Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-06-29T14:53:35.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Gravity-driven flow of Herschel–Bulkley fluid in a fracture and in a 2D porous medium

Published online by Cambridge University Press:  16 May 2017

V. Di Federico Open the ORCID record for V. Di Federico [Opens in a new window] ,
S. Longo Open the ORCID record for S. Longo [Opens in a new window] ,
S. E. King Open the ORCID record for S. E. King [Opens in a new window] ,
L. Chiapponi Open the ORCID record for L. Chiapponi [Opens in a new window] ,
D. Petrolo Open the ORCID record for D. Petrolo [Opens in a new window]  and
V. Ciriello
V. Di Federico*
Affiliation:
Dipartimento di Ingegneria Civile, Chimica, Ambientale e dei Materiali (DICAM), Università di Bologna, Viale Risorgimento, 2, 40136 Bologna, Italy
S. Longo
Affiliation:
Dipartimento di Ingegneria e Architettura (DIA), Università di Parma, Parco Area delle Scienze, 181/A, 43124 Parma, Italy
S. E. King
Affiliation:
School of Mathematics, University of Edinburgh, Edinburgh, EH9 3FD, UK
L. Chiapponi
Affiliation:
Dipartimento di Ingegneria e Architettura (DIA), Università di Parma, Parco Area delle Scienze, 181/A, 43124 Parma, Italy
D. Petrolo
Affiliation:
Dipartimento di Ingegneria e Architettura (DIA), Università di Parma, Parco Area delle Scienze, 181/A, 43124 Parma, Italy
V. Ciriello
Affiliation:
Dipartimento di Ingegneria Civile, Chimica, Ambientale e dei Materiali (DICAM), Università di Bologna, Viale Risorgimento, 2, 40136 Bologna, Italy
*
†Email address for correspondence: vittorio.difederico@unibo.it
Get access Rights & Permissions [Opens in a new window]

Abstract

New analytical models are introduced to describe the motion of a Herschel–Bulkley fluid slumping under gravity in a narrow fracture and in a porous medium. A useful self-similar solution can be derived for a fluid injection rate that scales as time $t$; an expansion technique is adopted for a generic injection rate that is power law in time. Experiments in a Hele-Shaw cell and in a narrow channel filled with glass ballotini confirm the theoretical model within the experimental uncertainty.

JFM classification

Geophysical and Geological Flows: Gravity currents Low-Reynolds-number flows: Hele-Shaw flows Non-Newtonian Flows: Non-Newtonian Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 821 , 25 June 2017 , pp. 59 - 84
Check for updates
Copyright
© 2017 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

