Granular rheology · Suspensions · Soft matter

Dimensionless Numbers for Particle Systems

Calculate and compare dimensionless groups for granular flows, immersed particles, sedimentation, capillary phenomena, and viscoelastic materials.

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Step 1

Physical quantities

Example values are provided. Replace them with your conditions; optional fields may be left blank.

Geometry and motion

Particle diameter, d

Characteristic length, L

Characteristic velocity, U

Particle–fluid relative velocity, U_r

Shear rate, γ̇ s⁻¹

Characteristic observation time, T used for De = λ/T

Gravitational acceleration, g m s⁻²

Particle properties

Particle density, ρ_p kg m⁻³

Cohesive force, F_c optional

Young’s modulus, E optional

Normal contact stiffness, k_n optional

Fluid and interface

Fluid density, ρ_f kg m⁻³

Dynamic viscosity, η_f

Surface tension, σ

Yield stress, τ_y optional

Stress and relaxation

Confining pressure, P

Shear stress, τ

Relaxation time, λ optional

Diffusion coefficient, D optional

Step 2

Dimensionless numbers

Calculating…

Definitions and conventions

Choose scales deliberately

Use particle diameter for particle-scale effects, droplet diameter for deformation, or layer depth for free-surface flow. The physical meaning changes with L and U.

Stokes number convention

This page uses the shear-rheology form St = ρ_pd²γ̇/η_f, without the 1/18 relaxation-time prefactor.

Weissenberg and Deborah

Wi uses shear time, whereas De compares relaxation time with the supplied characteristic observation time.

No universal regime boundary

Large and small values indicate competing scales, but transition values depend on geometry, material model, and the definition used.

Definitions follow commonly used forms in granular and suspension rheology, interfacial flow, and soft matter. See Jop, Forterre & Pouliquen for inertial-number rheology, Boyer, Guazzelli & Pouliquen for the viscous number, the JFM Galileo-number definition, and the JFM Weissenberg-number definition. Always compare the displayed convention with the source used in your work.