List of dimensionless quantities

This is a list of well-known dimensionless quantities illustrating their variety of forms and applications. The table also includes pure numbers, dimensionless ratios, or dimensionless physical constants; these topics are discussed in the article.

Name Standard symbol Definition Field of application
Abbe number V optics (dispersion in optical materials)
Activity coefficient chemistry (Proportion of "active" molecules or atoms)
Albedo climatology, astronomy (reflectivity of surfaces or bodies)
Archimedes number Ar fluid mechanics (motion of fluids due to density differences)
Arrhenius number chemistry (ratio of activation energy to thermal energy)[1]
Atomic weight M chemistry (mass of atom over one atomic mass unit, u, where carbon-12 is exactly 12 u)
Atwood number A fluid mechanics (onset of instabilities in fluid mixtures due to density differences)
Bagnold number Ba fluid mechanics, geology (ratio of grain collision stresses to viscous fluid stresses in flow of a granular material such as grain and sand)[2]
Basic reproduction number number of infections caused on average by an infectious individual over entire infectious period epidemiology
Bejan number
(fluid mechanics)
Be fluid mechanics (dimensionless pressure drop along a channel)[3]
Bejan number
Be thermodynamics (ratio of heat transfer irreversibility to total irreversibility due to heat transfer and fluid friction)[4]
Bingham number Bm fluid mechanics, rheology (ratio of yield stress to viscous stress)[1]
Biot number Bi heat transfer (surface vs. volume conductivity of solids)
Blake number Bl or B geology, fluid mechanics, porous media (inertial over viscous forces in fluid flow through porous media)
Blondeau number sport science, team sports[5]
Bodenstein number Bo or Bd chemistry (residence-time distribution; similar to the axial mass transfer Peclet number)[6]
Bond number Bo geology, fluid mechanics, porous media (buoyant versus capillary forces, similar to the Eötvös number) [7]
Brinkman number Br heat transfer, fluid mechanics (conduction from a wall to a viscous fluid)
Brownell–Katz number NBK fluid mechanics (combination of capillary number and Bond number) [8]
Capillary number Ca porous media, fluid mechanics (viscous forces versus surface tension)
Chandrasekhar number Q magnetohydrodynamics (ratio of the Lorentz force to the viscosity in magnetic convection)
Colburn J factors JM, JH, JD turbulence; heat, mass, and momentum transfer (dimensionless transfer coefficients)
Coefficient of kinetic friction mechanics (friction of solid bodies in translational motion)
Coefficient of static friction mechanics (friction of solid bodies at rest)
Coefficient of determination statistics (proportion of variance explained by a statistical model)
Coefficient of variation statistics (ratio of standard deviation to expectation)
Cohesion number Coh Chemical engineering, material science, mechanics (A scale to show the energy needed for detaching two solid particles)[9][10]
Correlation ρ or r statistics (measure of linear dependence)
Cost of transport COT energy efficiency, economics (ratio of energy input to kinetic motion)
Courant–Friedrich–Levy number C or 𝜈 mathematics (numerical solutions of hyperbolic PDEs)[11]
Damkohler number Da chemistry (reaction time scales vs. residence time)
Damping ratio mechanics (the level of damping in a system)
Darcy friction factor Cf or fD fluid mechanics (fraction of pressure losses due to friction in a pipe; four times the Fanning friction factor)
Darcy number Da porous media (ratio of permeability to cross-sectional area)
Dean number D turbulent flow (vortices in curved ducts)
Deborah number De rheology (viscoelastic fluids)
Decibel dB acoustics, electronics, control theory (ratio of two intensities or powers of a wave)
Drag coefficient cd aeronautics, fluid dynamics (resistance to fluid motion)
Dukhin number Du colloid science (ratio of electric surface conductivity to the electric bulk conductivity in heterogeneous systems)
Eckert number Ec convective heat transfer (characterizes dissipation of energy; ratio of kinetic energy to enthalpy)
Ekman number Ek geophysics (viscous versus Coriolis forces)
E economics (response of demand or supply to price changes)
Eötvös number Eo fluid mechanics (shape of bubbles or drops)
Ericksen number Er fluid dynamics (liquid crystal flow behavior; viscous over elastic forces)
Euler number Eu hydrodynamics (stream pressure versus inertia forces)
Euler's number e mathematics (base of the natural logarithm)
Excess temperature coefficient heat transfer, fluid dynamics (change in internal energy versus kinetic energy)[12]
