Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-23T21:56:56.379Z Has data issue: false hasContentIssue false
  • English
  • Français

Buoyant displacement flows in slightly non-uniform channels

Published online by Cambridge University Press:  22 April 2016

R. Mollaabbasi  and
S. M. Taghavi
R. Mollaabbasi
Affiliation:
Department of Chemical Engineering, Université Laval, Québec, QC G1V 0A6, Canada
S. M. Taghavi*
Affiliation:
Department of Chemical Engineering, Université Laval, Québec, QC G1V 0A6, Canada
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider displacement flows in slightly diverging or converging plane channels. The two fluids are miscible and buoyancy is significant. We assume that the channel is oriented close to horizontal. Employing a classical lubrication approximation, we simplify the governing equations to furnish a semi-analytical solution for the flux functions. Then, we demonstrate how the non-uniformity of the displacement flow geometry can affect the propagation of the interface between the heavy and light fluids in time, for various parameters studied, e.g. the viscosity ratio, a buoyancy number and rheological features. By setting the molecular diffusion effects to zero, certain solution behaviours at longer times can be practically predicted through the associated hyperbolic problem, using which it becomes possible to directly compute the interfacial features of interest, e.g. leading and trailing front heights and speeds. For a Newtonian displacement flow in a converging or uniform channel, as the buoyancy number increases from zero, we are able to classify three flow regimes based on the behaviour of the trailing front near the top of the channel: a no-back-flow regime, a stationary interface flow regime, and a sustained back-flow regime. For the case of a diverging channel flow, the sustained back-flow regime is replaced by an eventually stationary interface flow regime. In addition, as the displacement flow progresses, the leading front speed typically increases (decreases) in a converging (diverging) channel, while the opposite is usually true for the front height. For the no-back-flow regime (i.e. with small buoyancy), the solution of the displacement flow at long times in all the geometries considered converges to a similarity form, while no similarity form is found for the other flow regimes. As the displacement flow develops, frontal diffusive effects are reduced (enhanced) in a converging (diverging) channel and multiple fronts are progressively less (more) present in a converging (diverging) channel. Regarding non-Newtonian effects, a shear-thinning fluid displacing a Newtonian fluid exhibits an increasingly fast front that has a short height in a converging channel. When a yield stress is present in the displaced fluid, it is possible to find residual wall layers of displaced fluid that are completely static. These layers disappear at a certain critical downstream distance in a converging channel while they appear at a critical distance in a diverging channel. Finally, the combination of strong buoyant and yield-stress effects can modify the destiny of a second front that follows the leading front.

JFM classification

Low-Reynolds-number flows: Lubrication theory Multiphase and Particle-laden Flows: Multiphase flow Non-Newtonian Flows: Non-Newtonian Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 795 , 25 May 2016 , pp. 876 - 913
Check for updates
Copyright
© 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.)

