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Viscous flow around three-dimensional macroscopic cavities in a granular material

Published online by Cambridge University Press:  26 November 2021

Osamu Sano Open the ORCID record for Osamu Sano [Opens in a new window] ,
Timir Karmakar Open the ORCID record for Timir Karmakar [Opens in a new window]  and
G.P. Raja Sekhar Open the ORCID record for G.P. Raja Sekhar [Opens in a new window]
Osamu Sano
Affiliation:
Tokyo University of Agriculture and Technology, Fuchu, Tokyo 183-8538, Japan
Timir Karmakar
Affiliation:
Department of Mathematics, National Institute of Technology Meghalaya, Shillong, Meghalaya 793003, India
G.P. Raja Sekhar*
Affiliation:
Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India
*
†Email address for correspondence: rajas@iitkgp.ac.in
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Abstract

Viscous flow around spherical macroscopic cavities in a granular material is investigated. The Stokes equation inside and the Darcy–Brinkman equation outside the cavities are considered. In particular, the interaction of two equally sized cavities positioned in tandem is examined in detail, where the asymptotic effect of the other cavity is taken into account. The present analysis gives a reasonable estimate on the volume flow into the cavity and the local enhancement of stresses. This is applicable to predict the microscale waterway formation in that material, onset of landslides, collapse of cliffs and river banks, etc.

JFM classification

Geophysical and Geological Flows: Hydraulic control Granular media: Avalanches Mathematical Foundations: General fluid mechanics