Fanning friction factor f fluid mechanics (fraction of pressure losses due to friction in a pipe; 1/4th the Darcy friction factor)[13]
Feigenbaum constants ,
chaos theory (period doubling)[14]
Fine-structure constant quantum electrodynamics (QED) (coupling constant characterizing the strength of the electromagnetic interaction)
f-number f optics, photography (ratio of focal length to diameter of aperture)
Föppl–von Kármán number virology, solid mechanics (thin-shell buckling)
Fourier number Fo heat transfer, mass transfer (ratio of diffusive rate versus storage rate)
Fresnel number F optics (slit diffraction)[15]
Froude number Fr fluid mechanics (wave and surface behaviour; ratio of a body's inertia to gravitational forces)
Gain electronics (signal output to signal input)
Gain ratio bicycling (system of representing gearing; length traveled over length pedaled)[16]
Galilei number Ga fluid mechanics (gravitational over viscous forces)
Golden ratio mathematics, aesthetics (long side length of self-similar rectangle)
Görtler number G fluid dynamics (boundary layer flow along a concave wall)
Graetz number Gz heat transfer, fluid mechanics (laminar flow through a conduit; also used in mass transfer)
Grashof number Gr heat transfer, natural convection (ratio of the buoyancy to viscous force)
Hatta number Ha chemical engineering (adsorption enhancement due to chemical reaction)
Hagen number Hg heat transfer (ratio of the buoyancy to viscous force in forced convection)
Havnes parameter In Dusty plasma physics, ratio of the total charge carried by the dust particles to the charge carried by the ions , with the number density of particles
Helmholtz number The most important parameter in duct acoustics. If is the dimensional frequency, then is the corresponding free field wavenumber and is the corresponding dimensionless frequency [17]
Hydraulic gradient i fluid mechanics, groundwater flow (pressure head over distance)
Iribarren number Ir wave mechanics (breaking surface gravity waves on a slope)
Jakob number Ja chemistry (ratio of sensible to latent energy absorbed during liquid-vapor phase change)[18]
Karlovitz number Ka turbulent combustion (characteristic chemical time scale to Kolmogorov time scale)
Keulegan–Carpenter number KC fluid dynamics (ratio of drag force to inertia for a bluff object in oscillatory fluid flow)
Knudsen number Kn gas dynamics (ratio of the molecular mean free path length to a representative physical length scale)
Kt/V Kt/V medicine (hemodialysis and peritoneal dialysis treatment; dimensionless time)
Kutateladze number Ku fluid mechanics (counter-current two-phase flow)[19]
Laplace number La fluid dynamics (free convection within immiscible fluids; ratio of surface tension to momentum-transport)
Lewis number Le heat and mass transfer (ratio of thermal to mass diffusivity)
Lift coefficient CL aerodynamics (lift available from an airfoil at a given angle of attack)
Lockhart–Martinelli parameter two-phase flow (flow of wet gases; liquid fraction)[20]
Love numbers h, k, l geophysics (solidity of earth and other planets)
Lundquist number S plasma physics (ratio of a resistive time to an Alfvén wave crossing time in a plasma)
Mach number M or Ma gas dynamics (compressible flow; dimensionless velocity)
Magnetic Reynolds number Rm magnetohydrodynamics (ratio of magnetic advection to magnetic diffusion)
Manning roughness coefficient n open channel flow (flow driven by gravity)[21]
Marangoni number Mg fluid mechanics (Marangoni flow; thermal surface tension forces over viscous forces)
Markstein number fluid dynamics, combustion (turbulent combustion flames)
Morton number Mo fluid dynamics (determination of bubble/drop shape)
Nusselt number Nu heat transfer (forced convection; ratio of convective to conductive heat transfer)
Ohnesorge number Oh fluid dynamics (atomization of liquids, Marangoni flow)
Péclet number Pe heat transfer (advectiondiffusion problems; total momentum transfer to molecular heat transfer)
Peel number NP coating (adhesion of microstructures with substrate)[22]
Perveance K charged particle transport (measure of the strength of space charge in a charged particle beam)
pH chemistry (the measure of the acidity or basicity of an aqueous solution)
Pi mathematics (ratio of a circle's circumference to its diameter)
Pierce parameter Traveling wave tube
Pixel px digital imaging (smallest addressable unit)
Beta (plasma physics) Plasma (physics) and Fusion power. Ratio of plasma thermal pressure to magnetic pressure, controlling the level of turbulence in a magnetised plasma.
