The Archimedes number is a dimensionless quantity used in fluid mechanics to measure the relative importance of buoyancy (gravitational) forces compared with viscous resistance in a flow where density differences drive motion. The name honors the ancient Greek scientist Archimedes because the number quantifies the same buoyant effects underlying his principle. It appears most often in studies of particulate suspensions, sedimentation, bubble rise, and fluidization in viscous fluids.
Definition and formula
One common form of the Archimedes number, Ar, is written for a solid particle (or bubble) of characteristic length L in a surrounding fluid as Ar = g L^3 ρ_ℓ (ρ - ρ_ℓ) / μ^2, where g is gravitational acceleration, ρ is the density of the particle or dispersed phase, ρ_ℓ is the density of the continuous liquid, and μ is the dynamic viscosity of the liquid. This expression gives a nondimensional measure of buoyancy-driven forcing relative to viscous damping. Equivalent algebraic forms use the kinematic viscosity ν = μ/ρ_ℓ and yield alternative but consistent representations depending on which properties are treated as primary.
Physical meaning and uses
The Archimedes number indicates whether motion induced by density differences will be strongly resisted by viscosity. Small Ar values correspond to creeping, viscous-dominated motion with steady, laminar behavior; large Ar values imply buoyancy dominates and can produce wake formation, unsteady motion, or turbulence around particles or bubbles. Engineers and scientists use Ar to scale experiments, predict settling velocities, design fluidized beds, and analyze bubble and droplet dynamics in multiphase flows. It therefore plays a role in civil, chemical, and environmental engineering as well as in basic fluid dynamics research.
Relation to other dimensionless numbers
The Archimedes number is related to, but distinct from, other nondimensional groups. It is analogous to the Grashof number, which compares buoyancy and viscous forces in free convection, and it is often discussed alongside the Reynolds number and Galileo number when characterizing particle motion. Because definitions can vary by subfield and convention, users should check the precise formulation applied in a given study or handbook. For background on the role of material properties in these ratios, see general discussions of fluids and density-driven flows.
History and notable facts
The term derives from the conceptual link to Archimedes’ buoyancy principle rather than from a single historical derivation in modern fluid mechanics. It became a routine nondimensional parameter in twentieth-century studies of multiphase and particulate flows as researchers sought simple similarity criteria for laboratory models and engineering correlations. Modern texts treat Ar as part of the standard toolbox when describing buoyancy-versus-viscosity regimes.
Practical considerations and examples
In practice, the Archimedes number helps predict qualitative behavior: for example, a heavy solid particle in a viscous liquid with small L will typically have low Ar and descend slowly with a smooth wake, whereas a larger particle or a bubble with higher buoyant contrast yields larger Ar and may exhibit oscillations or vortex shedding. Because material properties (densities, viscosity) and the choice of length scale all enter the expression, the same physical situation can be represented by different numerical Ar values if definitions differ—so consistency is essential when comparing results or using empirical correlations.
