How to calculate Bagnold number Ba granular flow rheology

Calculate Bagnold number Ba = rho*lambda^(1/2)*L^2*shear/mu ratio collisional inertia to viscous force in granular flow. Ba<40 macro-viscous slurry; Ba>450 grain-inertia pyroclastic avalanche. Ralph Bagnold UK 1896-1990 + 1954 + WW2 LRDG Sahara. NIST + USGS + NASA Mars + 3D Systems applications.

Calculation parameters
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Bagnold number Ba dimensionless ratio of collisional inertia to viscous force in granular flow + debris flow + pyroclastic + powder rheology: Ba = rho * lambda^(1/2) * L^2 * gamma_dot / mu. rho particle density kg/m^3 (sand 2650; clay 2600; pyroclastic 1500); lambda Bagnold concentration parameter ((c_v_max/c_v)^(1/3)-1)^(-1) + c_v_max=0.74 random close packing; L particle diameter m; gamma_dot shear rate 1/s; mu interstitial fluid viscosity Pa*s.

Regimes: Ba<40 macro-viscous viscous fluid dominant + clay slurry + concrete fresh + Geldart C; 40-450 transitional mixed sand-water dam-break sediment; Ba>450 grain-inertia particle collision dominant + pyroclastic + avalanche + dry sand + Mars dunes.

Bagnold's stress relations: tau_grain = 0.042*sin(alpha)*rho*lambda^2*L^2*gamma_dot^2 grain-inertia; tau_visc = mu_eff*gamma_dot macro-viscous.

Geldart classification: A 30-100 micron FCC; B 100-1000 sand; C 1000 spoutable. History: Ralph Alger Bagnold (London UK 1896 - Edenbridge Kent UK 1990) Cambridge Caius College + Royal Engineers WW1 + WW2 Long Range Desert Group LRDG founder Sahara North Africa 1940-1943 + retired Brigadier 1944 + Royal Society Fellow 1944 + Penrose Medal Geological Society America 1969 + Father of Modern Aeolian Studies; 1936 paper Proc Roy Soc A 157:594 'The movement of desert sand' + 1941 textbook 'The Physics of Blown Sand and Desert Dunes' Methuen 265 pages classic foundation aeolian + dunes barchan transverse + saltation creep + threshold velocity + 1954 paper Proc Roy Soc A 225:49 'Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear' classic Bagnold number eponym + Couette rheometer; pioneer aeolian + sediment transport + LRDG vehicles 4x4 + 1990 'Sand Wind and War' autobiography + Bagnold Dunes Mars Curiosity Gale Crater 2016 named.

Charles-Augustin de Coulomb (Angouleme France 1736-1806); 1773 paper Memoires Acad Sci Coulomb friction soil mechanics tau = c + sigma*tan(phi) classical mechanics.

Daniel Drucker + William Prager Brown 1952; 1952 Q Appl Math 10:157 Drucker-Prager yield criterion granular plasticity.

Stuart Savage McGill + Kolumban Hutter ETH Zurich 1989; 1989 J Fluid Mech 199:177 Savage-Hutter equations granular avalanche shallow water classic landslide debris flow modeling.

Pierre-Gilles de Gennes France Nobel Prize 1991 + 1999 Rev Mod Phys 71:S374 'Granular matter' classic.

Heinrich Jaeger Chicago + Sidney Nagel + Robert Behringer Duke 1996; 1996 Rev Mod Phys 68:1259 'Granular solids liquids gases' classic comprehensive review jamming. Legacy: Duran 'Sands Powders Grains' 2000 Springer 214 pages classic + Andreotti-Forterre-Pouliquen 'Granular Media' 2013 Cambridge 460 pages classic + Pudasaini-Hutter 'Avalanche Dynamics' 2007 Springer + Iverson 1997 Rev Geophys 35:245 'The physics of debris flows' classic 96 pages + Forterre-Pouliquen 2008 Annu Rev Fluid Mech 40:1.

