Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-28T02:37:25.805Z Has data issue: false hasContentIssue false
  • English
  • Français

Granular flow on a rotating and gravitating elliptical body

Published online by Cambridge University Press:  14 April 2021

Kumar Gaurav Open the ORCID record for Kumar Gaurav [Opens in a new window] ,
Deepayan Banik Open the ORCID record for Deepayan Banik [Opens in a new window] ,
Ishan Sharma Open the ORCID record for Ishan Sharma [Opens in a new window]  and
Pravir Dutt
Kumar Gaurav*
Affiliation:
Mechanics & Applied Mathematics Group, Kanpur, Uttar Pradesh208016, India Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, Uttar Pradesh208016, India
Deepayan Banik
Affiliation:
Mechanics & Applied Mathematics Group, Kanpur, Uttar Pradesh208016, India Department of Aerospace Engineering, Indian Institute of Technology, Kanpur, Uttar Pradesh208016, India
Ishan Sharma
Affiliation:
Mechanics & Applied Mathematics Group, Kanpur, Uttar Pradesh208016, India Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, Uttar Pradesh208016, India
Pravir Dutt
Affiliation:
Department of Mathematics, Indian Institute of Technology, Kanpur, Uttar Pradesh208016, India
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate two-dimensional shallow granular flows on a rotating and gravitating elliptical body. This is motivated by regolith flow on small planetary bodies – also called minor planets – which is influenced by the rotation of the body, as well as its irregular topography and complex gravity field. Governing equations are obtained in an elliptic coordinate system attached to the body by extending the framework employed for terrestrial avalanches to incorporate effects of rotation, varying gravity and a curvilinear surface. Additionally, we introduce criteria to monitor grain shedding and to track flow initiation and cessation. We delineate different types of regolith motion that are governed by the rotation rate and surface roughness of the body. We find that grains migrate towards the minor and major axis of the body at low and high rotation rates, respectively. Grains are shed when the basal pressure vanishes, and shedding is encouraged by Coriolis effects during prograde flow. We observe the coexistence of regions of static and mobile regolith and their reorganization owing to the merging or division of flows. We also probe the formation and destruction of dunes – bulges arising from local grain accumulation – and find several aspects of their motion to be different from terrestrial situations. We then perform discrete element simulations that display a good match with theoretical predictions. Finally, we consider the evolution of a bi-disperse regolith. We find that big and small grains occupy, respectively, the top and bottom of the dunes formed on the surface, which is reminiscent of observations on asteroids like Itokawa.

JFM classification

Granular media: Granular media Geophysical and Geological Flows: Shallow water flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 916 , 10 June 2021 , A40
Check for updates
Copyright
© The Author(s), 2021. Published by 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

