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Latin Characters
| a | Inter-molecular distance | m |
| aw | Adhesion | Pa |
| aξ | Lack of velocity fluctuations | — |
| A | Hamaker constant | J |
| Ab | Cross-sectional area of the powder bed | m2 |
| Aeff | Normalized effective projected area | m2 |
| AI | Arching index | — |
| Ap | Cross-sectional area of plate | m2 |
| Aproj | Projected area of indent | m2 |
| AR | Aspect ratio | — |
| ARST | Cross-sectional area of Schulze ring shear cell | m2 |
| ATD | Average time difference | s |
| AU | Cross-sectional area of uniaxial compaction tester | m2 |
| AJT | Cross-sectional area of Jenike cell | m2 |
| b | Body force | N |
| B | Distance of barycentre of mass from axis of rotation | m |
| BIM | Ball indentation method | — |
| Bo | Bond number | — |
| c | Cohesion | Pa |
| C | Constraint factor | — |
| CBeverloo | Flowability index | — |
| CI | Carr index | — |
| Co | Compressibility | — |
| Coh | Cohesion number | — |
| d | Particle size | m |
| d43 | Volume moment mean diameter | m |
| db | Indenter diameter | m |
| dc | Diameter of contact | m |
| dFe | Feret diameter | m |
| dFe⊥ | Feret diameter measured in perpendicular direction | m |
| dv | Particle equivalent volume sphere diameter | m |
| D | Aperture diameter | m |
| Dheap | Diameter of powder heap | m |
| e | Rod diameter | m |
| eeff | Effective coefficient of restitution | — |
| E | Young modulus | Pa |
| Ef | Flow energy | J |
| f | Inter-particle force | N |
| fc | Unconfined yield stress | Pa |
| ff | Flow-factor | — |
| F | Force | N |
| Fc | Contact force | N |
| FD | Fractal dimensions | — |
| Fg | Gravitational force | N |
| F | Contact force exerted on i by j | N |
| Fn | Normal load | N |
| Fnc | Non-contact force | N |
| Fr | Frounde number | — |
| Fs | Macroscopic shear force | N |
| Ft | Translational force | N |
| g | Gravitational acceleration | m s−2 |
| g0 | Radial distribution function | — |
| Gn | Solid fraction dependent functions in rapid flow regime | — |
| h | Height | m |
| hB | Burke constant | — |
| hc,max | Indentation unloading intercept with depth axis | m |
| hd | Indentation depth | m |
| hf | Final indentation depth | m |
| hsep | Separation distance | m |
| H | Indentation hardness | Pa |
| HR | Haussner ratio | — |
| i | Element under consideration | — |
| I | Moment of inertia/inertial number | kg m2 |
| I0 | Constant for μ(I) rheology | — |
| IA, IB | Modified Bessel functions of the first kind | — |
| k | Permeability | m2 |
| kA | Fraction of collisional dissipation contribution to shear viscosity | — |
| kB | Fraction of frictional contribution to shear viscosity | — |
| kc | Carman constant | — |
| kj | Constant for Johanson hang-up indiciser | ft |
| Ka | Active Rankine coefficient | — |
| Kp | Passive Rankine coefficient | — |
| l | Rod length | m |
| lij | Vector from mass centre of particle i to mass centre of particle j | m |
| ℓp | Unit vector along the major axis | — |
| L | Drum length | m |
| Lp | Perimeter length | m |
| Ly | Dimension of domain in y-direction | m |
| m | Mass | kg |
| M | Contact torque | N m |
| n | Number of particles | — |
| N | Macroscopic normal force | N |
| Nc | Applied force at failure | N |
| O | Order parameter | — |
| p | Particle under consideration | — |
| P | Pressure | Pa |
| Pa | Active pressure | Pa |
| Pp | Passive pressure | Pa |
| PFI | Powder flow index | — |
| PR | Packing ratio | — |