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 931 , 25 January 2022 , A20
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Aranson, I.S. & Tsimring, L.S. 2006 Patterns and collective behavior in granular media: theoretical concepts. Rev. Mod. Phys. 78 (2), 641–692.CrossRefGoogle Scholar
Batchelor, G.K. 1972 Sedimentation in a dilute dispersion of spheres. J. Fluid Mech. 52 (2), 245–268.CrossRefGoogle Scholar
Behringer, R.P. 2015 Jamming in granular materials. C R Phys. 16 (1), 10–25.CrossRefGoogle Scholar
Brinkman, H.C. 1947 A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. 1 (1), 27–34.CrossRefGoogle Scholar
Campbell, C.S. 1990 Rapid granular flows. Annu. Rev. Fluid Mech. 22 (1), 57–90.CrossRefGoogle Scholar
Childress, S. 1972 Viscous flow past a random array of spheres. J. Chem. Phys. 56 (6), 2527–2539.CrossRefGoogle Scholar
Chwang, A. & Wu, T. 1974 Hydromechanics of low-Reynolds-number flow. Part 1. Rotation of axisymmetric prolate bodies. J. Fluid Mech. 63 (3), 607–622.CrossRefGoogle Scholar
Chwang, A.T. & Wu, T. 1975 Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for stokes flows. J. Fluid Mech. 67 (4), 787–815.CrossRefGoogle Scholar
Davis, T., Healy, D., Bubeck, A. & Walker, R. 2017 Stress concentrations around voids in three dimensions: the roots of failure. J. Struct. Geol. 102, 193–207.CrossRefGoogle Scholar
Duran, J. 1997 Sables, poudres et grains: Introduction la physique des milieux granulaires. Eyrolles.Google Scholar
Forterre, Y. & Pouliquen, O. 2008 Flows of dense granular media. Annu. Rev. Fluid Mech. 40, 1–24.CrossRefGoogle Scholar
Freeze, R.A. & Cherry, J.A. 1979 Groundwater, p. 604. Englewood Cliffs.Google Scholar
Givler, R.C. & Altobelli, S.A. 1994 A determination of the effective viscosity for the Brinkman-Forchheimer flow model. J. Fluid Mech. 258, 355–370.CrossRefGoogle Scholar
Gray, J.M.N.T. 2018 Particle segregation in dense granular flows. Annu. Rev. Fluid Mech. 50, 407–433.CrossRefGoogle Scholar
Harr, M.E. 1962 Groundwater and Seepage. McGraw-Hill.Google Scholar
Hill, A.A. & Straughan, B. 2008 Poiseuille flow in a fluid overlying a porous medium. J. Fluid Mech. 603, 137–149.CrossRefGoogle Scholar
Howells, I.D. 1974 Drag due to the motion of a newtonian fluid through a sparse random array of small fixed rigid objects. J. Fluid Mech. 64 (3), 449–476.CrossRefGoogle Scholar
Kaneko, Y. & Sano, O. 2003 Experimental study on the interaction of two two-dimensional circular holes placed in a uniform viscous flow in a porous medium. Fluid Dyn. Res. 32 (1–2), 15.CrossRefGoogle Scholar
Kaneko, Y. & Sano, O. 2005 Collapse and merging of cavity regions in a granular material due to viscous flow. Phys. Fluids 17 (3), 033102.CrossRefGoogle Scholar
Kohr, M. & Pop, I.I. 2004 Viscous Incompressible Flow for Low Reynolds Numbers, vol. 16. Wit Pr/Computational Mechanics.Google Scholar
Kohr, M. & Raja Sekhar, G.P. 2007 Existence and uniqueness result for the problem of viscous flow in a granular material with a void. Quarterly of Applied Mathematics 65 (4), 683–704.CrossRefGoogle Scholar
Le Bars, M. & 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.CrossRefGoogle Scholar
Liu, A.J. & Nagel, S.R. 1998 Jamming is not just cool any more. Nature 396 (6706), 21–22.CrossRefGoogle Scholar
Lundgren, T.S. 1972 Slow flow through stationary random beds and suspensions of spheres. J. Fluid Mech. 51 (2), 273–299.CrossRefGoogle Scholar
Majmudar, T.S. & Behringer, R.P. 2005 Contact force measurements and stress-induced anisotropy in granular materials. Nature 435 (7045), 1079–1082.CrossRefGoogle ScholarPubMed
Martys, N., Bentz, D.P. & Garboczi, E.J. 1994 Computer simulation study of the effective viscosity in Brinkman's equation. Physics of Fluids 6 (4), 1434–1439.CrossRefGoogle Scholar
MiDi, G.D.R. 2004 On dense granular flows. The European Physical Journal E 14, 341–365.CrossRefGoogle Scholar
Momii, K., Jinno, K., Ueda, T., Motomura, H., Hirano, F. & Honda, T. 1989 Experimental study on groundwater flow in a borehole. J Groundwater Hydrol 31 (1), 13–18.CrossRefGoogle Scholar
Morris, J.F.P. 2020 Shear thickening of concentrated suspensions: recent developments and relation to other phenomena. Annu. Rev. Fluid Mech. 52, 121–144.CrossRefGoogle Scholar
Muskat, M. 1937 Flow of Homogeneous Fluids through Porous Media. McGraw-Hill.Google Scholar
Nield, D.A. & Bejan, A. 2013 Convection in Porous Media. Springer Science & Business Media.CrossRefGoogle Scholar
Ochoa-Tapia, J.A. & Whitaker, S. 1995 Momentum transfer at the boundary between a porous medium and a homogeneous fluid-I. Theoretical development. International Journal of Heat and Mass Transfer 38 (14), 2635–2646.CrossRefGoogle Scholar
Raja Sekhar, G.P. & Sano, O. 2000 Viscous flow past a circular/spherical void in porous media — an application to measurement of the velocity of groundwater by the single boring method. Journal of the Physical Society of Japan 69 (8), 2479–2484.CrossRefGoogle Scholar
Raja Sekhar, G.P. & Sano, O. 2003 Two-dimensional viscous flow in a granular material with a void of arbitrary shape. Physics of Fluids 15 (2), 554–567.CrossRefGoogle Scholar
Sano, O. 1983 Viscous flow past a cylindrical hole bored inside a porous media-with applications to measurement of the velocity of subterranean water by the single boring method. Nagare 2, 252–259.Google Scholar
Sano, O. 2008 Viscous flow past two spherical cavities in a porous media: application to drug delivery system. In Proceedings of the 22nd International Congress of Theoretical and Applied Mechanics (Adelaide Australia), p. 10523.Google Scholar
Sano, O. 2011 Flow-induced waterway in a heterogeneous granular material. Computer Physics Communications 182 (9), 1870–1874.CrossRefGoogle Scholar
Sano, O. 2020 Viscous flow and collapse of macroscopic cavities in a granular material in terms of a Darcylet. Philosophical Transactions of the Royal Society A 378 (2174), 20190527.CrossRefGoogle Scholar
Sano, O. & Nagata, Y. 2006 Collapse and growth of cavity regions in granular media due to viscous flow. Physics of Fluids 18 (12), 121507.CrossRefGoogle Scholar
Sano, O., Sano, N., Takagi, Y. & Yamada, Y. 2013 Onset of landslides due to the flow-induced waterway formation in three-dimension in a granular material. In Proceedings of the 14th Asian Congress of Fluid Mechanics, Hanoi and Halong, Vietnam, pp. 1254–1260.Google Scholar
Scheidegger, A.E. 1960 The Physics of Flow through Porous Media. University of Toronto Press.Google Scholar
Shimizu, H. 2012 3d cell arrangement and its pathologic change. Forma 27, S9–S19.Google Scholar
Shimizu, H., Suda, K. & Yokoyama, T. 1996 Difference in the three-dimensional structure of the sinusoids between hepatocellular carcinoma and cirrhotic liver. Pathology International 46 (12), 992–996.CrossRefGoogle ScholarPubMed
Shimizu, H. & Yokoyama, T. 1993 Three-dimensional structural changes of hepatic sinusoids in cirrhosis providing an increase in vascular resistance of portal hypertension. Pathology International 43 (11), 625–634.CrossRefGoogle ScholarPubMed
Tam, C.K.W. 1969 The drag on a cloud of spherical particles in low reynolds number flow. J. Fluid Mech. 38 (3), 537–546.CrossRefGoogle Scholar
Waite, C.L. & Roth, C.M. 2012 Nanoscale drug delivery systems for enhanced drug penetration into solid tumors: current progress and opportunities. Critical Reviews in Biomedical Engineering 40 (1), 21–41.CrossRefGoogle ScholarPubMed