Poisson's ratio elasticity (strain in transverse and longitudinal direction)
Porosity geology, porous media (void fraction of the medium)
Power factor pf electrical (real power to apparent power)
Power number Np electronics (power consumption by agitators; resistance force versus inertia force)
Prandtl number Pr heat transfer (ratio of viscous diffusion rate over thermal diffusion rate)
Prater number β reaction engineering (ratio of heat evolution to heat conduction within a catalyst pellet)[23]
Pressure coefficient CP aerodynamics, hydrodynamics (pressure experienced at a point on an airfoil; dimensionless pressure variable)
Q factor Q physics, engineering (damping of oscillator or resonator; energy stored versus energy lost)
Radian measure rad mathematics (measurement of planar angles, 1 radian = 180/π degrees)
Rayleigh number Ra heat transfer (buoyancy versus viscous forces in free convection)
Refractive index n electromagnetism, optics (speed of light in a vacuum over speed of light in a material)
Relative density RD hydrometers, material comparisons (ratio of density of a material to a reference material—usually water)
Relative permeability magnetostatics (ratio of the permeability of a specific medium to free space)
Relative permittivity electrostatics (ratio of capacitance of test capacitor with dielectric material versus vacuum)
Reynolds number Re fluid mechanics (ratio of fluid inertial and viscous forces)[1]
Richardson number Ri fluid dynamics (effect of buoyancy on flow stability; ratio of potential over kinetic energy)[24]
Rockwell scale mechanical hardness (indentation hardness of a material)
Rolling resistance coefficient Crr vehicle dynamics (ratio of force needed for motion of a wheel over the normal force)
Roshko number Ro fluid dynamics (oscillating flow, vortex shedding)
Rossby number Ro geophysics (ratio of inertial to Coriolis force)
Rouse number P or Z sediment transport (ratio of the sediment fall velocity and the upwards velocity of grain)
Schmidt number Sc mass transfer (viscous over molecular diffusion rate)[25]
Shape factor H boundary layer flow (ratio of displacement thickness to momentum thickness)
Sherwood number Sh mass transfer (forced convection; ratio of convective to diffusive mass transport)
Shields parameter or sediment transport (threshold of sediment movement due to fluid motion; dimensionless shear stress)
Sommerfeld number S hydrodynamic lubrication (boundary lubrication)[26]
Specific gravity SG (same as Relative density)
Stanton number St heat transfer and fluid dynamics (forced convection)
Stefan number Ste phase change, thermodynamics (ratio of sensible heat to latent heat)
Stokes number Stk or Sk particles suspensions (ratio of characteristic time of particle to time of flow)
Strain materials science, elasticity (displacement between particles in the body relative to a reference length)
Strouhal number St or Sr fluid dynamics (continuous and pulsating flow; nondimensional frequency)[27]
Stuart number N magnetohydrodynamics (ratio of electromagnetic to inertial forces)
Taylor number Ta fluid dynamics (rotating fluid flows; inertial forces due to rotation of a fluid versus viscous forces)
Transmittance T optics, spectroscopy (the ratio of the intensities of radiation exiting through and incident on a sample)
Ursell number U wave mechanics (nonlinearity of surface gravity waves on a shallow fluid layer)
Vadasz number Va porous media (governs the effects of porosity , the Prandtl number and the Darcy number on flow in a porous medium) [28]
van 't Hoff factor i quantitative analysis (Kf and Kb)
Wallis parameter j* multiphase flows (nondimensional superficial velocity)[29]
Wagner number Wa electrochemistry (ratio of kinetic polarization resistance to solution ohmic resistance in an electrochemical cell)[30]
Weaver flame speed number Wea combustion (laminar burning velocity relative to hydrogen gas)[31]
Weber number We multiphase flow (strongly curved surfaces; ratio of inertia to surface tension)
Weissenberg number Wi viscoelastic flows (shear rate times the relaxation time)[32]
Womersley number biofluid mechanics (continuous and pulsating flows; ratio of pulsatile flow frequency to viscous effects)[33]
Zel'dovich number fluid dynamics, Combustion (Measure of activation energy)


  1. ^ a b c "Table of Dimensionless Numbers" (PDF). Retrieved 2009-11-05.
  2. ^ Bagnold number Archived 2005-05-10 at the Wayback Machine
  3. ^ Bhattacharjee S.; Grosshandler W.L. (1988). "The formation of wall jet near a high temperature wall under microgravity environment". ASME MTD. 96: 711–6. Bibcode:1988nht.....1..711B.
  4. ^ Paoletti S.; Rispoli F.; Sciubba E. (1989). "Calculation of exergetic losses in compact heat exchanger passager". ASME AES. 10 (2): 21–9.
