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Stability of Poiseuille flow of a Bingham fluid overlying an anisotropic and inhomogeneous porous layer

Published online by Cambridge University Press:  11 July 2019

Sourav Sengupta  and
Sirshendu De Open the ORCID record for Sirshendu De [Opens in a new window]
Sourav Sengupta
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India
Sirshendu De*
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India
*
†Email address for correspondence: [email protected]
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Abstract

Modal and non-modal stability analyses are performed for Poiseuille flow of a Bingham fluid overlying an anisotropic and inhomogeneous porous layer saturated with the same fluid. In the case of modal analysis, the resultant Orr–Sommerfeld type eigenvalue problem is formulated and solved via the Chebyshev collocation method, using QZ decomposition. It is found that no unstable eigenvalues are present for the problem, indicating that the flow is linearly stable. Therefore, non-modal analysis is attempted in order to observe the short-time response. For non-modal analysis, the initial value problem is solved, and the response of the system to initial conditions is assessed. The aim is to evaluate the effects on the flow stability of porous layer parameters in terms of depth ratio (ratio of the fluid layer thickness $d$ to the porous layer thickness $d_{m}$), Bingham number, Darcy number and slip coefficient. The effects of anisotropy and inhomogeneity of the porous layer on flow transition are also investigated. In addition, the shapes of the optimal perturbations are constructed. The mechanism of transient growth is explored to comprehend the complex interplay of various factors that lead to intermediate amplifications. The present analysis is perhaps the first attempt at analysing flow stability of viscoplastic fluids over a porous medium, and would possibly lead to better and efficient designing of flow environments involving such flow.

JFM classification

Flow Control: Instability control Low-Reynolds-number flows: Porous media Instability: Transition to turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 874 , 10 September 2019 , pp. 573 - 605
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Copyright
© 2019 Cambridge University Press 