| q | Energy flux | N m−1 s−1 |
| Qij | Symmetric traceless order tensor | Pa |
| r | Asperity size | m |
| rcol | Radius of meniscus curvature | m |
| rro | Surface roughness length | m |
| R | Radius | m |
| R1 | Minor radius of the lid cross-sectional annular area | m |
| R2 | Major radius of the lid cross-sectional annular area | m |
| Rgas | Ideal gas constant | kg m s−2 K−1 |
| RH | Relative humidity | % |
| RI | Ratholing index | — |
| s | Particle surface area | m2 |
| S | Stiffness | N m−1 |
| Seff | Effective projected surface area of a particle | m2 |
| Ssph | Effective projected surface area of the sphere of the same volume to the particle | m2 |
| STD | Standard deviation | — |
| t | Time | s |
| T | Torque | N m |
| Tabs | Temperature | K |
| Tgran | Granular temperature | m2 s−2 |
| U | Velocity at boundary | m s−1 |
| Ucons | Consolidation velocity | m s−1 |
| Umf | Minimum fluidisation velocity | m s−1 |
| Uro | Rain-off velocity | m s−1 |
| Ut | Terminal velocity | m s−1 |
| v | Velocity | m s−1 |
| vax | Axial velocity | m s−1 |
| 〈vp,ave〉 | Average particle velocity | m s−1 |
| vp,fl | Fluctuating velocity | m s−1 |
| vp,i | Velocity of particle i | m s−1 |
| V | Particle volume | m3 |
| Vliq | Volume of liquid per contact | m3 |
| Vm | Molar volume | m3 |
| w | Weight | N |
| W | Mass flow rate | kg s−1 |
| x | Distance between crossbeam holders | m |
| y | Parameter of yield locus curvature | — |
| Y | Yield stress | Pa |
| Z | Height | m |
| a | Inter-molecular distance | m |
| aw | Adhesion | Pa |
| aξ | Lack of velocity fluctuations | — |
| A | Hamaker constant | J |
| Ab | Cross-sectional area of the powder bed | m2 |
| Aeff | Normalized effective projected area | m2 |
| AI | Arching index | — |
| Ap | Cross-sectional area of plate | m2 |
| Aproj | Projected area of indent | m2 |
| AR | Aspect ratio | — |
| ARST | Cross-sectional area of Schulze ring shear cell | m2 |
| ATD | Average time difference | s |
| AU | Cross-sectional area of uniaxial compaction tester | m2 |
| AJT | Cross-sectional area of Jenike cell | m2 |
| b | Body force | N |
| B | Distance of barycentre of mass from axis of rotation | m |
| BIM | Ball indentation method | — |
| Bo | Bond number | — |
| c | Cohesion | Pa |
| C | Constraint factor | — |
| CBeverloo | Flowability index | — |
| CI | Carr index | — |
| Co | Compressibility | — |
| Coh | Cohesion number | — |
| d | Particle size | m |
| d43 | Volume moment mean diameter | m |
| db | Indenter diameter | m |
| dc | Diameter of contact | m |
| dFe | Feret diameter | m |
| dFe⊥ | Feret diameter measured in perpendicular direction | m |
| dv | Particle equivalent volume sphere diameter | m |
| D | Aperture diameter | m |
| Dheap | Diameter of powder heap | m |
| e | Rod diameter | m |
| eeff | Effective coefficient of restitution | — |
| E | Young modulus | Pa |
| Ef | Flow energy | J |
| f | Inter-particle force | N |
| fc | Unconfined yield stress | Pa |
| ff | Flow-factor | — |
| F | Force | N |
| Fc | Contact force | N |
| FD | Fractal dimensions | — |
| Fg | Gravitational force | N |
| F | Contact force exerted on i by j | N |
| Fn | Normal load | N |
| Fnc | Non-contact force | N |
| Fr | Frounde number | — |
| Fs | Macroscopic shear force | N |
| Ft | Translational force | N |
| g | Gravitational acceleration | m s−2 |
| g0 | Radial distribution function | — |
| Gn | Solid fraction dependent functions in rapid flow regime | — |