Ancey, C. & Cochard, S. 2009 The dam-break problem for Herschel–Bulkley viscoplastic fluids down steep flumes. J. Non-Newtonian Fluid Mech. 158, 18–35.Google Scholar
Balmforth, N. J., Craster, R. V., Rust, A. C. & Sassi, R. 2006 Viscoplastic flow over an inclined surface. J. Non-Newtonian Fluid Mech. 139, 103–127.Google Scholar
Barletta, A. & de B. Alves, L. S. 2014 On Gill’s stability problem for non-Newtonian Darcy’s flow. Intl J. Heat Mass Transfer 79, 759–768.CrossRefGoogle Scholar
Cantelli, A. 2009 Uniform flow of modified Bingham fluids in narrow cross sections. J. Hydraul. Engng 135, 640–650.CrossRefGoogle Scholar
Carotenuto, C. & Minale, M. 2013 On the use of rough geometries in rheometry. J. Non-Newtonian Fluid Mech. 198, 39–47.CrossRefGoogle Scholar
Carreau, P. J. 1972 Rheological equations from molecular network theories. Trans. Soc. Rheol. 16 (1), 99–127.Google Scholar
Chambon, G., Ghemmour, A. & Naiim, M. 2014 Experimental investigation of viscoplastic free-surface flows in a steady uniform regime. J. Fluid Mech. 754, 332–364.Google Scholar
Chevalier, T., Chevalier, C., Clain, X., Dupla, J. C., Canou, J., Rodts, S. & Coussot, P. 2013 Darcy’s law for yield stress fluid flowing through a porous medium. J. Non-Newtonian Fluid Mech. 195, 57–66.Google Scholar
Chevalier, T., Rodts, S., Chateau, X., Chevalier, C. & Coussot, P. 2014 Breaking of non-Newtonian character in flows through a porous medium. Phys. Rev. E 89, 023002.Google Scholar
Ciriello, V., Longo, S., Chiapponi, L. & Di Federico, V. 2016 Porous gravity currents: a survey to determine the joint influence of fluid rheology and variations of medium properties. Adv. Water Resour. 92, 105–115.Google Scholar
Coussot, P. 2014 Yield stress fluid flows: a review of experimental data. J. Non-Newtonian Fluid Mech. 211, 31–49.CrossRefGoogle Scholar
Cristopher, R. H. & Middleman, S. 1965 Power-law flow through a packed tube. Ind. Engng Chem. Fundam. 4, 422–427.Google Scholar
Cross, M. M. 1965 Rheology of non-Newtonian fluids: a new flow equation for pseudoplastic systems. J. Colloid Sci. 20 (5), 417–437.Google Scholar
De Graef, V., Depypere, F., Minnaert, M. & Dewettinck, K. 2011 Chocolate yield stress as measured by oscillatory rheology. Food Res. Intl 44 (9), 2660–2665.CrossRefGoogle Scholar
Di Federico, V., Archetti, R. & Longo, S. 2012a Similarity solutions for spreading of a two-dimensional non-Newtonian gravity current. J. Non-Newtonian Fluid Mech. 177–178, 46–53.CrossRefGoogle Scholar
Di Federico, V., Archetti, R. & Longo, S. 2012b Spreading of axisymmetric non-Newtonian power-law gravity currents in porous media. J. Non-Newtonian Fluid Mech. 189–190, 31–39.CrossRefGoogle Scholar
Di Federico, V., Longo, S., Chiapponi, L., Archetti, R. & Ciriello, V. 2014 Radial gravity currents in vertically graded porous media: theory and experiments for Newtonian and power-law fluids. Adv. Water Resour. 70, 65–76.Google Scholar
Gratton, J., Minotti, F. & Mahajan, S. M. 1999 Theory of creeping gravity currents of a non-Newtonian liquid. Phys. Rev. E 60 (6), 6960–6967.Google Scholar
Herschel, W. H. & Bulkley, R. 1926 Konsistenzmessungen von Gummi–Benzollösungen. Kolloid-Zeitschrift 39 (4), 291–300.Google Scholar
Hewitt, D. R., Daneshi, M., Balmforth, N. J. & Martinez, D. M. 2016 Obstructed and channelized viscoplastic flow in a Hele-Shaw cell. J. Fluid Mech. 790, 173–204.CrossRefGoogle Scholar
Hogg, A. J. & Matson, G. P. 2009 Slumps of viscoplastic fluids on slopes. J. Non-Newtonian Fluid Mech. 158, 101–112.Google Scholar
Hogg, A. J., Ungarish, M. & Huppert, H. E. 2000 Particle-driven gravity currents: asymptotic and box model solutions. Eur. J. Mech. (B/Fluids) 19 (1), 139–165.CrossRefGoogle Scholar
Huang, X. & Garcia, M. H. 1998 A Herschel–Bulkley model for mud flow down a slope. J. Fluid Mech. 374, 305–333.CrossRefGoogle Scholar
King, S. E. & Woods, A. W. 2003 Dipole solutions for viscous gravity currents: theory and experiments. J. Fluid Mech. 483, 91–109.Google Scholar