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References

Association of Swiss Road and Traffic Engineers1999 Characteristic Coefficients of Soils. Swiss Standard SN 670 010b.Google Scholar
Auriault, J. L. 2009 On the domain of validity of Brinkman’s equation. Trans. Porous Med. 79, 215–223.10.1007/s11242-008-9308-7Google Scholar
Balhoff, M. T. & Thompson, K. E. 2004 Modeling the steady flow of yield-stress fluids in packed beds. AIChE J. 50, 3034–3048.10.1002/aic.10234Google Scholar
Balmforth, N. J., Frigaard, I. A. & Ovarlez, G. 2014 Yielding to stress: recent developments in viscoplastic fluid mechanics. Annu. Rev. Fluid Mech. 46, 121–146.10.1146/annurev-fluid-010313-141424Google Scholar
Bars, M. L. & Worster, M. G. 2006 Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification. J. Fluid Mech. 550, 149–173.10.1017/S0022112005007998Google Scholar
Beavers, G. S. & Joseph, D. D. 1967 Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30, 197–207.10.1017/S0022112067001375Google Scholar
Bentrad, H., Esmael, A., Nouar, C., Lefevre, A. & Messaoudene, N. A. 2017 Energy growth in Hagen–Poiseuille flow of Herschel–Bulkley fluid. J. Non-Newtonian Fluid Mech. 241, 43–59.10.1016/j.jnnfm.2017.01.007Google Scholar
Bird, R. B., Dai, G. C. & Yarusso, B. J. 1983 Rheology and flow of viscoplastic materials. Rev. Chem. Engng 1, 1–70.10.1515/revce-1983-0102Google Scholar
Brinkman, H. C. 1949 A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. 1, 27–34.10.1007/BF02120313Google Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 1637–1650.10.1063/1.858386Google Scholar
Caton, F. 2006 Linear stability of circular Couette flow of inelastic viscoplastic fluids. J. Non-Newtonian Fluid Mech. 134, 148–154.10.1016/j.jnnfm.2006.02.003Google Scholar
Chang, M. H., Chen, F. & Straughan, B. 2006 Instability of Poiseuille flow in a fluid overlying a porous layer. J. Fluid Mech. 564, 287–303.10.1017/S0022112006001583Google Scholar
Chang, T. Y., Chen, F. & Chang, M. H. 2017 Stability of plane Poiseuille–Couette flow in a fluid layer overlying a porous layer. J. Fluid Mech. 826, 376–395.10.1017/jfm.2017.442Google Scholar
Chhabra, R. P. & Richardson, J. F. 1999 Non-Newtonian Flow in the Process Industries – Fundamentals and Engineering Applications. Butterworth Heinemann.Google Scholar
Chen, F. & Hsu, L. H. 1991 Onset of thermal convection in an anisotropic and inhomogeneous porous layer underlying a fluid layer. J. Appl. Phys. 69, 6289.10.1063/1.348827Google Scholar
Chen, Y. L. & Zhu, K. Q. 2008 Couette–Poiseuille flow of Bingham fluids between two porous parallel plates with slip conditions. J. Non-Newtonian Fluid Mech. 153, 1–11.10.1016/j.jnnfm.2007.11.004Google Scholar
Das, B. M. 2013 Advanced Soil Mechanics, 4th edn. CRC Press.10.1201/b15955Google Scholar
Dash, R. K., Mehta, K. N. & Jayaraman, G. 1996 Casson fluid flow in a pipe filled with a homogeneous porous medium. Intl J. Engng Sci. 34, 1145–1156.10.1016/0020-7225(96)00012-2Google Scholar
Deepu, P., Anand, P. & Basu, S. 2015 Stability of Poiseuille flow in a fluid overlying an anisotropic and inhomogeneous porous layer. Phys. Rev. E 92, 23009.10.1103/PhysRevE.92.023009Google Scholar
Deepu, P., Kallurkar, S., Anand, P. & Basu, S. 2016 Stability of a liquid film flowing down an inclined anisotropic and inhomogeneous porous layer: an analytical description. J. Fluid Mech. 807, 135–154.10.1017/jfm.2016.613Google Scholar
Drazin, P. G. & Reiid, W. H. 2004 Hydrodynamic Stability, 2nd edn. Cambridge University Press.10.1017/CBO9780511616938Google Scholar
Durlofsky, L. & Brady, J. F. 1987 Analysis of the Brinkman equation as a model for flow in porous media. Phys. Fluids 30, 3329–3341.10.1063/1.866465Google Scholar
Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18, 487–488.10.1063/1.861156Google Scholar
Farrell, B. F. & Ioannou, P. J. 1993 Optimal excitation of three-dimensional perturbations in viscous constant shear flow. Phys. Fluids A 5, 1390–1400.10.1063/1.858574Google Scholar
Frigaard, I. A. 2001 Super-stable parallel flows of multiple visco-plastic fluids. J. Non-Newtonian Fluid Mech. 100, 49–75.10.1016/S0377-0257(01)00129-XGoogle 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, 133–150.10.1017/S0022112094004052Google Scholar