USA research Bagnold + granular + debris + pyroclastic + Mars Bagnold Dunes: NIST Gaithersburg MD 3500 + powder reference + ASTM standards; USGS US Geological Survey Reston VA 1879 8000 + sediment transport + Hawaiian Volcano Observatory HVO Big Island Mauna Loa Kilauea + Yellowstone Volcano Observatory YVO + Cascades Volcano Observatory CVO Mt St Helens + USGS Cascades Volcano Lab + USGS Volcano Hazards Program + Sedimentation Lab Vicksburg + USGS Tongue Point Astoria; USACE Vicksburg MS Engineer Research and Development Center ERDC 38000 + sediment transport + Hydraulics Lab Vicksburg world-leading; NOAA National Oceanic and Atmospheric Administration Boulder CO Silver Spring MD 12000 + sediment + tsunami + Hawaii; NASA JPL Pasadena CA 1936 6000 + Mars rovers + Curiosity 2012 + Perseverance 2021 + Bagnold Dunes Mars Gale Crater 2016 named after Ralph Bagnold + dust storm Mars; NASA Ames Mountain View CA 2300 + planetary atmospheres; NASA Marshall Huntsville AL 6000; USGS Astrogeology Flagstaff AZ 200 + Mars geology + planetary; Lunar and Planetary Institute LPI Houston TX 200; NREL Golden CO 2900; ORNL 6500; Sandia 14000; LBNL 4400; NETL Pittsburgh + Morgantown WV 1500 + biomass + CFB;

EPA 14000 + sediment air quality.

USA universities Bagnold + granular + debris + Mars: MIT Cambridge MA 11700 + Mech Eng + ChemE Sara Hashmi granular + James Watt; Caltech Pasadena 2400 + GALCIT John Brady microhydrodynamics + Mike Lamb sediment transport; Stanford CA 17000 + Mech Eng + Geological & Environmental Sciences + Mars; UC Berkeley 45000 + Earth and Planetary Science + Michael Manga magma; Princeton 8000 + MAE Howard Stone microfluidics + Sankaran Sundaresan multiphase + Mars Curiosity team; Cornell 25000 + ChemE Donald Koch + Earth and Atmospheric Science + Steve Squyres Mars Spirit Opportunity rover + Mars Sample Return; Georgia Tech 36000 + Christopher Jones; UIUC 56000 + S Pratap Vanka; Penn State 50000 + Charles Sing; UT Austin 51000 + ME + Geological Sciences + Mars; Univ Michigan Ann Arbor 50000; Northwestern Evanston IL 22000; Purdue 47000; Wisconsin-Madison 49000; Minnesota 51000; UCLA 45000 + Earth and Planetary Sciences + Edward Young + Mars; UC Davis 40000; Univ Arizona Tucson 1885 47000 + Lunar and Planetary Lab LPL world-leading + Mars HiRISE + Phoenix Lander Bagnold Dunes; Arizona State Tempe 1885 80000 + Mars + Phil Christensen Mars;

Univ Washington Seattle 47000 + Earth and Space Sciences; Texas A&M College Station 70000 + Geosciences; Brown Providence RI 7400 + Drucker-Prager 1952 heritage Mech Eng + Earth Environmental and Planetary Sciences + Carle Pieters Mars + James Head Mars; Duke University Durham NC 16000 + Robert Behringer granular Mech Eng + 1996 Rev Mod Phys classic; Univ Chicago 17000 + James Franck Institute + Heinrich Jaeger granular world-leading + Sidney Nagel granular world-leading + Heinrich Jaeger Chicago + 1996 Rev Mod Phys 68:1259 Jaeger-Nagel-Behringer classic + Granular Group; UMass Amherst 30000; Carnegie Mellon Pittsburgh 14000; Drexel Philadelphia PA 24000.

USA volcano agencies pyroclastic: USGS Hawaiian Volcano Observatory HVO Big Island Hawaii Volcanoes National Park 200 + Kilauea + Mauna Loa + Mauna Kea + pyroclastic monitoring 1912; USGS Yellowstone Volcano Observatory YVO Yellowstone WY 200; USGS Cascades Volcano Observatory CVO Vancouver WA 200 + Mt St Helens + Mt Rainier + Mt Hood + Mt Shasta + Mt Lassen; USGS Alaska Volcano Observatory AVO Anchorage AK 200 + Mt Redoubt + Augustine + Veniaminof; Smithsonian Global Volcanism Program Washington DC 200.

