Dimensionless quantity

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.

Example

"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	advection–diffusion 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 (K_f and K_b)
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

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