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| In [[dimensional analysis]], a '''dimensionless quantity''' or '''quantity of dimension one''' is a [[quantity]] without an associated [[dimensional analysis|physical dimension]]. It is thus a "pure" number, and as such always has a dimension of 1.<ref>{{cite web|url=http://www.iso.org/sites/JCGM/VIM/JCGM_200e_FILES/MAIN_JCGM_200e/01_e.html#L_1_8|title='''1.8''' (1.6) '''quantity of dimension one''' dimensionless quantity|work=International vocabulary of metrology — Basic and general concepts and associated terms (VIM)|publisher=[[International Organization for Standardization|ISO]]|year=2008|accessdate=2011-03-22}}</ref> Dimensionless quantities are widely used in [[mathematics]], [[physics]], [[engineering]], [[economics]], and in everyday life (such as in counting). Numerous well-known quantities, such as [[pi|π]], [[Euler's number|e]], and [[Golden ratio|φ]], are dimensionless. By contrast, non-dimensionless quantities are measured in units of length, area, time, etc.
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| Dimensionless quantities are often defined as [[product (mathematics)|product]]s or [[ratio]]s of [[quantity|quantities]] that are not dimensionless, but whose dimensions cancel out when their [[Exponentiation|powers]] are multiplied. This is the case, for instance, with the [[engineering strain]], a measure of deformation. It is defined as change in length over initial length but, since these quantities both have dimensions ''L'' (length), the result is a dimensionless quantity.
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| == Properties ==
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| {{unreferenced section|date=June 2013}}
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| * Even though a dimensionless quantity has no physical dimension associated with it, it can still have dimensionless [[Units of measurement|units]]. To show the quantity being measured (for example [[mass fraction (chemistry)|mass fraction]] or [[mole fraction]]), it is sometimes helpful to use the same units in both the numerator and denominator (kg/kg or mol/mol). The quantity may also be given as a ratio of two different units that have the same dimension (for instance, [[light year]]s over [[meter]]s). This may be the case when calculating [[slope]]s in graphs, or when making unit conversions. Such notation does not indicate the presence of physical dimensions, and is purely a notational convention. Other common dimensionless units are % (= 0.01), ‰ (= 0.001), ppm (= 10<sup>−6</sup>), ppb (= 10<sup>−9</sup>), [[Parts per trillion|ppt]] (= 10<sup>−12</sup>) and angle units (degrees, radians, grad). Units of number such as the [[dozen]] and the [[Gross (unit)|gross]] are also dimensionless.
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| * The ratio of two quantities with the same dimensions is dimensionless, and has the same value regardless of the units used to calculate them. For instance, if body '''A''' exerts a force of magnitude ''F'' on body '''B''', and '''B''' exerts a force of magnitude ''f'' on '''A''', then the ratio ''F''/''f'' is always equal to [[Newton's Third Law of Motion|1]], regardless of the actual units used to measure ''F'' and ''f''. This is a fundamental property of dimensionless proportions and follows from the assumption that the laws of physics are independent of the system of units used in their expression. In this case, if the ratio ''F''/''f'' was not always equal to 1, but changed if we switched from [[SI]] to [[CGS]], that would mean that [[Newton's Third Law]]'s truth or falsity would depend on the system of units used, which would contradict this fundamental hypothesis. This assumption that the laws of physics are not contingent upon a specific unit system is the basis for the [[Buckingham π theorem]]. A statement of this theorem is that any physical law can be expressed as an [[Identity (mathematics)|identity]] involving only dimensionless combinations (ratios or products) of the variables linked by the law (e. g., pressure and volume are linked by [[Boyle's Law]] – they are inversely proportional). If the dimensionless combinations' values changed with the systems of units, then the equation would not be an identity, and Buckingham's theorem would not hold.
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| ==Buckingham π theorem==
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| Another consequence of the [[Buckingham π theorem]] of [[dimensional analysis]] is that the [[Function (mathematics)|functional]] dependence between a certain number (say, ''n'') of [[variable (mathematics)|variables]] can be reduced by the number (say, ''k'') of [[independent variable|independent]] [[dimension]]s occurring in those variables to give a set of ''p'' = ''n'' − ''k'' independent, dimensionless [[quantity|quantities]]. For the purposes of the experimenter, different systems that share the same description by dimensionless [[quantity]] are equivalent.
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| ===Example===
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| The [[electric power|power]] consumption of a [[stirrer]] with a given shape is a function of the [[density]] and the [[viscosity]] of the fluid to be stirred, the size of the stirrer given by its [[diameter]], and the [[speed]] of the stirrer. Therefore, we have ''n'' = 5 variables representing our example.