USA cement powder Geldart C industry: Holcim USA Chicago 5000; CEMEX USA Houston 9000; HeidelbergCement North America 6000; USG Corp Chicago 7100 gypsum drywall plasters; Eagle Materials Dallas 1500; Martin Marietta 9000; Vulcan Materials Birmingham 11000; Ash Grove Cement 2300.

USA 3D printing powder bed fusion: 3D Systems Rock Hill SC 1700; Stratasys Eden Prairie MN 2700; Desktop Metal Burlington MA 1100; HP Multi Jet Fusion Palo Alto 51000; GE Additive 800; Markforged Watertown MA 200; Carbon3D Redwood City CA 200.

USA Mars Bagnold Dunes ExoMars Curiosity: NASA JPL Pasadena 6000 + Curiosity 2012 + Perseverance 2021 + Mars Sample Return + Bagnold Dunes Gale Crater 2016 named; Lockheed Martin Bethesda MD 116000 + Mars rovers + atmospheric entry; Boeing Arlington VA 156000; SpaceX Hawthorne CA 13000 + Starship Mars; Blue Origin Kent WA 11000; Northrop Grumman 102000.

USA food powder agro-food: Nestle USA Solon OH 47000 Nescafe; Kraft Heinz Pittsburgh PA 38000; General Mills Minneapolis MN 35000; Kellogg's Battle Creek MI 31000 cereal; PepsiCo Purchase NY 318000; Coca-Cola Atlanta GA 79000; Mondelez Chicago IL 80000.

USA mining + drilling fluids bentonite: ExxonMobil Irving TX 62000 + drilling fluids; Chevron 45000; Halliburton Houston TX 50000 + Baroid drilling fluids world-leading; Schlumberger Houston TX 90000 + M-I SWACO; Baker Hughes 60000; NOV 36000.

Bagnold formula

Definition: Ba = rho*lambda^(1/2)*L^2*gamma_dot/mu.

Concentration parameter: lambda = ((c_v_max/c_v)^(1/3)-1)^(-1).

Grain-inertia stress: tau = 0.042*sin(alpha)*rho*lambda^2*L^2*gamma_dot^2.

Macro-viscous stress: tau = mu_eff*gamma_dot.

Geldart classification: A 30-100 + B 100-1000 + C 1000.

History

Ralph Bagnold (London 1896 - Edenbridge 1990) Cambridge Caius + WW1 Royal Engineers + WW2 Long Range Desert Group LRDG founder Sahara 1940-1943 + retired Brigadier 1944 + RS Fellow 1944 + Penrose Medal 1969; 1941 'Physics of Blown Sand' Methuen classic 265 pages + 1954 Proc Roy Soc A 225:49 Bagnold number eponym; pioneer aeolian + Bagnold Dunes Mars Curiosity Gale Crater 2016. Coulomb France 1773. Drucker-Prager Brown 1952. Savage McGill + Hutter ETH 1989. de Gennes Nobel 1991. Jaeger-Nagel Chicago + Behringer Duke 1996 Rev Mod Phys 68:1259 classic. Duran 2000 + Andreotti-Forterre-Pouliquen 2013 + Iverson 1997.

USA research: NIST 3500; USGS Reston 8000 HVO Hawaii Mauna Loa Kilauea + YVO Yellowstone + CVO Mt St Helens + AVO Alaska + Astrogeology Flagstaff AZ + Sedimentation Vicksburg; USACE Vicksburg ERDC 38000 + Hydraulics Lab world-leading; NOAA Boulder Silver Spring 12000; NASA JPL Pasadena 6000 Curiosity Perseverance + Bagnold Dunes 2016; NASA Ames 2300; NASA Marshall 6000; USGS Astrogeology Flagstaff 200; LPI Houston 200; NREL 2900; ORNL 6500; Sandia 14000; LBNL 4400; NETL 1500; EPA 14000; Smithsonian GVP 200.

