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A two-layer model for buoyant displacement flows in a channel with wall slip

Published online by Cambridge University Press:  10 August 2018

S. M. Taghavi Open the ORCID record for S. M. Taghavi [Opens in a new window]
S. M. Taghavi*
Affiliation:
Department of Chemical Engineering, Université Laval, Québec, QC G1V 0A6, Canada
*
†Email address for correspondence: Seyed-Mohammad.Taghavi@gch.ulaval.ca
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Abstract

We study theoretically buoyant displacement flows of two generalized Newtonian fluids in a two-dimensional (2-D) channel with wall slip. We assume that a pseudo-interface separates two miscible (immiscible) fluids at the limit of negligible molecular diffusion (negligible surface tension). A heavy fluid displaces a light fluid at near-horizontal channel inclinations, implying that a stratified flow assumption is relevant. We develop a classical lubrication approximation model as a semi-analytical framework that includes a number of dimensionless parameters, such as a buoyancy number, the viscosity ratio, the non-Newtonian properties and the upper and lower wall slip coefficients. For specified interface heights and slopes, the reduced model can furnish the flux and velocity functions in displacing and displaced phases. We numerically solve the interface kinematic condition for four different wall slip cases: no slip (Case I), slip at the lower wall (Case II), slip at the upper wall (Case III) and slip at both walls (Case IV). The solutions for these cases deliver the interface propagation in time, for which leading and trailing displacement front heights, shapes and speeds and several key displacement features, such as front characteristic spreading lengths and short time behaviours, can be directly predicted by simplified analyses. The results reveal in detail how the presence of a channel wall slip may significantly affect the overall displacement flow and the interface evolution characteristics, for both Newtonian and non-Newtonian fluids. Regarding the latter, our analysis quantifies in particular the appearance and removal of static residual wall layers of the displaced phase, versus the wall slip cases.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Low-Reynolds-number flows: Lubrication theory Non-Newtonian Flows: Non-Newtonian Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 852 , 10 October 2018 , pp. 602 - 640
Copyright
© 2018 Cambridge University Press 

