List of dimensionless quantities explained

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

Biology and medicine

Name	Standard symbol	Definition	Field of application
	R _{0}	number of infections caused on average by an infectious individual over entire infectious period
		total mass of fat divided by total body mass, multiplied by 100	biology
Kt/V	Kt/V		medicine (hemodialysis and peritoneal dialysis treatment; dimensionless time)
		waist circumference divided by hip circumference	biology
		waist circumference divided by chest circumference	biology
		waist circumference divided by height	biology

Chemistry

Name	Standard symbol	Definition	Field of application
	\gamma	\gamma = ({a)/() }	chemistry (Proportion of "active" molecules or atoms)
	\alpha	\alpha = (E _{a})/(RT)	chemistry (ratio of activation energy to thermal energy)
	M		chemistry (mass of one atom divided by the atomic mass constant,)
	Bo or Bd	Bo = vL/ l{D} = Re Sc	chemistry (residence-time distribution; similar to the axial mass transfer Peclet number)^[1]
	Da	Da = k \tau	chemistry (reaction time scales vs. residence time)
	Ha	Ha = (N _{A0})/(N ^{phys} _{A 0})	chemical engineering (adsorption enhancement due to chemical reaction)
	Ja	Ja = (c _{p} (T _{s} - T _{sat)})/(\Delta H _{f})	chemistry (ratio of sensible to latent energy absorbed during liquid-vapor phase change)^[2]
pH	pH	pH = - log _{10} (a ^{+ })} _{rm{H}	chemistry (the measure of the acidity or basicity of an aqueous solution)
	i	i = 1 + \alpha (n - 1 )	quantitative analysis (Kf and Kb)
	Wa	Wa = (\kappa)/(l) (d η)/(d i)	electrochemistry (ratio of kinetic polarization resistance to solution ohmic resistance in an electrochemical cell)^[3]
	Wea	Wea = (w)/(w _{H}) 100	combustion (laminar burning velocity relative to hydrogen gas)^[4]

Physics

Fluids and heat transfer

See main article: Dimensionless numbers in fluid mechanics.