| h | Height | m |
| hB | Burke constant | — |
| hc,max | Indentation unloading intercept with depth axis | m |
| hd | Indentation depth | m |
| hf | Final indentation depth | m |
| hsep | Separation distance | m |
| H | Indentation hardness | Pa |
| HR | Haussner ratio | — |
| i | Element under consideration | — |
| I | Moment of inertia/inertial number | kg m2 |
| I0 | Constant for μ(I) rheology | — |
| IA, IB | Modified Bessel functions of the first kind | — |
| k | Permeability | m2 |
| kA | Fraction of collisional dissipation contribution to shear viscosity | — |
| kB | Fraction of frictional contribution to shear viscosity | — |
| kc | Carman constant | — |
| kj | Constant for Johanson hang-up indiciser | ft |
| Ka | Active Rankine coefficient | — |
| Kp | Passive Rankine coefficient | — |
| l | Rod length | m |
| lij | Vector from mass centre of particle i to mass centre of particle j | m |
| ℓp | Unit vector along the major axis | — |
| L | Drum length | m |
| Lp | Perimeter length | m |
| Ly | Dimension of domain in y-direction | m |
| m | Mass | kg |
| M | Contact torque | N m |
| n | Number of particles | — |
| N | Macroscopic normal force | N |
| Nc | Applied force at failure | N |
| O | Order parameter | — |
| p | Particle under consideration | — |
| P | Pressure | Pa |
| Pa | Active pressure | Pa |
| Pp | Passive pressure | Pa |
| PFI | Powder flow index | — |
| PR | Packing ratio | — |
| q | Energy flux | N m−1 s−1 |
| Qij | Symmetric traceless order tensor | Pa |
| r | Asperity size | m |
| rcol | Radius of meniscus curvature | m |
| rro | Surface roughness length | m |
| R | Radius | m |
| R1 | Minor radius of the lid cross-sectional annular area | m |
| R2 | Major radius of the lid cross-sectional annular area | m |
| Rgas | Ideal gas constant | kg m s−2 K−1 |
| RH | Relative humidity | % |
| RI | Ratholing index | — |
| s | Particle surface area | m2 |
| S | Stiffness | N m−1 |
| Seff | Effective projected surface area of a particle | m2 |
| Ssph | Effective projected surface area of the sphere of the same volume to the particle | m2 |
| STD | Standard deviation | — |
| t | Time | s |
| T | Torque | N m |
| Tabs | Temperature | K |
| Tgran | Granular temperature | m2 s−2 |
| U | Velocity at boundary | m s−1 |
| Ucons | Consolidation velocity | m s−1 |
| Umf | Minimum fluidisation velocity | m s−1 |
| Uro | Rain-off velocity | m s−1 |
| Ut | Terminal velocity | m s−1 |
| v | Velocity | m s−1 |
| vax | Axial velocity | m s−1 |
| 〈vp,ave〉 | Average particle velocity | m s−1 |
| vp,fl | Fluctuating velocity | m s−1 |
| vp,i | Velocity of particle i | m s−1 |
| V | Particle volume | m3 |
| Vliq | Volume of liquid per contact | m3 |
| Vm | Molar volume | m3 |
| w | Weight | N |
| W | Mass flow rate | kg s−1 |
| x | Distance between crossbeam holders | m |
| y | Parameter of yield locus curvature | — |
| Y | Yield stress | Pa |
| Z | Height | m |
Greek Characters
| α | Angle of inclination | rad |
| 〈α〉 | Particle angularity | — |
| αd | Dynamic angle of repose | rad |
| αS | Static angle of repose | rad |
| β | Angle between the powder surface and horizontal line | rad |
| βn | Fitting parameter | — |
| γ | Surface tension or energy | J m−2 |
| γ0 | Fitting parameter | — |
| γ°* | Dimensionless shear rate | — |
| | Shear strain rate | s−1 |
| Γ | Energy dissipated per unit volume and unit time | Pa s−1 |
| δ | Filling angle | rad |