Frigaard, I. & Nouar, C. 2003 On three-dimensional linear stability of Poiseuille flow of Bingham fluids. Phys. Fluids 15, 2843.10.1063/1.1602451Google Scholar
Henningson, D. S., Lundbladh, A. & Johansson, A. V. 1993 A mechanism for bypass transition from localized disturbances in wall-bounded shear flows. J. Fluid Mech. 250, 169–207.10.1017/S0022112093001429Google Scholar
Herzig, J. P., Leclerc, D. M. & Goff, P. L. 1970 Flow of suspensions through porous media – application to deep filtration. Ind. Engng Chem. 62, 8–35.10.1021/ie50725a003Google Scholar
Hill, A. A. & Straughan, B. 2008 Poiseuille flow in a fluid overlying a porous medium. J. Fluid Mech. 603, 137–149.10.1017/S0022112008000852Google Scholar
Hill, A. A. & Straughan, B. 2009 Poiseuille flow in a fluid overlying a highly porous material. Adv. Water Resour. 32, 1609–1614.10.1016/j.advwatres.2009.08.007Google Scholar
Kabouya, N. & Nouar, C. 2003 On the stability of a Bingham fluid flow in an annular channel. C. R. Méc. 331, 149–156.10.1016/S1631-0721(02)00015-3Google Scholar
Landry, M. P., Frigaard, I. A. & Martinez, D. M. 2006 Stability and instability of Taylor–Couette flows of a Bingham fluid. J. Fluid Mech. 560, 321–353.10.1017/S0022112006000620Google Scholar
Liu, R., Ding, Z. & Hu, K. X. 2018 Stabilities in plane Poiseuille flow of Herschel–Bulkley fluid. J. Non-Newtonian Fluid Mech. 251, 132–144.10.1016/j.jnnfm.2017.11.007Google Scholar
Liu, R. & Liu, Q. 2009 Instabilities of a liquid film flowing down an inclined porous plane. Phys. Rev. E 80, 036316.10.1103/PhysRevE.80.036316Google Scholar
Liu, R. & Liu, Q. S. 2014 Non-modal stability in Hagen–Poiseuille flow of a Bingham fluid. Phys. Fluids 26, 14102.10.1063/1.4861025Google Scholar
Lyubimova, T. P., Lyubimov, D. V., Baydina, D. T., Kolchanova, E. A. & Tsiberkin, K. B. 2016 Instability of plane-parallel flow of incompressible liquid over a saturated porous medium. Phys. Rev. E 94, 013104.10.1103/PhysRevE.94.013104Google Scholar
Madani, A., Martinez, D. M., Olson, J. A. & Frigaard, I. A. 2013 The stability of spiral Poiseuille flows of Newtonian and Bingham fluids in an annular gap. J. Non-Newtonian Fluid Mech. 193, 3–10.10.1016/j.jnnfm.2012.02.007Google Scholar
Malashetty, M. S. & Mahantesh, S. 2010 The onset of convection in a binary fluid saturated anisotropic porous layer. Intl J. Therm. Sci. 49, 867–878.10.1016/j.ijthermalsci.2009.12.008Google Scholar
Mandal, A. & Bera, A. 2015 Modeling of flow of oil-in-water emulsions through porous media. Petrol. Sci. 12, 273–281.10.1007/s12182-015-0025-xGoogle Scholar
Métivier, C., Frigaard, I. A. & Nouar, C. 2009 Nonlinear stability of the Bingham Rayleigh–Bénard Poiseuille flow. J. Non-Newtonian Fluid Mech. 158, 127–131.10.1016/j.jnnfm.2008.08.009Google Scholar
Métivier, C. & Magnin, A. 2011 The effect of wall slip on the stability of the Rayleigh–Bénard Poiseuille flow of viscoplastic fluids. J. Non-Newtonian Fluid Mech. 166, 839–846.10.1016/j.jnnfm.2011.04.017Google Scholar
Métivier, C., Nouar, C. & Brancher, J. P. 2005 Linear stability involving the Bingham model when the yield stress approaches zero. Phys. Fluids 17, 104106.10.1063/1.2101007Google Scholar
Métivier, C. & Nouar, C. 2008 On linear stability of Rayleigh–Bénard Poiseuille flow of viscoplastic fluids. Phys. Fluids 20, 104101.10.1063/1.2987435Google Scholar
Métivier, C. & Nouar, C. 2009 Linear stability of the Rayleigh–Bénard Poiseuille flow for thermodependent viscoplastic fluids. J. Non-Newtonian Fluid Mech. 163, 1–8.10.1016/j.jnnfm.2009.06.001Google Scholar
Métivier, C. & Nouar, C. 2011 Stability of a Rayleigh–Bénard Poiseuille flow for yield stress fluids – comparison between Bingham and regularized models. Intl J. Non-Linear Mech. 46, 1205–1212.10.1016/j.ijnonlinmec.2011.05.017Google Scholar
Métivier, C., Nouar, C. & Brancher, J. P. 2010 Weakly nonlinear dynamics of thermoconvective instability involving viscoplastic fluids. J. Fluid Mech. 660, 316–353.10.1017/S0022112010002788Google Scholar
Moyers-Gonzalez, M., Burghelea, T. I. & Mak, J. 2011 Linear stability analysis for plane-Poiseuille flow of an elastoviscoplastic fluid with internal microstructure for large Reynolds numbers. J. Non-Newtonian Fluid Mech. 166, 515–531.10.1016/j.jnnfm.2011.02.007Google Scholar