Name	Standard symbol	Definition	Field of application
	Ar	Ar = (g L ^{3} \rho _{\ell} (\rho - \rho _{\ell)})/(\mu ^{2})	fluid mechanics (motion of fluids due to density differences)
	As	As = (W)/(\alpha \rho d _{p} H)	heat transfer (ratio of heat generation of microwave dielectric heating to thermal diffusion^[5]
	A	A = (\rho _{1} - \rho _{2})/(\rho _{1} + \rho _{2})	fluid mechanics (onset of instabilities in fluid mixtures due to density differences)
	Ba	Ba = (\rho d ^{2} λ ^{1/2} • \gamma)/(\mu)	fluid mechanics, geology (ratio of grain collision stresses to viscous fluid stresses in flow of a granular material such as grain and sand)^[6]
Bejan number (fluid mechanics)	Be	Be = (\Delta P L ^{2})/(\mu \alpha)	fluid mechanics (dimensionless pressure drop along a channel)^[7]
Bejan number (thermodynamics)	Be	Be = (• S' _{gen,})/(• S' _{gen,} + • S' _{gen,})	thermodynamics (ratio of heat transfer irreversibility to total irreversibility due to heat transfer and fluid friction)^[8]
	Bm	Bm = (\tau _{y} L)/(\mu V)	fluid mechanics, rheology (ratio of yield stress to viscous stress)
	Bi	Bi = (h L _{C})/(k _{b})	heat transfer (surface vs. volume conductivity of solids)
	Bl or B	B = (u \rho)/(\mu ( 1- \epsilon) D)	geology, fluid mechanics, porous media (inertial over viscous forces in fluid flow through porous media)
	Bo	Bo = (\rho a L ^{2})/(\gamma)	geology, fluid mechanics, porous media (buoyant versus capillary forces, similar to the Eötvös number) ^[9]
	Br	Br = (\mu U ^{2})/(\kappa (T _{w} - T _{0)})	heat transfer, fluid mechanics (conduction from a wall to a viscous fluid)
	NBK	N _{BK} = (u \mu)/(k _{rw\sigma})	fluid mechanics (combination of capillary number and Bond number) ^[10]
	Ca	Ca = (\mu V)/(\gamma)	porous media, fluid mechanics (viscous forces versus surface tension)
	Q	Q = ({B _{0})/(^{2}) d ^{2 }{\mu} _{0} \rho \nu λ }	magnetohydrodynamics (ratio of the Lorentz force to the viscosity in magnetic convection)
	JM, JH, JD		turbulence heat, mass, and momentum transfer (dimensionless transfer coefficients)
	Cf or fD		fluid mechanics (fraction of pressure losses due to friction in a pipe; four times the Fanning friction factor)
	D	D = (\rho V d)/(\mu) \left( (d)/(2 R) \right) ^{1/2}	turbulent flow (vortices in curved ducts)
	De	De = (t _{c})/(t _{p})	rheology (viscoelastic fluids)
	cd	c _{d} = \dfrac{ 2 F _{d} }\,,	aeronautics, fluid dynamics (resistance to fluid motion)
	Ec	Ec = (V ^{2})/(c _{p\Delta} T)	convective heat transfer (characterizes dissipation of energy; ratio of kinetic energy to enthalpy)
	Ek	Ek = (\nu)/(2 D ^{2\Omega\sin\varphi})	geophysics (viscous versus Coriolis forces)
	Eo	Eo = (\Delta\rho g L ^{2})/(\sigma)	fluid mechanics (shape of bubbles or drops)
	Er	Er = (\mu v L)/(K)	fluid dynamics (liquid crystal flow behavior; viscous over elastic forces)
	Eu	Eu = (\Delta{)/(p}{\rho) V ^{2}}	hydrodynamics (stream pressure versus inertia forces)