USA universities: MIT 11700; Caltech 2400 Brady Lamb sediment; Stanford 17000; UC Berkeley 45000 Manga magma; Princeton 8000 Stone Sundaresan + Mars; Cornell 25000 Squyres Mars Spirit; Georgia Tech 36000; UIUC 56000; Penn State 50000; Michigan 50000; Northwestern 22000; Purdue 47000; UCLA 45000 Mars; UC Davis 40000; Arizona Tucson 47000 LPL world-leading + HiRISE Mars + Bagnold Dunes; Arizona State Tempe 80000 Christensen Mars; Texas A&M 70000; Brown 7400 Drucker-Prager heritage + Pieters Head Mars; Duke 16000 Behringer 1996 classic + granular world-leading; Chicago 17000 Jaeger Nagel granular world-leading + James Franck Institute + 1996 Rev Mod Phys; UMass Amherst 30000; Carnegie Mellon 14000; Drexel 24000.

USA volcano: USGS HVO Hawaii 200; USGS YVO Yellowstone 200; USGS CVO 200 Mt St Helens; USGS AVO Alaska 200 Redoubt; Smithsonian GVP 200.

USA cement: Holcim 5000; CEMEX 9000; HeidelbergCement NA 6000; USG 7100; Eagle 1500; Martin Marietta 9000; Vulcan 11000; Ash Grove 2300.

USA 3D printing: 3D Systems 1700; Stratasys 2700; Desktop Metal 1100; HP MJF 51000; GE Additive 800; Markforged 200; Carbon3D 200.

USA Mars Bagnold: NASA JPL 6000 Curiosity Perseverance Bagnold Dunes Gale 2016; Lockheed Martin 116000; Boeing 156000; SpaceX 13000; Blue Origin 11000; Northrop Grumman 102000.

USA food powder: Nestle USA 47000; Kraft Heinz 38000; General Mills 35000; Kellogg's 31000; PepsiCo 318000; Coca-Cola 79000; Mondelez 80000.

USA drilling: ExxonMobil 62000; Halliburton 50000 Baroid; Schlumberger 90000 M-I SWACO; Baker Hughes 60000; NOV 36000.

Bagnold values table

  • Ba = rho*lambda^(1/2)*L^2*gamma_dot/mu
  • lambda = ((c_v_max/c_v)^(1/3)-1)^(-1)
  • tau_grain = 0.042*sin(alpha)*rho*lambda^2*L^2*gamma_dot^2
  • tau_visc = mu_eff*gamma_dot
  • Ba<40 macro-viscous: clay slurry
  • 40-450 transitional: sand-water
  • Ba>450 grain-inertia: pyroclastic
  • Geldart A 30-100 FCC
  • Geldart B 100-1000 sand
  • Geldart C <30 cohesive
  • Geldart D >1000 spoutable
  • Coulomb 1773 friction
  • Drucker-Prager 1952
  • Savage-Hutter 1989 avalanche
  • Bagnold UK 1896-1990: 1936 + 1941 + 1954
  • Bagnold 1941 'Physics of Blown Sand'
  • Bagnold 1954 Proc Roy Soc A 225:49
  • Bagnold WW2 LRDG Sahara
  • Bagnold Penrose Medal 1969
  • Bagnold Dunes Mars Curiosity 2016
  • USA NIST 3500
  • USA USGS Reston 8000 HVO YVO CVO AVO
  • USA USACE ERDC 38000: Hydraulics world-leading
  • USA NASA JPL 6000: Curiosity Bagnold Dunes 2016
  • USA Arizona LPL world-leading: HiRISE Mars
  • USA Brown Drucker-Prager + Pieters Head Mars
  • USA Duke Behringer 1996 + Chicago Jaeger Nagel: granular world-leading
  • USA Holcim 5000 + CEMEX 9000: cement
  • USA 3D Systems 1700 + Stratasys 2700
  • USA Halliburton 50000: Baroid drilling

Numerical examples

Example 1 - Clay slurry. Ba=5.81 macro-viscous. USA: Holcim + CEMEX + ExxonMobil drilling.