REFERENCES

Abramowitz, M. & Stegun, I.A. 1965 Handbook of Mathematical Functions. Dover.Google Scholar
Adimurthi, A., Aggarwal, A. & Veerappa Gowda, G.D. 2016 Godunov-type numerical methods for a model of granular flow. J. Comput. Phys. 305, 1083–1118.CrossRefGoogle Scholar
Baker, J.L., Johnson, C.G. & Gray, J.M.N.T. 2016 Segregation-induced finger formation in granular free-surface flows. J. Fluid Mech. 809, 168–212.CrossRefGoogle Scholar
Ballouz, R.-L., Baresi, N., Crites, S.T., Kawakatsu, Y. & Fujimoto, M. 2019 Surface refreshing of Martian moon Phobos by orbital eccentricity-driven grain motion. Nat. Geosci. 12, 229–234.CrossRefGoogle Scholar
Barnouin-Jha, O.S., Cheng, A.F., Mukai, T., Abe, S., Hirata, N., Nakamura, R., Gaskell, R.W., Saito, J. & Clark, B.E. 2008 Small-scale topography of 25143 Itokawa from the Hayabusa laser altimeter. Icarus 198, 108–124.CrossRefGoogle Scholar
Chandrasekhar, S. 1969 Ellipsoidal Figures of Equilibrium. Yale University Press.Google Scholar
Cheng, A.F. 2002 Near earth asteroid rendezvous: mission summary. In Asteroids III (ed. W.F. Bottke, A. Cellino, P. Paolicchi & R.P. Binzel), pp. 351–366. University of Arizona Press.Google Scholar
Chujo, T., Mori, O., Kawaguchi, J. & Yano, H. 2018 Categorization of Brazil nut effect and its reverse under less-convective conditions for microgravity geology. Mon. Not. R. Astron. Soc. 474, 4447–4459.CrossRefGoogle Scholar
Cleary, P.W. 2007 Granular flows: fundamentals and applications in granular and complex materials. In World Scientific Lecture Notes in Complex Systems (ed. T. Aste, T. Di Matteo & A. Tordesillas), vol. 8, pp. 141–168. World Scientific Publishing.Google Scholar
Cundall, P.A. & Strack, O.D.L. 1979 A discrete numerical model for granular assemblies. Géotechnique 29, 47–65.CrossRefGoogle Scholar
Delbo, M., Libourel, G., Wilkerson, J., Murdoch, N., Michel, P., Ramesh, K.T., Ganino, C., Verati, C. & Marchi, S. 2014 Thermal fatigue as the origin of regolith on small asteroids. Nature 508, 233–236.CrossRefGoogle ScholarPubMed
Dellar, P.J. & Salmon, R. 2005 Shallow water equations with a complete coriolis force and topography. Phys. Fluids 17, 106601.CrossRefGoogle Scholar
Dobrovolskis, A.R. & Burns, J.A. 1980 Life near the roche limit: behavior of ejecta from satellites close to planets. Icarus 42, 422–441.CrossRefGoogle Scholar
Fujiwara, A., et al. 2006 The rubble-pile asteroid Itokawa as observed by Hayabusa. Science 312, 1330–1334.CrossRefGoogle ScholarPubMed
Gaurav, K. 2020 Granular avalanches on a rotating, self gravitating ellipse. Master's thesis, Indian Institute of Technology, Kanpur.Google Scholar
Gray, J.M.N.T. & Ancey, C. 2009 Segregation, recirculation and deposition of coarse particles near two-dimensional avalanche fronts. J. Fluid Mech. 629, 387–423.Google Scholar
Gray, J.M.N.T. & Chugunov, V.A. 2006 Particle-size segregation and diffusive remixing in shallow granular avalanches. J. Fluid Mech. 569, 365–398.CrossRefGoogle Scholar
Gray, J.M.N.T. & Kokelaar, B.P. 2010 Large particle segregation, transport and accumulation in granular free-surface flows. J. Fluid Mech. 652, 105–137.CrossRefGoogle Scholar
Gray, J.M.N.T., Tai, Y.C. & Noelle, S. 2003 Shock waves, dead zones and particle-free regions in rapid granular free-surface flows. J. Fluid Mech. 491, 161–181.CrossRefGoogle Scholar
Gray, J.M.N.T. & Thornton, A.R. 2005 A theory for particle size segregation in shallow granular free-surface flows. Proc. R. Soc. Lond. A 461, 1447–1473.Google Scholar
Gray, J.M.N.T., Wieland, M. & Hutter, K. 1999 Gravity-driven free surface flow of granular avalanches over complex basal topographies. Proc. R. Soc. Lond. A 455, 1841–1874.CrossRefGoogle Scholar
Guibout, V. & Scheeres, D.J. 2003 Stability of surface motion on a rotating ellipsoid. Icarus 338, 263–290.Google Scholar
Kokelaar, B.P., Bahia, R.S., Joy, K.H., Viroulet, S. & Gray, J.M.N.T. 2017 Granular avalanches on the Moon: mass-wasting conditions, processes, and features. J. Geophys. Res.: Planet 122, 1893–1925.CrossRefGoogle Scholar
Krohn, K., et al. 2014 Mass movement on Vesta at steep scarps and crater rims. Icarus 244, 120–132.CrossRefGoogle Scholar
Kurganov, A. & Tadmor, E. 2000 New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations. J. Comput. Phys. 160, 241–282.CrossRefGoogle Scholar
Liu, Z. & Koyi, H.A. 2013 Kinematics and internal deformation of granular slopes: insights from discrete element modelling. Landslides 10, 139–160.CrossRefGoogle Scholar
Maurel, C., Ballouz, R.-L., Richardson, D.C., Michel, P. & Schwartz, S.R. 2017 Numerical simulations of oscillation-driven regolith motion: Brazil-nut effect. Mon. Not. R. Astron. Soc. 464, 2866–2881.CrossRefGoogle Scholar