	\Theta _{r}	\Theta _{r} = (c _{p} (T - T _{e)})/(U ^{2 / 2} _{e})	heat transfer, fluid dynamics (change in internal energy versus kinetic energy)^[11]
	f		fluid mechanics (fraction of pressure losses due to friction in a pipe; 1/4th the Darcy friction factor)^[12]
	Fo	Fo = (\alpha t)/(L ^{2})	heat transfer, mass transfer (ratio of diffusive rate versus storage rate)
	Fr	Fr = (v)/(\sqrt{g\ell) }	fluid mechanics (wave and surface behaviour; ratio of a body's inertia to gravitational forces)
	Ga	Ga = (g L ^{3})/(\nu ^{2})	fluid mechanics (gravitational over viscous forces)
	G	G = (U _{e} \theta)/(\nu) \left( (\theta)/(R) \right) ^{1/2}	fluid dynamics (boundary layer flow along a concave wall)
	Gz	Gz = {D _{H} \over L} Re Pr	heat transfer, fluid mechanics (laminar flow through a conduit; also used in mass transfer)
	Gr	Gr _{L} = (g \beta (T _{s} - T _{infty} ) L ^{3})/(\nu ^{2})	heat transfer, natural convection (ratio of the buoyancy to viscous force)
	Hg	Hg =- (1)/(\rho) (d p)/(d x) (L ^{3})/(\nu ^{2})	heat transfer (ratio of the buoyancy to viscous force in forced convection)
	i	i = (d h)/(d l) = (h _{2} - h _{1})/(length)	fluid mechanics, groundwater flow (pressure head over distance)
	Ka	Ka = (t _{F})/(t _{η})	turbulent combustion (characteristic chemical time scale to Kolmogorov time scale)
	KC	K _{C} = (VT)/(L)	fluid dynamics (ratio of drag force to inertia for a bluff object in oscillatory fluid flow)
	Kn	Kn = (λ)/(L)	gas dynamics (ratio of the molecular mean free path length to a representative physical length scale)
	Ku	Ku = (U _{h} \rho ^{1 / 2} _{g})/(\left({\sigma g (\rho _{l} - \rho _{g)}) \right) ^{1/4} }	fluid mechanics (counter-current two-phase flow)^[13]
	La	La = (\sigma \rho L)/(\mu ^{2})	fluid dynamics (free convection within immiscible fluids; ratio of surface tension to momentum-transport)
	Le	Le = (\alpha)/(D) = (Sc)/(Pr)	heat and mass transfer (ratio of thermal to mass diffusivity)
	CL	C _{L} = (L)/(qS)	aerodynamics (lift available from an airfoil at a given angle of attack)
	\chi	\chi = (m _{\ell})/(\sqrt{ m _{g}) (\rho _{g})/(\rho _{\ell}) }	two-phase flow (flow of wet gases; liquid fraction)^[14]
	M or Ma	M = ({v)/() }	gas dynamics (compressible flow; dimensionless velocity)
	Rm	R _{m} = (U L)/(η)	magnetohydrodynamics (ratio of magnetic advection to magnetic diffusion)
	n		open channel flow (flow driven by gravity)^[15]
	Mg	Mg = - { (d \sigma)/(d T) }\frac	fluid mechanics (Marangoni flow; thermal surface tension forces over viscous forces)
	l{M}	l{M} = (l{L)/(_{b}{\delta}) _{L}}	fluid dynamics, combustion (turbulent combustion flames)
	Mo	Mo = (g \mu ^{4} _{c} \Delta \rho)/(\rho ^{2} _{c} \sigma ^{3})	fluid dynamics (determination of bubble/drop shape)
	Nu	Nu _{d} = (hd)/(k)	heat transfer (forced convection; ratio of convective to conductive heat transfer)
	Oh	Oh = (\mu)/(\sqrt{\rho \sigma L) } = \frac	fluid dynamics (atomization of liquids, Marangoni flow)