| δij | Kronecker delta in the continuum mechanics notation | — |
| ΔP | Pressure drop | Pa |
| Δγ | Work of adhesion | J |
| ϵ | Internal energy per unit volume | J m−3 |
| ɛ | Porosity | — |
| ɛ0 | Permittivity of free space | — |
| ɛr | Dielectric constant of material | — |
| ζ | Dissipation function | — |
| η | Dynamic viscosity | Pa s |
| ηs | Shear viscosity | Pa s |
| ηs,fr | Frictional shear viscosity | Pa s |
| θ | Angle of inclination | rad |
| θwet | Wetting angle | rad |
| κ | Coordination number | — |
| κc | Surface curvature | m−1 |
| λ | Cohesive length | m |
| λσ | Horizontal to vertical stress ratio | — |
| μ | Friction coefficient | — |
| μd | Dynamic friction coefficient | — |
| μ(I) | Friction coefficient under given inertial number | — |
| μk | Kinematic friction coefficient | — |
| μr | Rolling friction coefficient | — |
| μs | Static friction coefficient | — |
| μ2 | Upper limit of the friction coefficient | — |
| ν | Poisson ratio | — |
| ξ | Standard deviation of velocity fluctuations | m2 s−1 |
| ρ | Density | kg m−3 |
| ρa | Air density | kg m−3 |
| ρaer | Aerated density | kg m−3 |
| ρb | Bulk density | kg m−3 |
| ρf | Fluid density | kg m−3 |
| ρp | Particle density | kg m−3 |
| ρpour | Poured density | kg m−3 |
| ρs | Solid (true) density | kg m−3 |
| ρtap | Tapped density | kg m−3 |
| σ | Cauchy stress tensor | Pa |
| σ | Normal stress | Pa |
| σ0 | Fitting parameter | — |
| σ1 | Major principal stress | Pa |
| σ1,b | Maximum consolidation stress at bed bottom | Pa |
| σ2 | Minor principal stress (in 2D) | Pa |
| σf | Dynamic cohesion index | — |
| σkin | Kinematic stress | Pa |
| σm | Normal stress | Pa |
| σm0 | Normal stress at top of powder bed increment | Pa |
| σp | Pre-shear normal stress | Pa |
| σs | Applied normal stress during shearing | Pa |
| σt | Tensile strength | Pa |
| | Centre of consolidation Mohr circle on abscissa | Pa |
 | Stress matrix in plane deformation | Pa |
| τ | Shear stress | Pa |
| τp | Steady state shear stress | Pa |
| τs | Corrected shear stress at incipient flow | Pa |
| τs′ | Shear stress at incipient flow | Pa |
| ϕ | Angle of inclination | rad |
| ϕe | Effective angle of internal friction | rad |
| ϕi | Static angle of internal friction | rad |
| ϕw | Angle of wall friction | rad |
| ϕw′ | Slope of angle of wall yield locus | rad |
| Φ | Angle of repose | rad |
| χ | Solid fraction | — |
| ψ | Sphericity factor | — |
| ω | Rotational velocity | rad s−1 |
| α | Angle of inclination | rad |
| 〈α〉 | Particle angularity | — |
| αd | Dynamic angle of repose | rad |
| αS | Static angle of repose | rad |
| β | Angle between the powder surface and horizontal line | rad |
| βn | Fitting parameter | — |
| γ | Surface tension or energy | J m−2 |
| γ0 | Fitting parameter | — |
| γ°* | Dimensionless shear rate | — |
| | Shear strain rate | s−1 |
| Γ | Energy dissipated per unit volume and unit time | Pa s−1 |
| δ | Filling angle | rad |
| δij | Kronecker delta in the continuum mechanics notation | — |
| ΔP | Pressure drop | Pa |
| Δγ | Work of adhesion | J |
| ϵ | Internal energy per unit volume | J m−3 |
| ɛ | Porosity | — |
| ɛ0 | Permittivity of free space | — |
| ɛr | Dielectric constant of material | — |
| ζ | Dissipation function | — |
| η | Dynamic viscosity | Pa s |
| ηs | Shear viscosity | Pa s |
| ηs,fr | Frictional shear viscosity | Pa s |
| θ | Angle of inclination | rad |
| θwet | Wetting angle | rad |
| κ | Coordination number | — |