Miyamoto, H., et al. 2007 Regolith migration and sorting on asteroid Itokawa. Science 316, 1011–1014.CrossRefGoogle ScholarPubMed
Morland, L.W. 1992 Flow of viscous fluids through a porous deformable matrix. Surv. Geophys. 13, 209–268.CrossRefGoogle Scholar
Murdoch, N., Sanchez, P., Schwartz, S.R. & Miyamoto, H. 2015 Asteroid surface geophysics. In Asteroids IV (ed. W.F. Bottke P. Michel & F.E. DeMeo), Space Science Series, pp. 767–792. University of Arizona Press.Google Scholar
Murray, C.D. & Dermott, S.F. 1999 Solar System Dynamics. Cambridge University Press.Google Scholar
Nedderman, R.M. 1992 Statics and Kinematics of Granular Materials, 1st edn. Cambridge University Press.CrossRefGoogle Scholar
Ogden, R.W. 1997 Non-linear Elastic Deformations, 1st edn. Dover.Google Scholar
Pedlosky, J. 1987 Geophysical Fluid Dynamics, 2nd edn. Springer.CrossRefGoogle Scholar
Plimpton, S. 1995 Fast parallel algorithms for short–range molecular dynamics. J. Comput. Phys. 117, 1–19.CrossRefGoogle Scholar
Pudasaini, S.P. & Fischer, J.T. 2020 A mechanical model for phase separation in debris flow. Intl J. Multiphase Flow 129, 432–448.CrossRefGoogle Scholar
Pudasaini, S.P. & Hutter, K. 2003 Rapid shear flows of dry granular masses down curved and twisted channels. J. Fluid Mech. 495, 193–208.CrossRefGoogle Scholar
Pudasaini, S.P. & Hutter, K. 2007 Avalanche Dynamics, Dynamics of Rapid Flows of Dense Granular Avalanches. Springer.Google Scholar
Roessler, T. & Katterfeld, A. 2019 DEM parameter calibration of cohesive bulk materials using a simple angle of repose test. Particuology 45, 105–115.CrossRefGoogle Scholar
Sánchez, P. & Scheeres, D.J. 2011 Simulating asteroid rubble piles with a self-gravitating soft-sphere distinct element method model. Astrophys. J. 727 (2), 120.CrossRefGoogle Scholar
Sanchez, P. & Scheeres, D.J. 2020 Cohesive regolith on fast rotating asteroids. Icarus 338, 113443.CrossRefGoogle Scholar
Savage, S.B. & Hutter, K. 1989 The motion of a finite mass of granular material down a rough incline. J. Fluid Mech. 199, 177–215.CrossRefGoogle Scholar
Savage, S.B. & Lun, C.K.K. 1988 Particle-size segregation in inclined chute flow of dry cohesionless granular solids. J. Fluid Mech. 189, 311–335.CrossRefGoogle Scholar
Scheeres, D.J. 2015 Landslides and mass shedding on spinning spheroidal asteroids. Icarus 247, 1–17.CrossRefGoogle Scholar
Scheeres, D.J., et al. 2016 The geophysical environment of Bennu. Icarus 276, 116–140.CrossRefGoogle Scholar
Scheeres, D.J., Van wal, S., Olikara, Z. & Baresi, N. 2019 Dynamics in the Phobos environment. Adv. Space Res. 63, 476–495.CrossRefGoogle Scholar
Schwartz, S.R., Richardson, D.C. & Michel, P. 2012 An implementation of the soft-sphere discrete element method in a high-performance parallel gravity tree-code. Granul. Matt. 14 (3), 363–380.CrossRefGoogle Scholar
Sharma, I. 2017 Shapes and Dynamics of Granular Minor Planets. Springer International Publisher.CrossRefGoogle Scholar
Sharma, I., Jenkins, J.T. & Burns, J.A. 2006 Tidal encounters of ellipsoidal granular asteroids with planets. Icarus 183, 312–330.CrossRefGoogle Scholar
Sharma, I., Jenkins, J.T. & Burns, J.A. 2009 Dynamical passage to approximate equilibrium shapes for spinning, gravitating rubble asteroids. Icarus 200, 304–322.CrossRefGoogle Scholar
Tanbakouei, S., Trigo-Rodríguez, J.M., Sort, J., Michel, P., Blum, J., Nakamura, T. & Williams, I. 2019 Mechanical properties of particles from the surface of asteroid 25143 Itokawa. Astron. Astrophys. 629, A119.CrossRefGoogle Scholar
Thornton, A.R. & Gray, J.M.N.T. 2008 Breaking size segregation waves and particle recirculation in granular avalanches. J. Fluid Mech. 596, 261–284.CrossRefGoogle Scholar
Van wal, S. & Scheeres, D.J. 2017 Lift-off velocity on Solar-System small bodies. J. Guid. Control Dyn. 40, 1990–2005.CrossRefGoogle Scholar
Veverka, J., et al. 2000 NEAR at Eros: imaging and spectral results. Science 289, 2088–2097.CrossRefGoogle ScholarPubMed
Yeomans, D.K., et al. 2000 Radio science results during the NEAR-Shoemaker spacecraft rendezvous with Eros. Science 289, 2085–2087.Google ScholarPubMed
Yu, Y., Michel, P., Hirabayashi, M. & Richardson, D.C. 2019 The expansion of debris flow shed from the primary of 65803 Didymos. Mon. Not. R. Astron. Soc. 484, 1057–1071.CrossRefGoogle Scholar
Yu, Y., Michel, P., Hirabayashi, M., Schwartz, S.R., Zhang, Y., Richardson, D.C. & Liu, X. 2018 The dynamical complexity of surface mass shedding from a top-shaped asteroid near the critical spin limit. Astron. J. 156, 59–76.CrossRefGoogle Scholar