	Pe	Pe _{d} = (du\rho c _{p})/(k) = Re _{d} Pr	heat transfer (advection–diffusion problems; total momentum transfer to molecular heat transfer)
	Pe	Pe _{d} = (du)/(D) = Re _{d} Sc	mass transfer (advection–diffusion problems; total momentum transfer to diffusive mass transfer)
	Pr	Pr = (\nu)/(\alpha) = (c _{p} \mu)/(k)	heat transfer (ratio of viscous diffusion rate over thermal diffusion rate)
	CP	C _{p} = {p - p _{infty} \over (1)/(2) \rho _{infty} V ^{2 }} _{inf ty}	aerodynamics, hydrodynamics (pressure experienced at a point on an airfoil; dimensionless pressure variable)
	Ra	Ra _{x} = (g \beta)/(\nu \alpha) (T _{s} - T _{infin)} x ^{3}	heat transfer (buoyancy versus viscous forces in free convection)
	Re	Re _{L} = (vL\rho)/(\mu)	fluid mechanics (ratio of fluid inertial and viscous forces)^[16]
	Ri	Ri = (gh)/(u ^{2}) = (1)/(Fr ^{2})	fluid dynamics (effect of buoyancy on flow stability; ratio of potential over kinetic energy)^[17]
	Ro	Ro = {f L ^{2} \over \nu} = St Re	fluid dynamics (oscillating flow, vortex shedding)
	Sc	Sc _{D} = (\nu)/(D)	mass transfer (viscous over molecular diffusion rate)^[18]
	H	H = (\delta ^{*})/(\theta)	boundary layer flow (ratio of displacement thickness to momentum thickness)
	Sh	Sh _{D} = (K L)/(D)	mass transfer (forced convection; ratio of convective to diffusive mass transport)
	S	S = \left( (r)/(c) \right) ^{2} (\mu N)/(P)	hydrodynamic lubrication (boundary lubrication)^[19]
	St	St = (h)/(c _{p} \rho V) = (Nu)/(Re Pr)	heat transfer and fluid dynamics (forced convection)
	Stk or Sk	Stk = (\tau U _{o})/(d _{c})	particles suspensions (ratio of characteristic time of particle to time of flow)
	St or Sr	St = {\omega L\over v}	fluid dynamics (continuous and pulsating flow; nondimensional frequency)^[20]
	N	N = (B ^{2} L _{c} \sigma)/(\rho U) = (Ha ^{2})/(Re)	magnetohydrodynamics (ratio of electromagnetic to inertial forces)
	Ta	Ta = (4 \Omega ^{2} R ^{4})/(\nu ^{2})	fluid dynamics (rotating fluid flows; inertial forces due to rotation of a fluid versus viscous forces)
	U	U = (H λ ^{2})/(h ^{3})	wave mechanics (nonlinearity of surface gravity waves on a shallow fluid layer)
	Va	Va = (\phi Pr)/(Da)	porous media (governs the effects of porosity \phi , the Prandtl number and the Darcy number on flow in a porous medium) ^[21]
	j^*	j ^{*} = R \left( (\omega \rho)/(\mu) \right) ^{(1)/(2)}	multiphase flows (nondimensional superficial velocity)^[22]
	We	We = (\rho v ^{2} l)/(\sigma)	multiphase flow (strongly curved surfaces; ratio of inertia to surface tension)
	Wi	Wi = • \gamma λ	viscoelastic flows (shear rate times the relaxation time)^[23]
	\alpha	\alpha = R \left( (\omega \rho)/(\mu) \right) ^{(1)/(2)}	biofluid mechanics (continuous and pulsating flows; ratio of pulsatile flow frequency to viscous effects)^[24]
	\beta	\beta = (E)/(RT _{f}) (T _{f-T} _{o})/(T _{f})	fluid dynamics, Combustion (Measure of activation energy)