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| Those ''n'' = 5 variables are built up from ''k'' = 3 dimensions:
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| * Length: ''L'' (m)
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| * Time: ''T'' (s)
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| * Mass: ''M'' (kg)
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| According to the π-theorem, the ''n'' = 5 variables can be reduced by the ''k'' = 3 dimensions to form ''p'' = ''n'' − ''k'' = 5 − 3 = 2 independent dimensionless numbers, which are, in case of the stirrer:
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| * [[Reynolds number]] (a dimensionless number describing the fluid flow regime)
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| * [[Power number]] (describing the stirrer and also involves the density of the fluid)
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| ==Standards efforts==
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| The [[International Committee for Weights and Measures]] contemplated defining the unit of 1 as the 'uno', but the idea was dropped.<ref>{{Cite web|url=http://www.bipm.fr/utils/common/pdf/CCU15.pdf|format=PDF|title=BIPM Consultative Committee for Units (CCU), 15th Meeting|date=17–18 April 2003|accessdate=2010-01-22}}</ref><ref>{{Cite web|format=PDF|url=http://www.bipm.fr/utils/common/pdf/CCU16.pdf|title=BIPM Consultative Committee for Units (CCU), 16th Meeting|accessdate=2010-01-22}}</ref><ref>{{cite journal |author=Dybkaer, René |title=An ontology on property for physical, chemical, and biological systems |journal=APMIS Suppl. |issue=117 |pages=1–210 |year=2004 |pmid=15588029 |url=http://www.iupac.org/publications/ci/2005/2703/bw1_dybkaer.html}}</ref>
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| ==Examples==
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| *Consider this example: Sarah says, "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.
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| * [[Plane angle]]s – An angle is measured as the ratio of the length of a circle's arc subtended by an angle whose vertex is the centre of the circle to some other length. The ratio—i.e., length divided by length—is dimensionless. When using [[radian]]s as the unit, the length that is compared is the length of the [[radius]] of the circle. When using [[degree (angle)|degree]] as the units, the arc's length is compared to 1/360 of the [[circumference]] of the circle.
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| *In the case of the dimensionless quantity π, being the ratio of a circle's circumference to its diameter, the number would be constant regardless of what unit is used to measure a circle's circumference and diameter (e.g., centimetres, miles, [[light-year]]s, etc.), as long as the same unit is used for both.
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| ==Dimensionless physical constants==
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| Certain [[fundamental physical constant]]s, such as the [[speed of light]] in a vacuum, the [[universal gravitational constant]], [[Planck's constant]] and [[Boltzmann's constant]] can be normalized to 1 if appropriate units for [[time]], [[length]], [[mass]], [[electric charge|charge]], and [[temperature]] are chosen. The resulting [[system of units]] is known as the [[natural units]]. However, not all [[physical constant]]s can be normalized in this fashion. For example, the values of the following constants are independent of the system of units and must be determined experimentally:
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| * α ≈ 1/137.036, the [[fine structure constant]] which is the [[coupling constant]] for the [[electromagnetic interaction]];
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| * β (or μ) ≈ 1836, the [[proton-to-electron mass ratio]]. This ratio is the [[rest mass]] of the [[proton]] divided by that of the [[electron]]. An analogous ratio can be defined for any [[elementary particle]];
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| * α<sub>s</sub>, the [[coupling constant]] for the [[strong force]];
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| * α<sub>G</sub> ≈ 1.75×10<sup>−45</sup>, the [[gravitational coupling constant]].