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References

Ahmed, R. M., Takach, N. E., Khan, U. M., Taoutaou, S., James, S., Saasen, A. & Godøy, R. 2009 Rheology of foamed cement. Cem. Concr. Res. 39 (4), 353–361.Google Scholar
Al-Housseiny, T. T., Tsai, P. A. & Stone, H. A. 2012 Control of interfacial instabilities using flow geometry. Nature Phys. 8 (10), 747–750.Google Scholar
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. 2013b Miscible density-unstable displacement flows in inclined tube. Phys. Fluids 25, 067101.Google Scholar
Alba, K., Taghavi, S. M. & Frigaard, I. A. 2013c 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
Alba, K., Taghavi, S. M. & Frigaard, I. A. 2014 Miscible heavy-light displacement flows in an inclined two-dimensional channel: a numerical approach. Phys. Fluids 26 (12), 122104.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
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
Amiri, A., Larachi, F. & Taghavi, S. M. 2016 Buoyant miscible displacement flows in vertical pipe. Phys. Fluids 28 (10), 102105.Google Scholar
Amiri, A., Larachi, F. & Taghavi, S. M. 2017 Displacement flows in periodically moving pipe: understanding multiphase flows hosted in oscillating geometry. Chem. Engng Sci. 170, 437–450.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
Barnes, H. A. 1995 A review of the slip (wall depletion) of polymer solutions, emulsions and particle suspensions in viscometers: its cause, character, and cure. J. Non-Newtonian Fluid Mech. 56 (3), 221–251.Google 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
Chattopadhyay, G., Usha, R. & Sahu, K. C. 2017 Core-annular miscible two-fluid flow in a slippery pipe: a stability analysis. Phys. Fluids 29 (9), 097106.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
Denn, M. M. 2001 Extrusion instabilities and wall slip. Annu. Rev. Fluid Mech. 33 (1), 265–287.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
Dimitriou, C.2013 The rheological complexity of waxy crude oils: yielding, thixotropy and shear heterogeneities. PhD thesis, Massachusetts Institute of Technology.Google Scholar
Eslami, A., Frigaard, I. A. & Taghavi, S. M. 2017 Viscoplastic fluid displacement flows in horizontal channels: numerical simulations. J. Non-Newtonian Fluid Mech. 249, 79–96.Google Scholar
Eslami, A. & Taghavi, S. M. 2017 Viscous fingering regimes in elasto-visco-plastic fluids. J. Non-Newtonian Fluid Mech. 243, 79–94.Google Scholar
Ferrás, L. L., Nóbrega, J. M. & Pinho, F. T. 2012 Analytical solutions for Newtonian and inelastic non-Newtonian flows with wall slip. J. Non-Newtonian Fluid Mech. 175, 76–88.Google Scholar
Freitas, J. F., Soares, E. J. & Thompson, R. L. 2013 Viscoplastic–viscoplastic displacement in a plane channel with interfacial tension effects. Chem. Engng Sci. 91, 54–64.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
Frigaard, I., Vinay, G. & Wachs, A. 2007 Compressible displacement of waxy crude oils in long pipeline startup flows. J. Non-Newtonian Fluid Mech. 147 (1–2), 45–64.Google Scholar
Ghosh, S., Usha, R. & Sahu, K. C. 2014a Double-diffusive two-fluid flow in a slippery channel: a linear stability analysis. Phys. Fluids 26 (12), 127101.Google Scholar
Ghosh, S., Usha, R. & Sahu, K. C. 2014b Linear stability analysis of miscible two-fluid flow in a channel with velocity slip at the walls. Phys. Fluids 26 (1), 014107.Google Scholar
Ghosh, S., Usha, R. & Sahu, K. C. 2015 Absolute and convective instabilities in double-diffusive two-fluid flow in a slippery channel. Chem. Engng Sci. 134, 1–11.Google Scholar
Ghosh, S., Usha, R. & Sahu, K. C. 2016 Stability of viscosity stratified flows down an incline: role of miscibility and wall slip. Phys. Fluids 28 (10), 104101.Google Scholar
Hallez, Y. & Magnaudet, J. 2008 Effects of channel geometry on buoyancy-driven mixing. Phys. Fluids 20, 053306.Google Scholar
Hallez, Y. & Magnaudet, J. 2009 A numerical investigation of horizontal viscous gravity currents. J. Fluid. Mech. 630, 71–91.Google Scholar
Härtel, C., Meiburg, E. & Necker, F. 2000 Analysis and direct numerical simulation of the flow at a gravity-current head. Part 1. Flow topology and front speed for slip and no-slip boundaries. J. Fluid Mech. 418, 189–212.Google Scholar
Hasnain, A. & Alba, K. 2017 Buoyant displacement flow of immiscible fluids in inclined ducts: a theoretical approach. Phys. Fluids 29, 052102.Google Scholar
Hatzikiriakos, S. G. 2012 Wall slip of molten polymers. Prog. Polym. Sci. 37 (4), 624–643.Google Scholar
Kraynik, A. M. 1988 Foam flows. Annu. Rev. Fluid Mech. 20 (1), 325–357.Google Scholar