Solids

Name	Standard symbol	Definition	Field of application
	\mu _{k}		mechanics (friction of solid bodies in translational motion)
	\mu _{s}		mechanics (friction of solid bodies at rest)
Dieterich-Ruina-Rice number	R _{u}	R _{u} = (W)/(L) ((b - a)\bar{\sigma)/() }	mechanics, friction, rheology, geophysics (stiffness ratio for frictional contacts)^[25]
	\gamma	\gamma = (Y r ^{2})/(\kappa)	virology, solid mechanics (thin-shell buckling)
	–		mechanical hardness (indentation hardness of a material)
	Crr	C _{rr} = (F)/(N _{f})	vehicle dynamics (ratio of force needed for motion of a wheel over the normal force)

Optics

Name	Standard symbol	Definition	Field of application
	V	V = (n _{d} -1)/(n _{F} - n _{C})	optics (dispersion in optical materials)
f-number	N	N = (f)/(D)	optics, photography (ratio of focal length to diameter of aperture)
	F	F = (a ^{2})/(L λ)	optics (slit diffraction)^[26]
	n	n = (c)/(v)	electromagnetism, optics (speed of light in vacuum over speed of light in a material)
	T	T = (I)/(I _{0})	optics, spectroscopy (the ratio of the intensities of radiation exiting through and incident on a sample)

Mathematics and statistics

Name	Standard symbol	Definition	Field of application
	R ^{2}		statistics (proportion of variance explained by a statistical model)
	(\sigma)/(\mu)	(\sigma)/(\mu)	statistics (ratio of standard deviation to expectation)
	ρ or r	(\operatorname{E)/([(X - \mu) _{X)(Y-\mu} _{Y)]}{\sigma} _{X} \sigma _{Y}}	statistics (measure of linear dependence)
	C or '	C = (u\Delta t)/(\Delta x)	mathematics (numerical solutions of hyperbolic PDEs)^[27]
	e	e = \displaystyle\sum\limits ^{inf ty} _{n =0} \dfrac{ 1 }{n!} ≈ 2 . 71828	mathematics (base of the natural logarithm)
	\alpha , \delta	\alpha ≈ 2 . 50290 , \delta ≈ 4 . 66920	chaos theory (period doubling)^[28]
	\varphi	\varphi = (1+ \sqrt{ 5)/() } \approx 1.61803	mathematics, aesthetics (long side length of self-similar rectangle)
Pi	\pi	\pi = (C)/(D) ≈ 3 . 14159	mathematics (ratio of a circle's circumference to its diameter)
Radian measure	rad	arclength / radius	mathematics (measurement of planar angles, 1 radian = 180/π degrees)
Steradian measure	sr		measurement of solid angles

