Dimensionless quantity

quantity with no physical dimension

In dimensional analysis, a dimensionless quantity (or more precisely, a quantity with the dimensions of 1) is a quantity without any physical units and thus a pure number. Such a number is typically defined as a product or ratio of quantities which do have units, in such a way that all the units cancel out.



"out of every 10 apples I gather, 1 is rotten." -- the rotten-to-gathered ratio is (1 apple) / (10 apples) = 0.1 = 10%, which is a dimensionless quantity.

List of dimensionless quantities


There are infinitely many dimensionless quantities and they are often called numbers. Some of those that are used most often have been given names, as in the following list of examples (alphabetical order):

Name Field of application
Abbe number optics (dispersion in optical materials)
Albedo climatology, astronomy (reflectivity of surfaces or bodies)
Archimedes number motion of fluids due to density differences
Bagnold number flow of grain, sand, etc. [1] Archived 2005-05-10 at the Wayback Machine
Biot number surface vs. volume conductivity of solids
Bodenstein number residence-time distribution
Bond number capillary action driven by buoyancy [2] Archived 2012-03-05 at the Wayback Machine
Brinkman number heat transfer by conduction from the wall to a viscous fluid
Brownell Katz number combination of capillary number and Bond number
Capillary number fluid flow influenced by surface tension
Coefficient of static friction friction of solid bodies at rest
Coefficient of kinetic friction friction of solid bodies in translational motion
Colburn j factor dimensionless heat transfer coefficient
Courant-Friedrich-Levy number numerical solutions of hyperbolic PDEs[3] Archived 2008-06-05 at the Wayback Machine
Damköhler numbers reaction time scales vs. transport phenomena
Darcy friction factor fluid flow
Dean number vortices in curved ducts
Deborah number rheology of viscoelastic fluids
Decibel ratio of two intensities of sound
Drag coefficient flow resistance
e mathematics
Eckert number convective heat transfer
Ekman number geophysics (frictional (viscous) forces)
Elasticity (economics) widely used to measure how demand or supply responds to price changes
Eötvös number determination of bubble/drop shape
Euler number hydrodynamics (pressure forces vs. inertia forces)
Fanning friction factor fluid flow in pipes [4] Archived 2013-12-20 at the Wayback Machine
Feigenbaum constants chaos theory (period doubling) [5]
Fine structure constant quantum electrodynamics (QED)
Foppl–von Karman number thin-shell buckling
Fourier number heat transfer
Fresnel number slit diffraction [6]
Froude number wave and surface behaviour
Gain electronics (signal output to signal input)
Galilei number gravity-driven viscous flow
Graetz number heat flow
Grashof number free convection
Hatta number adsorption enhancement due to chemical reaction
Hagen number forced convection
Karlovitz number turbulent combustion
Knudsen number continuum approximation in fluids
Kt/V medicine
Laplace number free convection within immiscible fluids
Lewis number ratio of mass diffusivity and thermal diffusivity
Lockhart-Martinelli parameter flow of wet gases [7] Archived 2009-11-15 at the Wayback Machine
Lift coefficient lift available from an airfoil at a given angle of attack
Mach number gas dynamics
Magnetic Reynolds number magnetohydrodynamics
Manning roughness coefficient open channel flow (flow driven by gravity) [8]PDF (109 KiB)
Marangoni number Marangoni flow due to thermal surface tension deviations
Morton number determination of bubble/drop shape
Nusselt number heat transfer with forced convection
Ohnesorge number atomization of liquids, Marangoni flow
Péclet number advectiondiffusion problems
Peel number adhesion of microstructures with substrate [9] Archived 2005-10-26 at the Wayback Machine
Pi mathematics (ratio of a circle's circumference to its diameter)
Poisson's ratio elasticity (load in transverse and longitudinal direction)
Power factor electronics (real power to apparent power)
Power number power consumption by agitators
Prandtl number forced and free convection
Pressure coefficient pressure experienced at a point on an airfoil
Radian measurement of angles
Rayleigh number buoyancy and viscous forces in free convection
Refractive index electromagnetism, optics
Reynolds number flow behavior (inertia vs. viscosity)
Richardson number effect of buoyancy on flow stability [10] Archived 2015-03-02 at the Wayback Machine
Rockwell scale mechanical hardness
Rossby number inertial forces in geophysics
Schmidt number fluid dynamics (mass transfer and diffusion) [11] Archived 2010-01-24 at the Wayback Machine
Sherwood number mass transfer with forced convection
Sommerfeld number boundary lubrication [12] Archived 2016-03-16 at the Wayback Machine
Stanton number heat transfer in forced convection
Stefan number heat transfer during phase change
Stokes number particle dynamics
Strain materials science, elasticity
Strouhal number continuous and pulsating flow [13] Archived 2009-03-25 at the Wayback Machine
Taylor number rotating fluid flows
van 't Hoff factor quantitative analysis (Kf and Kb)
Weaver flame speed number laminar burning velocity relative to hydrogen gas [14] Archived 2017-11-03 at the Wayback Machine
Weber number multiphase flow with strongly curved surfaces
Weissenberg number viscoelastic flows [15] Archived 2006-11-01 at the Wayback Machine
Womersley number continuous and pulsating flows [16] Archived 2009-03-25 at the Wayback Machine

Other websites