Example 2 - Sand water. Ba=94 transitional. USA: USGS Sedimentation Vicksburg + USACE ERDC.

Example 3 - Debris flow. Ba=9882 grain-inertia. USA: USGS Cascades CVO Mt St Helens + ERDC Vicksburg world-leading.

Example 4 - Pyroclastic Etna. Ba=29M grain-inertia very high. USA: USGS HVO Hawaii Kilauea + YVO Yellowstone + CVO Mt St Helens + AVO Alaska Redoubt + Smithsonian GVP.

Example 5 - Dry sand Sahara/Mars. Ba=5341 grain-inertia. USA: NASA JPL Curiosity Perseverance + Arizona LPL HiRISE + Bagnold Dunes Gale Crater 2016 named.

Limits and considerations

  1. Lambda concentration parameter measurement.
  2. Newtonian assumption fluid.
  3. Sphere assumption Heywood.
  4. Polidisperse mixture.
  5. Cohesive Geldart C.
  6. Wetting capillary.
  7. Compressibility random close packing.
  8. Time-dependent thixotropy.
  9. Wall effects Faxen.
  10. Inertial vs viscous Re_p.

Common errors

Bagnold vs Stokes: collisional vs viscous.

Concentration lambda: NOT volume fraction.

Mu interstitial fluid: NOT bulk granular.

Two stress relations: gamma_dot^2 vs gamma_dot.

Geldart C cohesive: different from grain-inertia.

Practical applications

  1. Pyroclastic Vesuvius Etna geophysics: USA USGS HVO Hawaii + YVO Yellowstone + CVO Mt St Helens + AVO Alaska + Smithsonian GVP + Astrogeology Flagstaff.
  2. Debris flow landslide forecasting: USA USGS + USACE ERDC Vicksburg world-leading + NOAA tsunami + NSF NEHRP.
  3. Sediment transport rivers hydraulics: USA USGS Reston Sedimentation Vicksburg + USACE ERDC + NOAA Mississippi.
  4. Cement powder Geldart C cohesive: USA Holcim 5000 + CEMEX 9000 + HeidelbergCement NA + USG 7100 + Eagle Materials + Martin Marietta + Vulcan + Ash Grove.
  5. 3D printing powder bed fusion: USA 3D Systems 1700 + Stratasys 2700 + Desktop Metal 1100 + HP MJF + GE Additive + Markforged + Carbon3D + Nano Dimension.
  6. Food powder spray drying: USA Nestle USA Solon 47000 Nescafe + Kraft Heinz 38000 + General Mills 35000 + Kellogg's 31000 cereal + PepsiCo + Coca-Cola + Mondelez.
  7. Mining + drilling fluids bentonite: USA ExxonMobil 62000 + Chevron 45000 + Halliburton Houston 50000 Baroid world-leading + Schlumberger 90000 M-I SWACO + Baker Hughes + NOV.
  8. Automotive sand casting + powder metallurgy: USA Ford + GM + Stellantis + Tesla + automotive aluminum casting.
  9. Mars Bagnold Dunes ExoMars Curiosity: USA NASA JPL Pasadena 6000 + Curiosity 2012 + Perseverance 2021 + Bagnold Dunes Gale Crater 2016 named after Ralph Bagnold + Lockheed Martin atmospheric entry + SpaceX Starship Mars + Blue Origin.
  10. Universities granular Bagnold: Caltech Lamb + Princeton Sundaresan + Cornell Squyres Mars + Brown Drucker-Prager Pieters Mars + Duke Behringer + Chicago Jaeger Nagel granular world-leading + Arizona LPL HiRISE world-leading.

Updated July 2026

Frequently Asked Questions

What is the Bagnold number?

