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.   Biot number                surface vs. volume conductivity of solids Bodenstein number          residence-time distribution Bond number                capillary action driven by buoyancy   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  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   Feigenbaum constants       chaos theory (period doubling)  Fine structure constant    quantum electrodynamics (QED) Foppl–von Karman number    thin-shell buckling Fourier number             heat transfer Fresnel number             slit diffraction  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   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)   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   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   Rockwell scale             mechanical hardness Rossby number              inertial forces in geophysics Schmidt number             fluid dynamics (mass transfer and diffusion)   Sherwood number            mass transfer with forced convection Sommerfeld number          boundary lubrication    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   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   Weber number               multiphase flow with strongly curved surfaces Weissenberg number         viscoelastic flows   Womersley number           continuous and pulsating flows  

 Other websites 
 Biographies of 16 scientists with dimensionless numbers of heat and mass transfer named after them 
 How Many Fundamental Constants Are There? by John Baez
 Michael Sheppard, Systematic Search for Expressions of Dimensionless Constants using the NIST database of Physical Constants, 2007

Category:Mathematics