear analysis based on the weighted residual method shows excellent agreement with the Orr–Sommerfeld approach. For displacement flows in unstable parameter ranges we do observe growth of interfacial waves that saturate nonlinearly and disperse. The observed waves have similar characteristics to those observed experimentally in pipe flow displacements. Although the focus in this study is on displacement flows, the formulation laid out can be easily used for similar two-layer flows, e.g. co-extrusion flows.

JFM classification

Instability: Instability Non-Newtonian Flows: Non-Newtonian Flows Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 731 , 25 September 2013 , pp. 509 - 544
Copyright
©2013 Cambridge University Press 

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References

Alba, K., Laure, P. & Khayat, R. E. 2011 Transient two-layer thin-film flow inside a channel. Phys. Rev. E 84, 026320.CrossRefGoogle ScholarPubMed
Alba, K., Taghavi, S. M. & Frigaard, I. A. 2012 Miscible density-stable displacement flows in inclined tube. Phys. Fluids 24, 123102.Google Scholar
Amaouche, M., Djema, A. & Abderrahmane, H. A. 2012 Film flow for power-law fluids: modelling and linear stability. Eur. Phys. J. B 34, 70–84.Google Scholar
Amaouche, M., Djema, A. & Bourdache, L. 2009 A modified Shkadov’s model for thin film flow of a power law fluid over an inclined surface. C. R. Mec. 337, 48–52.Google Scholar
Amaouche, M., Mehidi, N. & Amatousse, N. 2007 Linear stability of a two-layer film flow down an inclined channel: a second-order weighted residual approach. Phys. Fluids 19, 084106.Google Scholar
Balmforth, N. J. & Liu, J. J. 2004 Roll waves in mud. J. Fluid Mech. 519, 33–54.Google Scholar
Beckett, F., Mader, H. M., Phillips, J. C., Rust, A. & Witham, F. 2011 An experimental study of low Reynolds number exchange flow of two Newtonian fluids in a vertical pipe. J. Fluid Mech. 682, 652–670.Google Scholar
Debacq, M., Fanguet, V., Hulin, J. P., Salin, D. & Perrin, B. 2001 Self similar concentration profiles in buoyant mixing of miscible fluids in a vertical tube. Phys. Fluids 13, 3097.Google Scholar
Debacq, M., Hulin, J. P., Salin, D., Perrin, B. & Hinch, E. J. 2003 Buoyant mixing of miscible fluids of varying viscosities in vertical tube. Phys. Fluids 15, 3846.Google Scholar
d’Olce, M. 2008 Instabilités de cisaillement dans lécoulement concentrique de deux fluides miscibles. PhD thesis, These de l’Universite Pierre et Marie Curie, Orsay, France.Google Scholar
d’Olce, M., Martin, J., Rakotomalala, N. & Salin, D. 2008 Pearl and mushroom instability patterns in two miscible fluids core annular flows. Phys. Fluids 20, 024104.Google Scholar
d’Olce, M., Martin, J., Rakotomalala, N., Salin, D. & Talon, L. 2009 Convective/absolute instability in miscible core-annular flow. Part 1. Experiments. J. Fluid Mech. 618, 305–322.CrossRefGoogle Scholar
Frigaard, I. A. 2001 Super-stable parallel flows of multiple visco-plastic fluids. J. Non-Newtonian Fluid Mech. 100, 49–76.Google Scholar
Frigaard, I. A., Howison, S. D. & Sobey, I. J. 1994 On the stability of Poiseuille flow of a Bingham fluid. J. Fluid Mech. 263, 133–150.Google Scholar
Frigaard, I. A. & Nouar, C. 2003 On three-dimensional linear stability of Poiseuille flow of Bingham fluids. Phys. Fluids 15, 2843–2851.Google Scholar
Gabard, C. 2001 Etude de la stabilité de films liquides sur les parois d’une conduite verticale lors de l’ecoulement de fluides miscibles non-newtoniens. PhD thesis, These de l’Universite Pierre et Marie Curie, Orsay, France.Google Scholar
Gabard, C. & Hulin, J.-P. 2003 Miscible displacements of non-Newtonian fluids in a vertical tube. Eur. Phys. J. E 11, 231–241.Google Scholar
Huen, C. K., Frigaard, I. A. & Martinez, D. M. 2007 Experimental studies of multi-layer flows using a visco-plastic lubricant. J. Non-Newtonian Fluid Mech. 142, 150–161.Google Scholar
Huppert, H. E. & Hallworth, M. A. 2007 Bi-directional flows in constrained systems. J. Fluid Mech. 578, 95–112.Google Scholar
Khomami, B. 1990 Interfacial stability and deformation of two stratified power law fluids in plane Poiseuille flow Part I. Stability analysis. J. Non-Newtonian Fluid Mech. 36, 289–303.Google Scholar