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| ==List of dimensionless quantities==
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| All numbers are dimensionless [[quantity|quantities]]. Certain dimensionless quantities of some importance are given below:
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| {| class="wikitable sortable" border="1"
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| ! scope="col" | Name
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| ! scope="col" | Standard symbol
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| ! scope="col" class="unsortable" | Definition
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| ! scope="col" | Field of application
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| | [[Abbe number]] || ''V'' || <math>V = \frac{ n_d - 1 }{ n_F - n_C }</math>|| [[optics]] ([[dispersion (optics)|dispersion]] in optical materials)
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| | [[Activity coefficient]] || <math>\gamma</math> || <math> \gamma= \frac {{a}}{{x}} </math>|| [[chemistry]] (Proportion of "active" molecules or atoms)
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| | [[Albedo]] || <math>\alpha</math> ||<math>\alpha= (1-D) \bar \alpha(\theta_i) + D \bar{ \bar \alpha}</math>|| [[climatology]], [[astronomy]] ([[reflectivity]] of surfaces or bodies)
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| | [[Archimedes number]] || Ar || <math> \mathrm{Ar} = \frac{g L^3 \rho_\ell (\rho - \rho_\ell)}{\mu^2}</math>|| [[fluid mechanics]] (motion of [[fluid]]s due to [[density]] differences)
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| | [[Arrhenius equation|Arrhenius number]] || <math>\alpha</math> || <math>\alpha = \frac{E_a}{RT} </math> || [[chemistry]] (ratio of [[activation energy]] to [[thermal energy]])<ref name="berkley" />
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| | [[Atomic weight]] || ''M'' || || [[chemistry]] ([[mass]] of [[atom]] over one [[atomic mass unit]], u, where [[carbon-12]] is exactly 12 u)
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| | [[Atwood number]] || A || <math>\mathrm{A} = \frac{\rho_1 - \rho_2} {\rho_1 + \rho_2} </math> || [[fluid mechanics]] (onset of instabilities in [[fluid]] mixtures due to [[density]] differences)
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| | [[Bagnold number]] || Ba ||<math>\mathrm{Ba} = \frac{\rho d^2 \lambda^{1/2} \gamma}{\mu}</math>|| [[fluid mechanics]], [[geology]] (ratio of grain collision stresses to viscous fluid [[Stress (mechanics)|stresses]] in flow of a [[granular material]] such as [[grain]] and [[sand]])<ref>[http://www2.umt.edu/Geology/faculty/hendrix/g432/g432_L6.htm Bagnold number]</ref>
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| | [[Bejan number]]<br/><small>([[fluid mechanics]])</small>|| Be || <math>\mathrm{Be} = \frac{\Delta P L^2} {\mu \alpha}</math> || [[fluid mechanics]] (dimensionless [[pressure]] drop along a [[Channel (geography)|channel]])<ref>{{cite journal |author=Bhattacharjee S., Grosshandler W.L. |title=The formation of wall jet near a high temperature wall under microgravity environment |journal=ASME MTD |volume=96 |pages=711–6 |year=1988 }}</ref>
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| | [[Bejan number]]<br/><small>([[thermodynamics]])</small>|| Be || <math>\mathrm{Be} = \frac{\dot S'_{\mathrm{gen},\, \Delta T}}{\dot S'_{\mathrm{gen},\, \Delta T}+ \dot S'_{\mathrm{gen},\, \Delta p}}</math> || [[thermodynamics]] (ratio of [[heat transfer]] [[Irreversible process|irreversibility]] to total irreversibility due to heat transfer and [[fluid]] friction)<ref>{{cite journal |author=Paoletti S., Rispoli F., Sciubba E. |title=Calculation of exergetic losses in compact heat exchanger passager |journal=ASME AES |volume=10 |issue=2 |pages=21–9 |year=1989 }}</ref>
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| | [[Herschel–Bulkley fluid#Channel flow|Bingham number]] || Bm ||<math>\mathrm{Bm} = \frac{ \tau_y L }{ \mu V }</math>|| [[fluid mechanics]], [[rheology]] (ratio of yield stress to viscous stress)<ref name="berkley" />