References

Alba, K., Taghavi, S. M., de Bruyn, J. R. & Frigaard, I. A. 2013a Incomplete fluid–fluid displacement of yield-stress fluids. Part 2: Highly inclined pipes. J. Non-Newtonian Fluid Mech. 201, 80–93.Google Scholar
Alba, K., Taghavi, S. M. & Frigaard, I. A. 2012 Miscible density-stable displacement flows in inclined tube. Phys. Fluids 24, 123102.Google Scholar
Alba, K., Taghavi, S. M. & Frigaard, I. A. 2013b A weighted residual method for two-layer non-Newtonian channel flows: steady-state results and their stability. J. Fluid Mech. 731, 509–544.Google Scholar
Al-Housseiny, T.2014 Exploring fluid mechanics in energy processes: viscous flows, interfacial instabilities and elastohydrodynamics. PhD thesis, Princeton University, Princeton, USA.Google Scholar
Al-Housseiny, T. T., Christov, I. C. & Stone, H. A. 2013 Two-phase fluid displacement and interfacial instabilities under elastic membranes. Phys. Rev. Lett. 111 (3), 034502.Google Scholar
Al-Housseiny, T. T. & Stone, H. A. 2013 Controlling viscous fingering in tapered Hele-Shaw cells. Phys. Fluids 25 (9), 092102.Google Scholar
Al-Housseiny, T. T., Tsai, P. A. & Stone, H. A. 2012 Control of interfacial instabilities using flow geometry. Nat. Phys. 8 (10), 747–750.Google Scholar
Allouche, M., Frigaard, I. A. & Sona, G. 2000 Static wall layers in the displacement of two visco-plastic fluids in a plane channel. J. Fluid Mech. 424, 243–277.Google Scholar
Aris, R. 1956 On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. Lond. A 235, 67–78.Google Scholar
Balmforth, N. J., Frigaard, I. A. & Ovarlez, G. 2014 Yielding to stress: recent developments in viscoplastic fluid mechanics. Annu. Rev. Fluid Mech. 46, 121–146.Google Scholar
Ben Amar, M. & Poire, E. C. 1999 Pushing a non-Newtonian fluid in a Hele-Shaw cell: from fingers to needles. Phys. Fluids 11 (7), 1757–1767.CrossRefGoogle Scholar
Benjamin, T. J. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31 (2), 209–248.Google Scholar
Birman, V. K., Battandier, B. A., Meiburg, E. & Linden, P. F. 2007 Lock-exchange flows in sloping channels. J. Fluid Mech. 577, 53–77.CrossRefGoogle Scholar
Bittleston, S. H., Ferguson, J. & Frigaard, I. A. 2002 Mud removal and cement placement during primary cementing of an oil well; laminar non-Newtonian displacements in an eccentric Hele-Shaw cell. J. Engng Maths 43, 229–253.Google Scholar
Buka, A., Palffy-Muhoray, P. & Racz, Z. 1987 Viscous fingering in liquid crystals. Phys. Rev. A 36 (8), 3984–3989.Google Scholar
Burns, J. C. & Parkes, T. 1967 Peristaltic motion. J. Fluid Mech. 29 (04), 731–743.Google Scholar
Chen, C.-Y. & Meiburg, E. 1996 Miscible displacements in capillary tubes. Part 2. Numerical simulations. J. Fluid Mech. 326, 57–90.Google Scholar
Coussot, P. 1999 Saffman–Taylor instability in yield-stress fluids. J. Fluid Mech. 380, 363–376.Google Scholar
De Sousa, D. A., Soares, E. J., de Queiroz, R. S. & Thompson, R. L. 2007 Numerical investigation on gas-displacement of a shear-thinning liquid and a visco-plastic material in capillary tubes. J. Non-Newtonian Fluid Mech. 144 (2–3), 149–159.Google Scholar
Dias, E. O. & Miranda, J. A. 2013 Taper-induced control of viscous fingering in variable-gap Hele-Shaw flows. Phys. Rev. E 87 (5), 053015.CrossRefGoogle ScholarPubMed
Dimakopoulos, Y. & Tsamopoulos, J. 2003 Transient displacement of a viscoplastic material by air in straight and suddenly constricted tubes. J. Non-Newtonian Fluid Mech. 112 (1), 43–75.Google Scholar
Dimakopoulos, Y. & Tsamopoulos, J. 2007 Transient displacement of Newtonian and viscoplastic liquids by air in complex tubes. J. Non-Newtonian Fluid Mech. 142 (1–3), 162–182.Google Scholar
Ebrahimi, B., Taghavi, S. M. & Sadeghy, K. 2015 Two-phase viscous fingering of immiscible thixotropic fluids: a numerical study. J. Non-Newtonian Fluid Mech. 218, 40–52.CrossRefGoogle Scholar