Geography, geology and geophysics

Name	Standard symbol	Definition	Field of application
	\alpha	\alpha = ( 1- D) \bar \alpha(\theta _{i)} + D \bar{ \bar \alpha}	climatology, astronomy (reflectivity of surfaces or bodies)
Love numbers	h, k, l		geophysics (solidity of earth and other planets)
Porosity	\phi	\phi = (V _{V})/(V _{T})	geology, porous media (void fraction of the medium)
	Ro	Ro = (U)/(Lf)	geophysics (ratio of inertial to Coriolis force)

Sport

Name	Standard symbol	Definition	Field of application
	B _{\kappa}	B _{\kappa} = (t _{g} v _{f})/(l _{mf})	sport science, team sports^[29]
Gain ratio	–		bicycling (system of representing gearing; length traveled over length pedaled)^[30]
	GD	Goaldifference = goalsscored - goalsconceded	Association football^[31]
	RpW ratio	RpWratio = (runsscored)/(wicketslost) ÷ (runsconceded)/(wicketstaken)	cricket^[32]
	–	Various, e.g. (Gameswon)/(Gamesplayed) or (Pointswon)/(Pointscontested)	Various sports

Other fields

Name	Standard symbol	Definition	Field of application
		(actualelectricalenergyoutput)/(maximumpossibleelectricalenergyoutput)	energy
Cohesion number	Coh	Coh = (1)/(\rho g) \left ( (\Gamma ^{5})/({E ^{*}) ^{2{R} ^{*}} ^{8}} \right ) ^{(1)/(3)}	Chemical engineering, material science, mechanics (A scale to show the energy needed for detaching two solid particles)^[33] ^[34]
	COT	COT = (E)/(mgd)	energy efficiency, economics (ratio of energy input to kinetic motion)
	\zeta	\zeta = (c)/(2 \sqrt{km) }	mechanics, electrical engineering (the level of damping in a system)
	Da	Da = (K)/(d ^{2})	porous media (ratio of permeability to cross-sectional area)
Decibel	dB		acoustics, electronics, control theory (ratio of two intensities or powers of a wave)
	Du	Du = (\kappa ^{\sigma})/({\Kappa _{m}) a}	colloid science (ratio of electric surface conductivity to the electric bulk conductivity in heterogeneous systems)
Elasticity (economics)	E	E _{x,y} = (\partial ln (x))/(\partial ln (y)) = (\partial x)/(\partial y) (y)/(x)	economics (response of demand or supply to price changes)
	\alpha	\alpha = (e ^{2})/(4 \pi\varepsilon _{0} \hbar c)	quantum electrodynamics (QED) (coupling constant characterizing the strength of the electromagnetic interaction)
Gain	–		electronics (signal output to signal input)
	P _{H}	P _{H} = (Z _{d} n _{d})/(n _{i})	In Dusty plasma physics, ratio of the total charge Z _{d} carried by the dust particles d to the charge carried by the ions i , with n the number density of particles
	He	He = (\omega a)/(c _{0}) = k _{0a}	The most important parameter in duct acoustics. If \omega is the dimensional frequency, then k _{0} is the corresponding free field wavenumber and He is the corresponding dimensionless frequency ^[35]
	Ir	Ir = (\tan \alpha)/(\sqrt{H/L _{0}) }	wave mechanics (breaking surface gravity waves on a slope)
		(averageload)/(peakload)	energy
	S	S = (\mu _{0LV} _{A})/(η)	plasma physics (ratio of a resistive time to an Alfvén wave crossing time in a plasma)
	NP	N _{P} = (Restoringforce)/(Adhesiveforce)	coating (adhesion of microstructures with substrate)^[36]
	K	{K} = ({I)/() }\,\frac (1-\gamma^2f_e)	charged particle transport (measure of the strength of space charge in a charged particle beam)
	C	C ^{3= (Z _{c} I _{K})/(4 V _{K})}	Traveling wave tube
Pixel	px		digital imaging (smallest addressable unit)
Beta (plasma physics)	\beta	\beta = (n k _{B} T)/(B ^{2 / 2 \mu} _{0})	Plasma (physics) and Fusion power. Ratio of plasma thermal pressure to magnetic pressure, controlling the level of turbulence in a magnetised plasma.
	\nu	\nu =- (d \varepsilon _{trans})/(d \varepsilon _{axial})	elasticity (strain in transverse and longitudinal direction)
	pf	pf = (P)/(S)	electrical (real power to apparent power)
	Np	N _{p} = {P\over \rho n ^{3} d ^{5}}	fluid mechanics, power consumption by rotary agitators; resistance force versus inertia force)
	β	\beta = (- \Delta H _{r} D ^{e} _{TA} C _{AS})/(λ ^{e} T _{s})	reaction engineering (ratio of heat evolution to heat conduction within a catalyst pellet)^[37]
	Q	Q = 2 \pi f _{r} (EnergyStored)/(PowerLoss)	physics, engineering (Damping ratio of oscillator or resonator; energy stored versus energy lost)
	RD	RD = (\rho _{substance})/(\rho _{reference})	hydrometers, material comparisons (ratio of density of a material to a reference material—usually water)
	\mu _{r}	\mu _{r} = (\mu)/(\mu _{0})	magnetostatics (ratio of the permeability of a specific medium to free space)
	\varepsilon _{r}	\varepsilon _{r} = (C _{x})/(C _{0})	electrostatics (ratio of capacitance of test capacitor with dielectric material versus vacuum)
	P or Z	P = (w _{s})/(\kappa u _{*})	sediment transport (ratio of the sediment fall velocity and the upwards velocity of grain)
	\tau _{*} or \theta	\tau _{\ast} = (\tau)/((\rho _{s} - \rho) g D)	sediment transport (threshold of sediment movement due to fluid motion; dimensionless shear stress)
	SG		(same as Relative density)
	Ste	Ste = (c _{p} \Delta T)/(L)	phase change, thermodynamics (ratio of sensible heat to latent heat)
	\epsilon	\epsilon = \cfrac{\partial{F}}{\partial{X}} - 1	materials science, elasticity (displacement between particles in the body relative to a reference length)