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| | [[Biot number]] || Bi ||<math>\mathrm{Bi} = \frac{h L_C}{k_b}</math>|| [[heat transfer]] (surface vs. volume [[thermal conductivity|conductivity]] of solids)
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| | [[Blake number]] || Bl or B ||<math>\mathrm{B} = \frac{u \rho}{\mu (1 - \epsilon) D}</math> || [[geology]], [[fluid mechanics]], [[porous media]] (inertial over [[Viscosity|viscous forces]] in fluid flow through porous media)
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| | [[Bodenstein number]] || Bo or Bd || <math>\mathrm{Bo} = vL/\mathcal{D} = \mathrm{Re}\, \mathrm{Sc} </math> || [[chemistry]] ([[residence time|residence-time]] distribution; similar to the axial [[mass transfer]] [[Peclet number]])<ref>{{cite doi|10.1016/S0008-6223(97)00175-9}}</ref>
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| | [[Bond number]] || Bo ||<math>\mathrm{Bo} = \frac{\rho a L^2}{\gamma}</math>|| [[geology]], [[fluid mechanics]], [[porous media]] ([[buoyancy|buoyant]] versus [[capilary]] forces, similar to the [[Eötvös number]]) <ref>[http://ising.phys.cwru.edu/plt/PapersInPdf/181BridgeCollapse.pdf Bond number]</ref>
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| | [[Brinkman number]] || Br ||<math> \mathrm{Br} = \frac {\mu U^2}{\kappa (T_w - T_0)}</math>|| [[heat transfer]], [[fluid mechanics]] ([[Thermal conductivity|conduction]] from a wall to a [[viscosity|viscous]] [[fluid]])
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| | [[Brownell–Katz number]] || N<sub>BK</sub> || <math>\mathrm{N}_\mathrm{BK} = \frac{u \mu}{k_\mathrm{rw}\sigma} </math> || [[fluid mechanics]] (combination of [[capillary number]] and [[Bond number]]) <ref>http://www.onepetro.org/mslib/servlet/onepetropreview?id=00020506</ref>
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| | [[Capillary number]] || Ca || <math>\mathrm{Ca} = \frac{\mu V}{\gamma} </math> || [[porous media]], [[fluid mechanics]] ([[viscous forces]] versus [[surface tension]])
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| | [[Chandrasekhar number]] || Q ||<math> \mathrm{Q} = \frac{{B_0}^2 d^2}{\mu_0 \rho \nu \lambda} </math> || [[magnetohydrodynamics]] (ratio of the [[Lorentz force]] to the [[viscosity]] in magnetic [[convection]])
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| | [[Chilton and Colburn J-factor analogy|Colburn J factors]] || ''J''<sub>M</sub>, ''J''<sub>H</sub>, ''J''<sub>D</sub> || || [[turbulence]]; [[heat transfer|heat]], [[mass transfer|mass]], and [[fluid mechanics|momentum]] transfer (dimensionless transfer coefficients)
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| | [[Coefficient of friction|Coefficient of kinetic friction]] || <math>\mu_k</math> || || [[mechanics]] (friction of solid bodies in translational motion)
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| | [[Coefficient of static friction]] || <math>\mu_s</math> || || [[mechanics]] (friction of solid bodies at rest)
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| | [[Coefficient of determination]] || <math>R^2</math> || || [[statistics]] (proportion of variance explained by a statistical model)
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| | [[Coefficient of variation]] || <math>\frac{\sigma}{\mu}</math> || <math>\frac{\sigma}{\mu}</math> || [[statistics]] (ratio of standard deviation to expectation)
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| | [[Correlation]] || ''ρ'' or ''r'' || <math>\frac{{\mathbb E}[(X-\mu_X)(Y-\mu_Y)]}{\sigma_X \sigma_Y}</math> or <math> \frac{\sum_{k=1}^n (x_k-\bar x)(y_k-\bar y)}{\sqrt{\sum_{k=1}^n (x_k-\bar x)^2 \sum_{k=1}^n (y_k-\bar y)^2}} </math> where <math>\bar x = \sum_{k=1}^n x_k/n</math> and similarly for <math>\bar y</math> || [[statistics]] (measure of linear dependence)
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| | [[Courant–Friedrich–Levy number]] || ''C'' or ''𝜈'' || <math>C = \frac {u\,\Delta t} {\Delta x}</math> || [[mathematics]] (numerical solutions of [[hyperbolic PDE]]s)<ref>[http://www.cnrm.meteo.fr/aladin/newsletters/news22/J_Vivoda/Texte.html Courant–Friedrich–Levy number]</ref>
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| | [[Damkohler number]] || Da ||<math> \mathrm{Da} = k \tau</math>|| [[chemistry]] (reaction time scales vs. residence time)