Lauga, E. & Stone, H. A. 2003 Effective slip in pressure-driven stokes flow. J. Fluid Mech. 489, 55–77.Google Scholar
Leal, G. 2007 Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes. Cambridge University Press.Google Scholar
Li, S., Lowengrub, J. S., Fontana, J. & Palffy-Muhoray, P. 2009 Control of viscous fingering patterns in a radial Hele-Shaw cell. Phys. Rev. Lett. 102 (17), 174501.Google Scholar
Lindner, A., Coussot, P. & Bonn, D. 2000 Viscous fingering in a yield stress fluid. Phys. Rev. Lett. 85, 314–317.Google Scholar
Liu, Y. & de Bruyn, J. R. 2018 Start-up flow of a yield-stress fluid in a vertical pipe. J. Non-Newtonian Fluid Mech. 257, 50–58.Google Scholar
Malham, I. B., Jarrige, N., Martin, J., Rakotomalala, N., Talon, L. & Salin, D. 2010 Lock-exchange experiments with an autocatalytic reaction front. J. Chem. Phys. 133 (24), 244505.Google Scholar
Martin, J., Rakotomalala, N., Talon, L. & Salin, D. 2011 Viscous lock-exchange in rectangular channels. J. Fluid Mech. 673, 132–146.Google Scholar
Matson, G. P. & Hogg, A. J. 2012 Viscous exchange flows. Phys. Fluids 24 (2), 023102.Google Scholar
Mollaabbasi, R. & Taghavi, S. M. 2016 Buoyant displacement flows in slightly non-uniform channels. J. Fluid Mech. 795, 876–913.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
Navier, C. L. M. H. 1823 Mémoire sur les lois du mouvement des fluides. Mem. Acad. Sci. Inst. Fr 6 (1827), 389–416.Google Scholar
Nelson, E. B. & Guillot, D. 2006 Well Cementing, 2nd edn. Schlumberger Educational Services.Google Scholar
Nickerson, C. S. & Kornfield, J. A. 2005 A cleat geometry for suppressing wall slip. J. Rheol. 49 (4), 865–874.Google Scholar
Nirmalkar, N., Chhabra, R. P. & Poole, R. J. 2013 Laminar forced convection heat transfer from a heated square cylinder in a bingham plastic fluid. Intl J. Heat Mass Transfer 56 (1–2), 625–639.Google Scholar
Panaseti, P. & Georgiou, G. C. 2017 Viscoplastic flow development in a channel with slip along one wall. J. Non-Newtonian Fluid Mech. 248, 8–22.Google Scholar
Petitjeans, P. & Maxworthy, T. 1996 Miscible displacements in capillary tubes. Part 1. Experiments. J. Fluid Mech. 326, 37–56.Google Scholar
Phillips, D. A., Forsdyke, I. N., McCracken, I. R. & Ravenscroft, P. D. 2011 Novel approaches to waxy crude restart. Part 2. An investigation of flow events following shut down. J. Pet. Sci. Engng 77 (3–4), 286–304.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
Poumaere, A., Moyers-González, M., Castelain, C. & Burghelea, T. 2014 Unsteady laminar flows of a carbopol® gel in the presence of wall slip. J. Non-Newtonian Fluid Mech. 205, 28–40.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
Redapangu, P. R., Sahu, K. C. & Vanka, S. P. 2013 A lattice Boltzmann simulation of three-dimensional displacement flow of two immiscible liquids in a square duct. J. Fluids Engng 135 (12), 121202.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
Schowalter, W. R. 1988 The behavior of complex fluids at solid boundaries. J. Non-Newtonian Fluid Mech. 29, 25–36.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), 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.Google 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.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. 696, 175–214.Google Scholar
Taghavi, S. M., Mollaabbasi, R. & St-Hilaire, Y. 2017 Buoyant miscible displacement flows in rectangular channels. J. Fluid Mech. 826, 676–713.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
Thompson, P. A. & Troian, S. M. 1997 A general boundary condition for liquid flow at solid surfaces. Nature 389 (6649), 360.Google Scholar
Thompson, R. L. & Soares, E. J. 2016 Viscoplastic dimensionless numbers. J. Non-Newtonian Fluid Mech. 238, 57–64.Google Scholar
Vayssade, A., Lee, C., Terriac, E., Monti, F., Cloitre, M. & Tabeling, P. 2014 Dynamical role of slip heterogeneities in confined flows. Phys. Rev. E 89 (5), 052309.Google Scholar
Voronov, R. S., Papavassiliou, D. V. & Lee, L. L. 2008 Review of fluid slip over superhydrophobic surfaces and its dependence on the contact angle. Ind. Engng Chem. Res. 47 (8), 2455–2477.Google Scholar
Walling, E., Mollaabbasi, R. & Taghavi, S. M. 2018 Buoyant miscible displacement flows in a nonuniform Hele-Shaw cell. Phys. Rev. Fluids 3 (3), 034003.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
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
Zhu, Y. & Granick, S. 2001 Rate-dependent slip of Newtonian liquid at smooth surfaces. Phys. Rev. Lett. 87 (9), 096105.Google Scholar