Bibliography

• Web site: iso.org . ISO 80000-11:2019 Quantities and units — Part 11: Characteristic numbers . 2023-08-31.

Notes and References

1. Becker . A. . Hüttinger . K. J. . 10.1016/S0008-6223(97)00175-9 . 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 . 1998 .
2. Book: Incropera, Frank P. . Fundamentals of heat and mass transfer . limited . 376 . 2007 . John Wiley & Sons, Inc. 9780470055540 .
3. Book: Popov . Konstantin I. . Djokić . Stojan S. . Grgur . Branimir N. . Fundamental Aspects of Electrometallurgy . 2002 . . Boston, MA . 978-0-306-47564-1 . 101–102.
4. Book: Kuneš . J.. Technology and Mechanical Engineering. 10.1016/B978-0-12-416013-2.00008-7. Dimensionless Physical Quantities in Science and Engineering. 353–390. 2012. 978-0-12-416013-2.
5. Asakuma . Y. . 2020 . A dimensionless number for microwave non-equilibrium local heating through surfactant desorption . Colloids and Surfaces A: Physicochemical and Engineering Aspects . 591 . 124560 . 10.1016/j.colsurfa.2020.124560.
6. http://www2.umt.edu/Geology/faculty/hendrix/g432/g432_L6.htm Bagnold number
7. Bhattacharjee S. . Grosshandler W.L. . The formation of wall jet near a high temperature wall under microgravity environment . ASME MTD . 96 . 711–6 . 1988 . 1988nht.....1..711B .
8. Paoletti S. . Rispoli F. . Sciubba E. . Calculation of exergetic losses in compact heat exchanger passager . ASME AES . 10 . 2 . 21–9 . 1989 .
9. http://ising.phys.cwru.edu/plt/PapersInPdf/181BridgeCollapse.pdf Bond number
10. Web site: Home . OnePetro . 2015-05-04 . 2015-05-08.
11. Book: Schetz, Joseph A.. Boundary Layer Analysis. limited. 1993. Prentice-Hall, Inc.. Englewood Cliffs, NJ. 0-13-086885-X. 132–134.
12. Web site: Fanning friction factor . 2015-10-07 . https://web.archive.org/web/20131220032423/http://www.engineering.uiowa.edu/~cee081/Exams/Final/Final.htm . 2013-12-20 . dead .
13. Tan . R. B. H. . Sundar . R. . 10.1016/S0009-2509(01)00247-0 . On the froth–spray transition at multiple orifices . Chemical Engineering Science . 56 . 21–22 . 6337 . 2001 . 2001ChEnS..56.6337T .
14. https://web.archive.org/web/20040215214253/http://www.flowprogramme.co.uk/publications/guidancenotes/GN40.pdf Lockhart–Martinelli parameter
15. Web site: Manning coefficient . 10 June 2013 .  
16. Web site: Table of Dimensionless Numbers . 2009-11-05.
17. http://apollo.lsc.vsc.edu/classes/met455/notes/section4/2.html Richardson number
18. http://www.ent.ohiou.edu/~hbwang/fluidynamics.htm Schmidt number
19. http://epubl.luth.se/avslutade/0348-8373/41/ Sommerfeld number
20. http://www.engineeringtoolbox.com/strouhal-number-d_582.html Strouhal number
21. Straughan . B. . A sharp nonlinear stability threshold in rotating porous convection . 10.1098/rspa.2000.0657 . Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences . 457 . 2005 . 87–88 . 2001 . 2001RSPSA.457...87S . 122753376 .
22. Petritsch . G. . Mewes . D. . 10.1016/S0029-5493(99)00005-9 . Experimental investigations of the flow patterns in the hot leg of a pressurized water reactor . Nuclear Engineering and Design . 188 . 75–84 . 1999 .
23. http://physics.ucsd.edu/~des/Shear1999.pdf Weissenberg number
24. http://www.seas.upenn.edu/courses/belab/LabProjects/2001/be310s01m2.doc Womersley number
25. Barbot . S. . 10.1016/j.tecto.2019.228171 . Slow-slip, slow earthquakes, period-two cycles, full and partial ruptures, and deterministic chaos in a single asperity fault . Tectonophysics . 768 . 228171 . 2019 . 2019Tectp.76828171B . free .
26. http://www.ilt.fraunhofer.de/default.php?web=1&id=100050&lan=eng&dat=2 Fresnel number
27. http://www.cnrm.meteo.fr/aladin/newsletters/news22/J_Vivoda/Texte.html Courant–Friedrich–Levy number
28. http://www.drchaos.net/drchaos/Book/node44.html Feigenbaum constants
29. 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 . 2021 . 17 . 2 . 145–154 . 10.1515/jqas-2020-0009 . 224929098 .
30. http://sheldonbrown.com/gain.html Gain Ratio – Sheldon Brown
31. Web site: goal difference. Cambridge Dictionary.
32. Web site: World Test Championship Playing Conditions: What's different?. International Cricket Council. 11 August 2021.
33. Behjani. Mohammadreza Alizadeh. Rahmanian. Nejat. Ghani. Nur Fardina bt Abdul. Hassanpour. Ali. An investigation on process of seeded granulation in a continuous drum granulator using DEM. Advanced Powder Technology. 28. 10. 2456–2464. 10.1016/j.apt.2017.02.011. 2017.
34. 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. en. 140. 06024. 10.1051/epjconf/201714006024. 2100-014X. 2017EPJWC.14006024A. free.
35. S.W. RIENSTRA, 2015, Fundamentals of Duct Acoustics, Von Karman Institute Lecture Notes
36. Van Spengen . W. M. . Puers . R. . De Wolf . I. . 10.1109/TDMR.2003.820295 . The prediction of stiction failures in MEMS . IEEE Transactions on Device and Materials Reliability . 3 . 4 . 167 . 2003 .
37. Book: Davis . Mark E. . Davis . Robert J. . Fundamentals of Chemical Reaction Engineering . 2012 . Dover . 978-0-486-48855-4 . 215.