Mehidi, N. & Amatousse, N. 2009 Modélisation dun écoulement coaxial en conduite circulaire de deux fluides visqueux. C. R. Mec. 337, 112–118.Google Scholar
Moyers-Gonzalez, M., Frigaard, I. A. & Nouar, C. 2004 Nonlinear stability of a visco-plastically lubricated shear flow. J. Fluid Mech. 506, 117–146.Google Scholar
Nouar, C., Bottaro, A. & Brancher, J.-P. 2007 Delaying transition to turbulence in channel flow: revisiting the stability of shear-thinning fluids. J. Fluid Mech. 592, 177–194.Google Scholar
Pinarbasi, A. & Liakopoulos, A. 1993 Stability analysis of interfaces in two-layer, inelastic-fluid flows: applications to coextrusion processes. In Transport Phenomena in Nonconventional Manufacturing and Materials Processing, pp. 113–125. ASME Press.Google Scholar
Ruyer-Quil, C., Chakrabortya, S. & Dandapata, B. S. 2012 Wavy regime of a power-law film flow. J. Fluid. Mech. 692, 220–256.Google Scholar
Ruyer-Quil, C. & Manneville, P. 1998 Modelling film flows down an inclined planes. Eur. Phys. J. B 6, 277–292.Google Scholar
Ruyer-Quil, C. & Manneville, P. 2000 Improved modelling of flows down inclined planes. Eur. Phys. J. B 15, 357–369.Google Scholar
Sahu, K. C., Ding, H., Valluri, P. & Matar, O. K. 2009 Pressure-driven miscible two-fluid channel flow with density gradients. Phys. Fluids 21, 043603.Google Scholar
Sahu, K. C., Valluri, P., Spelt, P. D. M & Matar, O. K. 2007 Linear instability of pressure-driven channel flow of a Newtonian and a Herschel–Bulkley fluid. Phys. Fluids 19, 122101.Google Scholar
Schramm, G. 1981 Introduction to Practical Viscometry. Haake Buchler.Google Scholar
Seon, T., Hulin, J.-P., Salin, D., Perrin, B. & Hinch, E. J. 2004 Buoyant mixing of miscible fluids in tilted tubes. Phys. Fluids 16, L103–L106.Google Scholar
Seon, T., Hulin, J.-P., Salin, D., Perrin, B. & Hinch, E. J. 2005 Buoyancy driven miscible front dynamics in tilted tubes. Phys. Fluids 17, 031702.Google Scholar
Seon, T., Hulin, J.-P., Salin, D., Perrin, B. & Hinch, E. J. 2006 Laser-induced fluorescence measurements of buoyancy driven mixing in tilted tubes. Phys. Fluids 18, 041701.CrossRefGoogle Scholar
Seon, T., Znaien, J., Salin, D., Hulin, J.-P., Hinch, E. J. & Perrin, B. 2007a Front dynamics and macroscopic diffusion in buoyant mixing in a tilted tube. Phys. Fluids 19, 125105.Google Scholar
Seon, T., Znaien, J., Salin, D., Hulin, J.-P., Hinch, E. J. & Perrin, B. 2007b Transient buoyancy-driven front dynamics in nearly horizontal tubes. Phys. Fluids 19, 123603.Google Scholar
Shkadov, V. Y. 1967 Wave modes in the flow of thin layer of a viscous liquid under the action of gravity. Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza 1, 43–51.Google Scholar
Stevenson, D. S. & Blake, S. 1998 Modelling the dynamics and thermodynamics of volcanic degassing. Bull. Volcanol. 38 (4), 307–317.CrossRefGoogle Scholar
Su, Y. Y. & Khomami, B. 1991 Stability of multilayer power law and second-order fluids in plane Poiseuille flow. Chem. Engng Commun. 109, 209–223.Google Scholar
Taghavi, S. M., Alba, K. & Frigaard, I. A. 2012a Buoyant miscible displacement flows at moderate viscosity ratios and low Atwood numbers in near-horizontal ducts. Chem. Engng Sci. 69, 404–418.Google Scholar
Taghavi, S. M., Alba, K., Seon, T., Wielage-Burchard, K., Martinez, D. M. & Frigaard, I. A. 2012b Miscible displacement flows in near-horizontal ducts at low Atwood number. J. Fluid Mech. 175, 696.Google Scholar
Taghavi, S. M., Seon, T., Martinez, D. M. & Frigaard, I. A. 2009 Buoyancy-dominated displacement flows in near-horizontal channels: the viscous limit. J. Fluid. Mech. 639, 1–35.Google Scholar
Taghavi, S. M., Seon, T., Martinez, D. M. & Frigaard, I. A. 2010 Influence of an imposed flow on the stability of a gravity current in a near horizontal duct. Phys. Fluids 22, 031702.Google Scholar
Taghavi, S. M., Seon, T., Wielage-Burchard, K., Martinez, D. M. & Frigaard, I. A. 2011 Stationary residual layers in buoyant Newtonian displacement flows. Phys. Fluids 23, 044105.Google Scholar
Yee, H. C., Warming, R. F. & Harten, A. 1985 Implicit total variation diminishing (TVD) schemes for steady-state calculations. J. Comput. Phys. 57, 327–360.Google Scholar
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