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| | [[Damping ratio]] || <math>\zeta</math> ||<math> \zeta = \frac{c}{2 \sqrt{km}}</math>|| [[mechanics]] (the level of [[damping]] in a system)
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| | [[Darcy friction factor]] || ''C''<sub>f</sub> or ''f''<sub>D</sub> || || [[fluid mechanics]] (fraction of [[pressure]] losses due to [[friction]] in a [[pipe (fluid conveyance)|pipe]]; four times the [[Fanning friction factor]])
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| | [[Darcy number]] || Da ||<math> \mathrm{Da} = \frac{K}{d^2}</math>|| [[porous media]] (ratio of [[Hydraulic conductivity|permeability]] to cross-sectional area)
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| | [[Dean number]] || D || <math>\mathrm{D} = \frac{\rho V d}{\mu} \left( \frac{d}{2 R} \right)^{1/2}</math> || [[Turbulence|turbulent flow]] ([[Vortex|vortices]] in curved ducts)
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| | [[Deborah number]] || De || <math> \mathrm{De} = \frac{t_\mathrm{c}}{t_\mathrm{p}}</math> || [[rheology]] ([[viscoelastic]] fluids)
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| | [[Decibel]] || ''dB'' || || [[acoustics]], [[electronics]], [[control theory]] (ratio of two [[Intensity (physics)|intensities]] or [[power (physics)|powers]] of a [[wave]])
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| | [[Drag coefficient]] || ''c''<sub>d</sub> || <math>c_\mathrm{d} = \dfrac{2 F_\mathrm{d}}{\rho v^2 A}\, ,</math> || [[aeronautics]], [[fluid dynamics]] (resistance to fluid motion)
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| | [[Dukhin number]] || Du || <math> \mathrm{Du} = \frac{\kappa^{\sigma}}{{\Kappa_m} a}</math> || [[Interface and colloid science|colloid science]] (ratio of electric [[surface conductivity]] to the electric bulk conductivity in [[heterogeneous]] systems)
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| | [[Eckert number]] || Ec || <math> \mathrm{Ec} = \frac{V^2}{c_p\Delta T} </math> || [[convective heat transfer]] (characterizes [[dissipation]] of [[energy]]; ratio of [[kinetic energy]] to [[enthalpy]])
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| | [[Ekman number]] || Ek || <math>\mathrm{Ek} = \frac{\nu}{2D^2\Omega\sin\varphi} </math> || [[geophysics]] ([[viscosity|viscous]] versus [[Coriolis force]]s)
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| | [[Elasticity (economics)|Elasticity]]<br/><small>([[economics]])</small> || ''E'' || || [[economics]] (response of [[demand]] or [[supply and demand|supply]] to [[price]] changes)
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| | [[Eötvös number]] || Eo || <math>\mathrm{Eo}=\frac{\Delta\rho \,g \,L^2}{\sigma}</math> || [[fluid mechanics]] (shape of [[Liquid bubble|bubbles]] or [[drop (liquid)|drops]])
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| | [[Ericksen number]] || Er || <math>\mathrm{Er}=\frac{\mu v L}{K}</math> || [[fluid dynamics]] ([[liquid crystal]] flow behavior; [[viscous]] over [[Elasticity (physics)|elastic]] forces)
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| | [[Euler number (physics)|Euler number]] || Eu || <math> \mathrm{Eu}=\frac{\Delta{}p}{\rho V^2} </math> || [[hydrodynamics]] (stream [[pressure]] versus [[inertia]] forces)
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| | [[Euler's number]] || ''e'' || <math> e = \sum_{k = 0}^n \frac{1}{k!} \approx 2.71828 </math> || [[mathematics]] ([[logarithm|base]] of the [[natural logarithm]])
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| | [[Excess temperature coefficient]] || <math>\Theta_r</math> ||<math>\Theta_r = \frac{c_p (T-T_e)}{U_e^2/2}</math>|| [[heat transfer]], [[fluid dynamics]] (change in [[internal energy]] versus [[kinetic energy]])<ref>{{cite book|last=Schetz|first=Joseph A.|title=Boundary Layer Analysis|year=1993|publisher=Prentice-Hall, Inc.|location=Englewood Cliffs, NJ|isbn=0-13-086885-X|pages=132–134}}</ref>
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| | [[Fanning friction factor]] || ''f'' || || [[fluid mechanics]] (fraction of [[pressure]] losses due to [[friction]] in a [[pipe (fluid conveyance)|pipe]]; 1/4<sup>th</sup> the [[Darcy friction factor]])<ref>[http://www.engineering.uiowa.edu/~cee081/Exams/Final/Final.htm Fanning friction factor]</ref>