Fast, P., Kondic, L., Shelley, M. J. & Palffy-Muhoray, P. 2001 Pattern formation in non-Newtonian Hele-Shaw flow. Phys. Fluids 13 (5), 1191–1212.Google Scholar
Frigaard, I. A., Leimgruber, S. & Scherzer, O. 2003 Variational methods and maximal residual wall layers. J. Fluid Mech. 483, 37–65.Google Scholar
Frigaard, I. A. & Ryan, D. P. 2004 Flow of a visco-plastic fluid in a channel of slowly varying width. J. Non-Newtonian Fluid Mech. 123 (1), 67–83.Google Scholar
Hallez, Y. & Magnaudet, J. 2008 Effects of channel geometry on buoyancy-driven mixing. Phys. Fluids 20, 053306.CrossRefGoogle Scholar
Hallez, Y. & Magnaudet, J. 2009 A numerical investigation of horizontal viscous gravity currents. J. Fluid Mech. 630, 71–91.CrossRefGoogle Scholar
Hasegawa, E. & Izuchi, H. 1983 On steady flow through a channel consisting of an uneven wall and a plane wall: Part 1. Case of no relative motion in two walls. Bull. JSME 26 (214), 514–520.Google Scholar
Heussler, F. H. C., Oliveira, R. M., John, M. O. & Meiburg, E. 2014 Three-dimensional Navier–Stokes simulations of buoyant, vertical miscible Hele-Shaw displacements. J. Fluid Mech. 752, 157–183.Google Scholar
Huh, D., Fujioka, H., Tung, Y. C., Futai, N., Paine, R., Grotberg, J. B. & Takayama, S. 2007 Acoustically detectable cellular-level lung injury induced by fluid mechanical stresses in microfluidic airway systems. Proc. Natl Acad. Sci. USA 104 (48), 18886–18891.Google Scholar
Ignes-Mullol, J., Zhao, H. & Maher, J. V. 1995 Velocity fluctuations of fracturelike disruptions of associating polymer solutions. Phys. Rev. E 51 (2), 1338–1343.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, 295–323.Google Scholar
Kondic, L., Shelley, M. J. & Palffy-Muhoray, P. 1998 Non-Newtonian Hele-Shaw flow and the Saffman–Taylor instability. Phys. Rev. Lett. 80, 1433–1436.Google Scholar
Lemaire, E., Levitz, P., Daccord, G. & Van Damme, H. 1991 From viscous fingering to viscoelastic fracturing in colloidal fluids. Phys. Rev. Lett. 67 (15), 2009–2012.CrossRefGoogle ScholarPubMed
Lindner, A., Coussot, P. & Bonn, D. 2000 Viscous fingering in a yield stress fluid. Phys. Rev. Lett. 85, 314–317.Google Scholar
Lipscomb, G. G. & Denn, M. M. 1984 Flow of Bingham fluids in complex geometries. J. Non-Newtonian Fluid Mech. 14, 337–346.Google Scholar
Moyers-Gonzalez, M., Alba, K., Taghavi, S. M. & Frigaard, I. A. 2013 A semi-analytical closure approximation for pipe flows of two Herschel–Bulkley fluids with a stratified interface. J. Non-Newtonian Fluid Mech. 193, 49–67.Google Scholar
Nelson, E. B. & Guillot, D. 2006 Well Cementing, 2nd edn. Schlumberger Educational Services.Google Scholar
Nguyen, S., Folch, R., Verma, V. K., Henry, H. & Plapp, M. 2010 Phase-field simulations of viscous fingering in shear-thinning fluids. Phys. Fluids 22 (10), 103102.Google Scholar
Nishimura, T., Arakawa, S., Murakami, S. & Kawamura, Y. 1989 Oscillatory viscous flow in symmetric wavy-walled channels. Chem. Engng Sci. 44 (10), 2137–2148.Google Scholar
Nittmann, J., Daccord, G. & Stanley, H. E. 1985 Fractal growth of viscous fingers: quantitative characterization of a fluid instability phenomenon. Nature 314, 141–144.Google Scholar
Oviedo-Tolentino, F., Romero-Méndez, R., Hernández-Guerrero, A. & Girón-Palomares, B. 2008 Experimental study of fluid flow in the entrance of a sinusoidal channel. Intl J. Heat Fluid Flow 29 (5), 1233–1239.Google Scholar
Petitjeans, P. & Maxworthy, T. 1996 Miscible displacements in capillary tubes. Part 1. Experiments. J. Fluid Mech. 326, 37–56.Google Scholar
Pihler-Puzovic, D., Illien, P., Heil, M. & Juel, A. 2012 Suppression of complex fingerlike patterns at the interface between air and a viscous fluid by elastic membranes. Phys. Rev. Lett. 108 (6), 074502.Google Scholar