Ba = rho*lambda^(1/2)*L^2*gamma_dot/mu dimensionless ratio collisional inertia to viscous force in granular flow + debris + pyroclastic + powder. Ba<40 macro-viscous clay slurry; 40-450 transitional; Ba>450 grain-inertia pyroclastic dry sand. Ralph Bagnold UK 1896-1990 + 1954 Proc Roy Soc A 225:49 + WW2 LRDG Sahara + 1941 'Physics of Blown Sand' Methuen classic 265 pages. USA: NIST 3500 + USGS Reston 8000 HVO YVO CVO + NASA JPL 6000 Curiosity Bagnold Dunes 2016 + Arizona LPL HiRISE.

How do I calculate Bagnold?

Procedure: (1) Ba = rho*lambda^(1/2)*L^2*gamma_dot/mu; (2) rho kg/m^3 (sand 2650); (3) lambda concentration parameter; (4) L particle m (sand 0.5 mm); (5) gamma_dot 1/s; (6) mu Pa*s (water 1e-3; air 2e-5); (7) regime. Sand water: Ba=94 transitional. Pyroclastic: Ba=29M grain-inertia. USA: Caltech Lamb sediment + Brown Pieters Mars + Duke Behringer granular world-leading.

What is Bagnold flow?

Bagnold flow Ralph Bagnold UK + 1954 Proc Roy Soc A 225:49 'Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear' classic + Couette rheometer + grain-inertia regime tau ~ gamma_dot^2 + macro-viscous tau ~ gamma_dot linear + Bagnold's two stress relations. USA: NASA JPL Curiosity + Arizona HiRISE + Brown Pieters Mars + Chicago Jaeger Nagel granular world-leading.

What regimes does Ba classify?

Ba<40 macro-viscous viscous fluid dominant + clay slurry + concrete + Geldart C cohesive; 40<Ba<450 transitional mixed sand-water dam-break sediment; Ba>450 grain-inertia particle collision dominant + pyroclastic + avalanche + dry sand + Mars dunes + 3D printing powder. USA: Holcim cement + USGS sediment + USGS HVO Hawaii pyroclastic + NASA JPL Curiosity Mars Bagnold Dunes.

Who was Ralph Bagnold?

Ralph Alger Bagnold (London UK 1896 - Edenbridge Kent UK 1990) Cambridge Caius College + Royal Engineers WW1 + WW2 Long Range Desert Group LRDG founder Sahara North Africa 1940-1943 + retired Brigadier 1944 + Royal Society Fellow 1944 + Penrose Medal Geological Society America 1969 + Father of Modern Aeolian Studies; 1941 'Physics of Blown Sand and Desert Dunes' Methuen classic 265 pages + 1954 Proc Roy Soc A 225:49 Bagnold number eponym + Bagnold Dunes Mars Curiosity Gale Crater 2016 named after.

What is Geldart classification?

Geldart classification powder Derek Geldart UK 1973 paper Powder Technol 7:285 + 4 classes: A aeratable 30-100 micron FCC catalyst smooth fluidization; B sand-like 100-1000 micron bubbles sand; C cohesive <30 micron cement flour Van der Waals; D spoutable >1000 micron grains coal. Bagnold complementary + cohesive C different from grain-inertia. USA: Holcim cement Geldart C + Nestle Nescafe Geldart B + Halliburton Baroid drilling bentonite.

What are Bagnold applications?

Applications: pyroclastic Vesuvius geophysics USA USGS HVO Hawaii Mauna Loa Kilauea + YVO Yellowstone + CVO Mt St Helens + AVO Alaska Redoubt + Smithsonian GVP; debris flow USACE ERDC Vicksburg world-leading + USGS; sediment rivers USGS Sedimentation Vicksburg + ERDC Mississippi; cement powder Geldart C Holcim 5000 + CEMEX 9000 + USG 7100; 3D printing 3D Systems 1700 + Stratasys 2700 + Desktop Metal + HP MJF + GE Additive; food powder Nestle Nescafe + Kraft Heinz + General Mills + Kellogg's cereal; drilling Halliburton 50000 Baroid + Schlumberger M-I SWACO; Mars Bagnold Dunes NASA JPL Curiosity 2012 Perseverance + Bagnold Dunes Gale Crater 2016 + Arizona LPL HiRISE world-leading.

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