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| | [[Feigenbaum constants]] || <math>\alpha</math>, <math>\delta</math> ||<math>\alpha \approx 2.50290,</math><br/><math>\ \delta \approx 4.66920 </math> || [[chaos theory]] ([[Period-doubling bifurcation|period doubling]])<ref>[http://www.drchaos.net/drchaos/Book/node44.html Feigenbaum constants]</ref>
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| | [[Fine structure constant]] || <math>\alpha</math> ||<math>\alpha = \frac{e^2}{2\varepsilon_0 hc}</math>|| [[quantum electrodynamics|quantum electrodynamics (QED)]] ([[coupling constant]] characterizing the strength of the [[electromagnetic interaction]])
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| | [[f-number]] || ''f'' || <math> f = \frac {{\ell}}{{D}}</math> || [[optics]], [[photography]] (ratio of [[focal length]] to [[diameter]] of [[aperture]])
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| | [[Föppl–von Kármán equations|Föppl–von Kármán number]] || <math>\gamma</math> || <math>\gamma = \frac{Y r^2}{\kappa}</math> || [[virology]], [[solid mechanics]] (thin-shell buckling)
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| | [[Fourier number]] || Fo || <math>\mathrm{Fo} = \frac{\alpha t}{L^2}</math> || [[heat transfer]], [[mass transfer]] (ratio of diffusive rate versus storage rate)
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| | [[Fresnel number]] || ''F'' || <math>\mathit{F} = \frac{a^{2}}{L \lambda}</math> || [[optics]] (slit [[diffraction]])<ref>[http://www.ilt.fraunhofer.de/default.php?web=1&id=100050&lan=eng&dat=2 Fresnel number]</ref>
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| | [[Froude number]] || Fr || <math>\mathrm{Fr} = \frac{v}{\sqrt{g\ell}}</math> || [[fluid mechanics]] ([[wave]] and [[surface wave|surface]] behaviour; ratio of a body's [[inertia]] to [[gravity|gravitational forces]])
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| | [[Gain]] || – || || [[electronics]] (signal output to signal input)
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| | [[Bicycle gearing|Gain ratio]] || – || || [[bicycle|bicycling]] (system of representing gearing; length traveled over length pedaled)<ref>[http://sheldonbrown.com/gain.html Gain Ratio - Sheldon Brown]</ref>
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| | [[Galilei number]] || Ga || <math>\mathrm{Ga} = \frac{g\, L^3}{\nu^2}</math> || [[fluid mechanics]] ([[gravity|gravitational]] over [[viscosity|viscous]] forces)
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| | [[Golden ratio]] || <math>\varphi</math> || <math>\varphi = \frac{1+\sqrt{5}}{2} \approx 1.61803</math> || [[mathematics]], [[aesthetics]] (long side length of self-similar [[rectangle]])
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| | [[Görtler vortices|Görtler number]] || G || <math>\mathrm{G} = \frac{U_e \theta}{\nu} \left( \frac{\theta}{R} \right)^{1/2}</math> || [[fluid dynamics]] ([[boundary layer flow]] along a concave wall)
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| | [[Graetz number]] || Gz || <math>\mathrm{Gz} = {D_H \over L} \mathrm{Re}\, \mathrm{Pr}</math> || [[heat transfer]], [[fluid mechanics]] ([[laminar flow]] through a conduit; also used in [[mass transfer]])
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| | [[Grashof number]] || Gr || <math> \mathrm{Gr}_L = \frac{g \beta (T_s - T_\infty ) L^3}{\nu ^2}</math> || [[heat transfer]], [[natural convection]] (ratio of the [[buoyancy]] to [[viscous]] force)
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| | [[Gravitational coupling constant]] || <math>\alpha_G</math> ||<math>\alpha_G=\frac{Gm_e^2}{\hbar c}</math>|| [[gravitation]] (attraction between two [[mass]]y [[elementary particles]]; analogous to the [[Fine structure constant]])
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| | [[Hatta number]] || Ha || <math>\mathrm{Ha} = \frac{N_{\mathrm{A}0}}{N_{\mathrm{A}0}^{\mathrm{phys}}} </math> || [[chemical engineering]] ([[adsorption]] enhancement due to [[chemical reaction]])