Pihler-Puzović, D., Périllat, R., Russell, M., Juel, A. & Heil, M. 2013 Modelling the suppression of viscous fingering in elastic-walled Hele-Shaw cells. J. Fluid Mech. 731, 162–183.Google Scholar
Putz, A., Frigaard, I. A. & Martinez, D. M. 2009 On the lubrication paradox and the use of regularisation methods for lubrication flows. J. Non-Newtonian Fluid Mech. 163 (1), 62–77.Google Scholar
Rakotomalala, N., Salin, D. & Watzky, P. 1997 Miscible displacement between two parallel plates: BGK lattice gas simulations. J. Fluid Mech. 338, 277–297.Google Scholar
Roustaei, A. & Frigaard, I. A. 2013 The occurrence of fouling layers in the flow of a yield stress fluid along a wavy-walled channel. J. Non-Newtonian Fluid Mech. 198, 109–124.Google Scholar
Roustaei, A. & Frigaard, I. A. 2015 Residual drilling mud during conditioning of uneven boreholes in primary cementing. Part 2: Steady laminar inertial flows. J. Non-Newtonian Fluid Mech. 226, 1–15.Google Scholar
Roustaei, A., Gosselin, A. & Frigaard, I. A. 2015 Residual drilling mud during conditioning of uneven boreholes in primary cementing. Part 1: Rheology and geometry effects in non-inertial flows. J. Non-Newtonian Fluid Mech. 220, 87–98.Google Scholar
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. 245, 312–329.Google Scholar
Schweizer, P. & Kistler, S. F. 2012 Liquid Film Coating: Scientific Principles and their Technological Implications. Springer.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 (12), 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 (3), 31702.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 (12), 123603.CrossRefGoogle Scholar
Shin, J. O., Dalziel, S. B. & Linden, P. F. 2004 Gravity currents produced by lock exchange. J. Fluid Mech. 521, 1–34.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.CrossRefGoogle Scholar
Taghavi, S. M., Alba, K., Moyers-Gonzalez, M. & Frigaard, I. A. 2012b Incomplete fluid–fluid displacement of yield stress fluids in near-horizontal pipes: experiments and theory. J. Non-Newtonian Fluid Mech. 167–168, 59–74.Google Scholar
Taghavi, S. M., Alba, K., Seon, T., Wielage-Burchard, K., Martinez, D. M. & Frigaard, I. A. 2012c Miscible displacement flows in near-horizontal ducts at low Atwood number. J. Fluid Mech. 696, 175–214.CrossRefGoogle 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
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, 268–294.Google Scholar
Taylor, G. I. 1953 Dispersion of soluble matter in a solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219, 186–203.Google Scholar
Vanaparthya, S. H. & Meiburg, E. 2008 Variable density and viscosity, miscible displacements in capillary tubes. Eur. J. Mech. (B/Fluids) 27 (3), 268–289.Google Scholar
Walton, I. C. & Bittleston, S. H. 1991 The axial flow of a Bingham plastic in a narrow eccentric annulus. J. Fluid Mech. 222, 39–60.Google Scholar
Wielage-Burchard, K. & Frigaard, I. A. 2011 Static wall layers in plane channel displacement flows. J. Non-Newtonian Fluid Mech. 166 (5), 245–261.Google Scholar
Wiklund, J., Stading, M. & Trägårdh, C. 2010 Monitoring liquid displacement of model and industrial fluids in pipes by in-line ultrasonic rheometry. J. Food Engng 99 (3), 330–337.Google Scholar
Wilson, M. 2012 Flow geometry controls viscous fingering. Phys. Today 65 (10), 15–16.Google Scholar
Yang, Z. & Yortsos, Y. C. 1997 Asymptotic solutions of miscible displacements in geometries of large aspect ratio. Phys. Fluids 9 (2), 286–298.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
Zhang, J. Y. & Frigaard, I. A. 2006 Dispersion effects in the miscible displacement of two fluids in a duct of large aspect ratio. J. Fluid Mech. 549 (1), 225–251.Google Scholar