  5. ^ Blondeau, J. "The influence of field size, goal size and number of players on the average number of goals scored per game in variants of football and hockey: the Pi-theorem applied to team sports". Journal of Quantitative Analysis in sports.
  6. ^ Becker, A.; Hüttinger, K. J. (1998). "Chemistry and kinetics of chemical vapor deposition of pyrocarbon—II pyrocarbon deposition from ethylene, acetylene and 1,3-butadiene in the low temperature regime". Carbon. 36 (3): 177. doi:10.1016/S0008-6223(97)00175-9.
  7. ^ Bond number Archived 2012-03-05 at the Wayback Machine
  8. ^ "Home". OnePetro. 2015-05-04. Retrieved 2015-05-08.
  9. ^ Behjani, Mohammadreza Alizadeh; Rahmanian, Nejat; Ghani, Nur Fardina bt Abdul; Hassanpour, Ali (2017). "An investigation on process of seeded granulation in a continuous drum granulator using DEM" (PDF). Advanced Powder Technology. 28 (10): 2456–2464. doi:10.1016/j.apt.2017.02.011.
  10. ^ Alizadeh Behjani, Mohammadreza; Hassanpour, Ali; Ghadiri, Mojtaba; Bayly, Andrew (2017). "Numerical Analysis of the Effect of Particle Shape and Adhesion on the Segregation of Powder Mixtures". EPJ Web of Conferences. 140: 06024. Bibcode:2017EPJWC.14006024A. doi:10.1051/epjconf/201714006024. ISSN 2100-014X.
  11. ^ Courant–Friedrich–Levy number Archived 2008-06-05 at the Wayback Machine
  12. ^ Schetz, Joseph A. (1993). Boundary Layer Analysis. Englewood Cliffs, NJ: Prentice-Hall, Inc. pp. 132–134. ISBN 0-13-086885-X.
  13. ^ "Fanning friction factor". Archived from the original on 2013-12-20. Retrieved 2015-10-07.
  14. ^ Feigenbaum constants
  15. ^ Fresnel number Archived 2011-10-01 at the Wayback Machine
  16. ^ Gain Ratio – Sheldon Brown
  17. ^ S.W. RIENSTRA, 2015, Fundamentals of Duct Acoustics, Von Karman Institute Lecture Notes
  18. ^ Incropera, Frank P. (2007). Fundamentals of heat and mass transfer. John Wiley & Sons, Inc. p. 376.
  19. ^ Tan, R. B. H.; Sundar, R. (2001). "On the froth–spray transition at multiple orifices". Chemical Engineering Science. 56 (21–22): 6337. doi:10.1016/S0009-2509(01)00247-0.
  20. ^ Lockhart–Martinelli parameter
  21. ^ "Manning coefficient" (PDF). (109 KB)
  22. ^ Van Spengen, W. M.; Puers, R.; De Wolf, I. (2003). "The prediction of stiction failures in MEMS". IEEE Transactions on Device and Materials Reliability. 3 (4): 167. doi:10.1109/TDMR.2003.820295.
  23. ^ Davis, Mark E.; Davis, Robert J. (2012). Fundamentals of Chemical Reaction Engineering. Dover. p. 215. ISBN 978-0-486-48855-4.
  24. ^ Richardson number Archived 2015-03-02 at the Wayback Machine
  25. ^ Schmidt number Archived 2010-01-24 at the Wayback Machine
  26. ^ Sommerfeld number
  27. ^ Strouhal number, Engineering Toolbox
  28. ^ Straughan, B. (2001). "A sharp nonlinear stability threshold in rotating porous convection". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 457 (2005): 87–88. Bibcode:2001RSPSA.457...87S. doi:10.1098/rspa.2000.0657.
  29. ^ Petritsch, G.; Mewes, D. (1999). "Experimental investigations of the flow patterns in the hot leg of a pressurized water reactor". Nuclear Engineering and Design. 188: 75–84. doi:10.1016/S0029-5493(99)00005-9.
  30. ^ Popov, Konstantin I.; Djokić, Stojan S.; Grgur, Branimir N. (2002). Fundamental Aspects of Electrometallurgy. Boston, MA: Springer. pp. 101–102. ISBN 978-0-306-47564-1.
  31. ^ Kuneš, J. (2012). "Technology and Mechanical Engineering". Dimensionless Physical Quantities in Science and Engineering. pp. 353–390. doi:10.1016/B978-0-12-416013-2.00008-7. ISBN 978-0-12-416013-2.
  32. ^ Weissenberg number Archived 2006-11-01 at the Wayback Machine
  33. ^ Womersley number Archived 2009-03-25 at the Wayback Machine