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| | [[Hagen number]] || Hg || <math> \mathrm{Hg} = -\frac{1}{\rho}\frac{\mathrm{d} p}{\mathrm{d} x}\frac{L^3}{\nu^2} </math> || [[heat transfer]] (ratio of the [[buoyancy]] to [[viscous]] force in [[forced convection]])
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| | [[Hydraulic gradient]] || ''i'' || <math>i = \frac{\mathrm{d}h}{\mathrm{d}l} = \frac{h_2 - h_1}{\mathrm{length}}</math> || [[fluid mechanics]], [[groundwater]] flow ([[hydraulic head|pressure head]] over distance)
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| | [[Iribarren number]] || Ir || <math>\mathrm{Ir} = \frac{\tan \alpha}{\sqrt{H/L_0}}</math> || [[wave|wave mechanics]] (breaking [[surface gravity wave]]s on a slope)
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| | [[Jakob number]] || Ja ||<math>\mathrm{Ja} = \frac{c_p (T_\mathrm{s} - T_\mathrm{sat}) }{\Delta H_{\mathrm{f}} }</math>|| [[chemistry]] (ratio of [[Sensible heat|sensible]] to [[Enthalpy of vaporization|latent energy]] absorbed during liquid-vapor [[phase transition|phase change]])<ref>{{cite book |last=Incropera |first=Frank P. |title= Fundamentals of heat and mass transfer |page=376 |year=2007 |publisher=John Wiley & Sons, Inc}}</ref>
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| |-
| |
| | [[Béla Karlovitz|Karlovitz number]] || Ka || <math>\mathrm{Ka} = k t_c</math> || [[Turbulence|turbulent]] [[combustion]] (characteristic flow time times flame stretch rate)
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| |-
| |
| | [[Keulegan–Carpenter number]] || K<sub>C</sub> || <math>\mathrm{K_C} = \frac{V\,T}{L}</math> || [[fluid dynamics]] (ratio of [[drag force]] to [[inertia]] for a bluff object in [[oscillation|oscillatory]] fluid flow)
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| |-
| |
| | [[Knudsen number]] || Kn || <math>\mathrm{Kn} = \frac {\lambda}{L}</math> || [[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 || <math>\mathrm{Ku} = \frac{U_h \rho_g^{1/2}}{\left({\sigma g (\rho_l - \rho_g)}\right)^{1/4}}</math> || [[fluid mechanics]] (counter-current [[two-phase flow]])<ref>{{cite doi|10.1016/S0009-2509(01)00247-0}}</ref>
| |
| |-
| |
| | [[Laplace number]] || La || <math>\mathrm{La} = \frac{\sigma \rho L}{\mu^2}</math> || [[fluid dynamics]] ([[free convection]] within [[Miscibility|immiscible]] fluids; ratio of [[surface tension]] to [[momentum]]-transport)
| |
| |-
| |
| | [[Lewis number]] || Le || <math>\mathrm{Le} = \frac{\alpha}{D} = \frac{\mathrm{Sc}}{\mathrm{Pr}}</math> || [[heat transfer|heat]] and [[mass transfer]] (ratio of [[thermal diffusivity|thermal]] to [[mass diffusivity]])
| |
| |-
| |
| | [[Lift coefficient]] || ''C''<sub>L</sub> || <math>C_\mathrm{L} = \frac{L}{q\,S}</math> || [[aerodynamics]] ([[lift (force)|lift]] available from an [[airfoil]] at a given [[angle of attack]])
| |
| |-
| |
| | [[Lockhart–Martinelli parameter]] || <math>\chi</math> || <math>\chi = \frac{m_\ell}{m_g} \sqrt{\frac{\rho_g}{\rho_\ell}}</math> || [[two-phase flow]] (flow of [[wet gas]]es; [[liquid]] fraction)<ref>[http://www.flowprogramme.co.uk/publications/guidancenotes/GN40.pdf Lockhart–Martinelli parameter]</ref>
| |
| |-
| |
| | [[Love number]]s || ''h'', ''k'', ''l'' || || [[geophysics]] ([[solidity]] of [[earth]] and other [[planet]]s)
| |
| |-
| |
| | [[Lundquist number]] || ''S'' || <math>S = \frac{\mu_0LV_A}{\eta}</math> || [[plasma physics]] (ratio of a resistive time to an [[Alfvén wave]] crossing time in a plasma)
| |
| |-
| |
| | [[Mach number]] || M or Ma ||<math> \mathrm{M} = \frac{{v}}{{v_\mathrm{sound}}}</math> || [[gas dynamics]] ([[compressible flow]]; dimensionless [[velocity]])
| |
| |-
| |
| | [[Magnetic Reynolds number]] || R<sub>m</sub> || <math>\mathrm{R}_\mathrm{m} = \frac{U L}{\eta}</math> || [[magnetohydrodynamics]] (ratio of magnetic [[advection]] to magnetic [[diffusion]])
| |
| |-
| |
| | [[Manning formula|Manning roughness coefficient]] || ''n'' || || [[open channel flow]] (flow driven by [[gravity]])<ref>{{PDFlink|[http://www.epa.gov/ORD/NRMRL/pubs/600r01043/600R01043chap2.pdf Manning coefficient]|109 KB}}</ref>
| |
| |-
| |
| | [[Marangoni number]] || Mg || <math>\mathrm{Mg} = - {\frac{\mathrm{d}\sigma}{\mathrm{d}T}}\frac{L \Delta T}{\eta \alpha} </math> || [[fluid mechanics]] ([[Marangoni flow]]; thermal [[surface tension]] forces over [[viscosity|viscous]] forces)