| κc | Surface curvature | m−1 |
| λ | Cohesive length | m |
| λσ | Horizontal to vertical stress ratio | — |
| μ | Friction coefficient | — |
| μd | Dynamic friction coefficient | — |
| μ(I) | Friction coefficient under given inertial number | — |
| μk | Kinematic friction coefficient | — |
| μr | Rolling friction coefficient | — |
| μs | Static friction coefficient | — |
| μ2 | Upper limit of the friction coefficient | — |
| ν | Poisson ratio | — |
| ξ | Standard deviation of velocity fluctuations | m2 s−1 |
| ρ | Density | kg m−3 |
| ρa | Air density | kg m−3 |
| ρaer | Aerated density | kg m−3 |
| ρb | Bulk density | kg m−3 |
| ρf | Fluid density | kg m−3 |
| ρp | Particle density | kg m−3 |
| ρpour | Poured density | kg m−3 |
| ρs | Solid (true) density | kg m−3 |
| ρtap | Tapped density | kg m−3 |
| σ | Cauchy stress tensor | Pa |
| σ | Normal stress | Pa |
| σ0 | Fitting parameter | — |
| σ1 | Major principal stress | Pa |
| σ1,b | Maximum consolidation stress at bed bottom | Pa |
| σ2 | Minor principal stress (in 2D) | Pa |
| σf | Dynamic cohesion index | — |
| σkin | Kinematic stress | Pa |
| σm | Normal stress | Pa |
| σm0 | Normal stress at top of powder bed increment | Pa |
| σp | Pre-shear normal stress | Pa |
| σs | Applied normal stress during shearing | Pa |
| σt | Tensile strength | Pa |
| | Centre of consolidation Mohr circle on abscissa | Pa |
| | Stress matrix in plane deformation | Pa |
| τ | Shear stress | Pa |
| τp | Steady state shear stress | Pa |
| τs | Corrected shear stress at incipient flow | Pa |
| τs′ | Shear stress at incipient flow | Pa |
| ϕ | Angle of inclination | rad |
| ϕe | Effective angle of internal friction | rad |
| ϕi | Static angle of internal friction | rad |
| ϕw | Angle of wall friction | rad |
| ϕw′ | Slope of angle of wall yield locus | rad |
| Φ | Angle of repose | rad |
| χ | Solid fraction | — |
| ψ | Sphericity factor | — |
| ω | Rotational velocity | rad s−1 |
Subscripts
| 10 | 10th percentile | — |
| 50 | 50th percentile | — |
| 90 | 90th percentile | — |
| ave. | Average | — |
| b | Bed | — |
| col | Collisional | — |
| d | Drag force or of equivalent drag force | — |
| eff | Effective | — |
| end | At the end of the test | — |
| eq | Equivalent | — |
| fl | Fluctuating | — |
| i | Case under consideration | — |
| kin | Kinetic | — |
| max | Maximum | — |
| min | Minimum | — |
| plug | Of plug | — |
| s | Of equivalent surface area | — |
| sat | Saturation | — |
| sv | Of equivalent specific surface area | — |
| tv | Of equivalent terminal velocity | — |
| v | Of equivalent volume | — |
| w | Wall | — |
| x | x component | — |
| y | y component | — |
| Ω | Of equivalent projected surface area | — |
| 10 | 10th percentile | — |
| 50 | 50th percentile | — |
| 90 | 90th percentile | — |
| ave. | Average | — |
| b | Bed | — |
| col | Collisional | — |
| d | Drag force or of equivalent drag force | — |
| eff | Effective | — |
| end | At the end of the test | — |
| eq | Equivalent | — |
| fl | Fluctuating | — |
| i | Case under consideration | — |
| kin | Kinetic | — |
| max | Maximum | — |
| min | Minimum | — |
| plug | Of plug | — |
| s | Of equivalent surface area | — |
| sat | Saturation | — |
| sv | Of equivalent specific surface area | — |
| tv | Of equivalent terminal velocity | — |
| v | Of equivalent volume | — |
| w | Wall | — |
| x | x component | — |
| y | y component | — |
| Ω | Of equivalent projected surface area | — |