Moyers-Gonzalez, M. A., Frigaard, I. A. & Nouar, C. 2004 Nonlinear stability of a visco-plastically lubricated viscous shear flow. J. Fluid Mech. 506, 117–146.10.1017/S0022112004008560Google Scholar
Nash, S. & Rees, D. A. S. 2017 The effect of microstructure on models for the flow of a Bingham fluid in porous media: one-dimensional flows. Trans. Porous Med. 116, 1073–1092.10.1007/s11242-016-0813-9Google Scholar
Nield, D. A. & Bejan, A. 2006 Convection in Porous Media. Springer.Google Scholar
Nouar, C. & Bottaro, A. 2010 Stability of the flow of a Bingham fluid in a channel: eigenvalue sensitivity, minimal defects and scaling laws of transition. J. Fluid Mech. 642, 349–372.10.1017/S0022112009991832Google Scholar
Nouar, C. & Frigaard, I. A. 2001 Nonlinear stability of Poiseuille flow of a Bingham fluid: theoretical results and comparison with phenomenological criteria. J. Non-Newtonian Fluid Mech. 100, 127–149.10.1016/S0377-0257(01)00144-6Google Scholar
Nouar, C., Kabouya, N., Dusek, J. & Mamou, M. 2007 Modal and non-modal linear stability of the plane Bingham–Poiseuille flow. J. Fluid Mech. 577, 211–239.10.1017/S0022112006004514Google Scholar
Ochoa-Tapia, J. A. & Whitaker, S. 1995a Momentum transfer at the boundary between a porous medium and a homogeneous fluid – I. Theoretical development. Intl J. Heat Mass Transfer 38, 2635–2646.10.1016/0017-9310(94)00346-WGoogle Scholar
Ochoa-Tapia, J. A. & Whitaker, S. 1995b Momentum transfer at the boundary between a porous medium and a homogeneous fluid – II. Comparison with experiment. Intl J. Heat Mass Transfer 38, 2647–2655.10.1016/0017-9310(94)00347-XGoogle Scholar
Pavlov, K. B., Romanov, A. S. & Simkhovich, S. L. 1974 Hydrodynamic stability of Poiseuille flow of a viscoplastic non-Newtonian fluid. Fluid Dyn. 9, 996–998.10.1007/BF01020033Google Scholar
Pavlov, K. B., Romanov, A. S. & Simkhovich, S. L. 1975 Stability of Poiseuille flow of a viscoplastic fluid with respect to perturbations of finite amplitude. Fluid Dyn. 10, 841–844.10.1007/BF01015462Google Scholar
Peng, J. & Zhu, K. Q. 2004 Linear stability of Bingham fluids in spiral Couette flow. J. Fluid Mech. 512, 21–45.10.1017/S0022112004009139Google Scholar
Pinarbasi, A. & Liakopoulos, A. 1995 Stability of two-layer Poiseuille flow of Carreau–Yasuda and Bingham-like fluids. J. Non-Newtonian Fluid Mech. 57, 227–241.10.1016/0377-0257(94)01330-KGoogle Scholar
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209–238.10.1017/S0022112093003738Google Scholar
Rees, D. A. S. 2015 Convection of a Bingham fluid in a porous medium. In Handbook of Porous Media, 3rd edn. (ed. Vafai, K.), vol. 17, pp. 559–595. CRC Press.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, 112103.10.1063/1.3502023Google 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, 122101.10.1063/1.2814385Google Scholar
Schmid, P. J. & Henningson, D. S. 1992 A new mechanism for rapid transition involving a pair of oblique waves. Phys. Fluids A 4, 1986–1989.10.1063/1.858367Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.10.1007/978-1-4613-0185-1Google Scholar
Sengupta, S. & De, S. 2019 Couette–Poiseuille flow of a Bingham fluid through a channel overlying a porous layer. J. Non-Newtonian Fluid Mech. 265, 28–40.10.1016/j.jnnfm.2019.01.002Google Scholar
Soleimani, M. & Sadeghy, K. 2010 Dean instability of Bingham fluids in tangential flow between two fixed concentric cylinders. J. Soc. Rheol. Japan 38, 125–132.10.1678/rheology.38.125Google Scholar
Soleimani, M. & Sadeghy, K. 2011 Instability of Bingham fluids in Taylor–Dean flow between two concentric cylinders at arbitrary gap spacings. Intl J. Non-Linear Mech. 46, 931–937.10.1016/j.ijnonlinmec.2011.04.003Google 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, 621–628.Google Scholar
Straughan, B. & Walker, D. W. 1996 Anisotropic porous penetrative convection. Proc. R. Soc. Lond. A 452, 97–115.Google Scholar
Trefethen, L. N. & Embree, M. 2005 Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press.10.1515/9780691213101Google Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578–584.10.1126/science.261.5121.578Google Scholar
Tripathi, D., Yadav, A., Beg, O. A. & Kumar, R. 2018 Study of microvascular non-Newtonian blood flow modulated by electroosmosis. Microvasc. Res. 117, 28–36.10.1016/j.mvr.2018.01.001Google Scholar