Lavrov, A. 2013 Redirection and channelization of power-law fluid flow in a rough-walled fracture. Chem. Engng Sci. 99, 81–88.Google Scholar
Liu, K. F. & Mei, C. C. 1989 Slow spreading of a sheet of Bingham fluid on an inclined plane. J. Fluid Mech. 207, 505–529.Google Scholar
Longo, S., Chiapponi, L. & Di Federico, V. 2016 On the propagation of viscous gravity currents of non-Newtonian fluids in channels with varying cross section and inclination. J. Non-Newtonian Fluid Mech. 235, 95–108.Google Scholar
Longo, S., Ciriello, V., Chiapponi, L. & Di Federico, V. 2015a Combined effect of rheology and confining boundaries on spreading of porous gravity currents. Adv. Water Resour. 79, 140–152.Google Scholar
Longo, S. & Di Federico, V. 2014 Axisymmetric gravity currents within porous media: first order solution and experimental validation. J. Hydrol. 519, 238–247.Google Scholar
Longo, S., Di Federico, V., Archetti, R., Chiapponi, L., Ciriello, V. & Ungarish, M. 2013a On the axisymmetric spreading of non-Newtonian power-law gravity currents of time-dependent volume: an experimental and theoretical investigation focused on the inference of rheological parameters. J. Non-Newtonian Fluid Mech. 201, 69–79.CrossRefGoogle Scholar
Longo, S., Di Federico, V. & Chiapponi, L. 2015b A dipole solution for power-law gravity currents in porous formations. J. Fluid Mech. 778, 534–551.CrossRefGoogle Scholar
Longo, S., Di Federico, V. & Chiapponi, L. 2015c Non-Newtonian power-law gravity currents propagating in confining boundaries. Environ. Fluid Mech. 15 (3), 515–535.Google Scholar
Longo, S., Di Federico, V. & Chiapponi, L. 2015d Propagation of viscous gravity currents inside confining boundaries: the effects of fluid rheology and channel geometry. Proc. R. Soc. Lond. A 471 (2178), 20150070.Google Scholar
Longo, S., Di Federico, V., Chiapponi, L. & Archetti, R. 2013b Experimental verification of power-law non-Newtonian axisymmetric porous gravity currents. J. Fluid Mech. 731, R2.Google Scholar
Lyle, S., Huppert, H. E., Hallworth, M., Bickle, M. & Chadwick, A. 2005 Axisymmetric gravity currents in a porous medium. J. Fluid Mech. 543, 293–302.Google Scholar
Magnin, A. & Piau, J. M. 1987 Shear rheometry of fluids with a yield stress. J. Non-Newtonian Fluid Mech. 23, 91–106.Google Scholar
Mei, C. C. & Yuhi, M. 2001 Slow flow of a Bingham fluid in a shallow channel of finite width. J. Fluid Mech. 431, 135–159.CrossRefGoogle Scholar
Nguyen, Q. D. & Boger, D. V. 1992 Measuring the flow properties of yield stress fluids. Annu. Rev. Fluid Mech. 24 (1), 47–88.Google Scholar
Ostwald, W. 1929 Ueber die rechnerische Darstellung des Strukturgebietes der Viskosität. Colloid Polym. Sci. 47 (2), 176–187.Google Scholar
Pascal, J. P. & Pascal, H. 1993 Similarity solutions to gravity flows of non-Newtonian fluids through porous media. Intl J. Non-Linear Mech. 28 (2), 157–167.Google Scholar
Perazzo, C. A. & Gratton, J. 2005 Exact solutions for two-dimensional steady flows of a power-law liquid on an incline. Phys. Fluids 17 (1), 013102.CrossRefGoogle Scholar
Sachdev, P. L. 2000 Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems. CRC.CrossRefGoogle Scholar
Uhlherr, P. H. T., Park, K. H., Tiu, C. & Andrews, J. R. G. 1984 Yield stress from fluid behaviour on an inclined plane. In Advances in Rheology, Mexico City, 1984, vol. 2, pp. 183–190. Springer.Google Scholar
Vola, D., Babik, F. & Latch, J.-C. 2004 On a numerical strategy to compute gravity currents of non-Newtonian fluids. J. Comput. Phys. 201, 397–420.Google Scholar
Wang, S. & Clarens, A. F. 2012 The effects of CO2-brine rheology on leakage processes in geologic carbon sequestration. Water Resour. Res. 48 (8), W08518.Google Scholar
Yasuda, K. Y., Armstrong, R. C. & Cohen, R. E. 1981 Shear flow properties of concentrated solutions of linear and star branched polystyrenes. Rheol. Acta 20 (2), 163–178.Google Scholar