| |
| |-
| |
| | [[Morton number]] || Mo || <math>\mathrm{Mo} = \frac{g \mu_c^4 \, \Delta \rho}{\rho_c^2 \sigma^3} </math> || [[fluid dynamics]] (determination of [[Liquid bubble|bubble]]/[[drop (liquid)|drop]] shape)
| |
| |-
| |
| | [[Nusselt number]] || Nu ||<math>\mathrm{Nu} =\frac{hd}{k}</math> || [[heat transfer]] (forced [[convection]]; ratio of [[convection|convective]] to [[heat conduction|conductive]] heat transfer)
| |
| |-
| |
| | [[Ohnesorge number]] || Oh || <math> \mathrm{Oh} = \frac{ \mu}{ \sqrt{\rho \sigma L }} = \frac{\sqrt{\mathrm{We}}}{\mathrm{Re}} </math> || [[fluid dynamics]] (atomization of liquids, [[Marangoni flow]])
| |
| |-
| |
| | [[Péclet number]] || Pe ||<math>\mathrm{Pe} = \frac{du\rho c_p}{k} = \mathrm{Re}\, \mathrm{Pr}</math>|| [[heat transfer]] ([[advection]]–[[diffusion]] problems; total [[momentum transfer]] to molecular heat transfer)
| |
| |-
| |
| | [[Peel number]] || ''N''<sub>P</sub> || <math>N_\mathrm{P} = \frac{\text{Restoring force}}{\text{Adhesive force}}</math> || [[coating]] ([[adhesion]] of [[microstructure]]s with [[substrate (chemistry)|substrate]])<ref>{{Cite doi|10.1109/TDMR.2003.820295}}</ref>
| |
| |-
| |
| | [[Perveance]] || ''K'' || <math>{K} = \frac{{I}}{{I_0}}\,\frac{{2}}{{\beta}^3{\gamma}^3} (1-\gamma^2f_e)</math> || [[Charged particle|charged particle transport]] (measure of the strength of space charge in a charged particle beam)
| |
| |-
| |
| | [[Pi]] || <math>\pi</math> || <math>\pi = \frac{C}{d} \approx 3.14159</math> || [[mathematics]] (ratio of a [[circle]]'s [[circumference]] to its [[diameter]])
| |
| |-
| |
| | [[Poisson's ratio]] || <math>\nu</math> || <math>\nu = -\frac{\mathrm{d}\varepsilon_\mathrm{trans}}{\mathrm{d}\varepsilon_\mathrm{axial}} </math> || [[elasticity (physics)|elasticity]] ([[Structural load|load]] in transverse and longitudinal direction)
| |
| |-
| |
| | [[Porosity]] || <math>\phi</math> || <math>\phi = \frac{V_\mathrm{V}}{V_\mathrm{T}}</math> || [[geology]], [[porous media]] (void fraction of the medium)
| |
| |-
| |
| | [[Power factor]] || ''P''/''S'' || || [[electronics]] (real power to apparent power)
| |
| |-
| |
| | [[Power number]] || ''N''<sub>''p''</sub> || <math> N_p = {P\over \rho n^3 d^5} </math> || [[electronics]] (power consumption by agitators; [[resistance force]] versus [[inertia|inertia force]])
| |
| |-
| |
| | [[Prandtl number]] || Pr ||<math>\mathrm{Pr} = \frac{\nu}{\alpha} = \frac{c_p \mu}{k}</math>|| [[heat transfer]] (ratio of [[viscosity|viscous diffusion]] rate over [[Thermal conductivity|thermal diffusion]] rate)
| |
| |-
| |
| | [[Prater number]] || ''β'' || <math>\beta = \frac{-\Delta H_r D_{TA}^e C_{AS}}{\lambda^e T_s}</math> || [[chemical reaction engineering|reaction engineering]] (ratio of heat evolution to [[heat conduction]] within a [[catalyst]] pellet)<ref>{{cite book |last1=Davis |first1=Mark E. |last2=Davis |first2=Robert J. |title=Fundamentals of Chemical Reaction Engineering |year=2012 |publisher=Dover |isbn=978-0486488554 |page=215}}</ref>
| |
| |-
| |
| | [[Pressure coefficient]] || ''C<sub>P</sub>'' || <math>C_p = {p - p_\infty \over \frac{1}{2} \rho_\infty V_\infty^2}</math> || [[aerodynamics]], [[hydrodynamics]] ([[pressure]] experienced at a point on an [[airfoil]]; dimensionless pressure variable)
| |
| |-
| |
| | [[Q factor]] || ''Q'' || || [[physics]], [[engineering]] ([[damping]] of [[oscillation|oscillator]] or [[resonator]]; [[energy]] stored versus energy lost)
| |
| |-
| |
| | [[Radian]] measure || rad ||<math>\text{arc length}/\text{radius}</math> || [[mathematics]] (measurement of planar [[angles]], 1 radian = 180/[[pi|π]] [[degree (angle)|degrees]])
| |
| |-
| |
| | [[Rayleigh number]] || Ra || <math>\mathrm{Ra}_{x} = \frac{g \beta} {\nu \alpha} (T_s - T_\infin) x^3 </math> || [[heat transfer]] ([[buoyancy]] versus [[viscous forces]] in [[free convection]])
| |
| |-
| |
| | [[Refractive index]] || ''n'' || <math>n=\frac{c}{v}</math> || [[electromagnetism]], [[optics]] ([[speed of light]] in a vacuum over speed of light in a material)
| |
| |-
| |
| | [[Relative density]] || ''RD'' || <math>RD = \frac{\rho_\mathrm{substance}}{\rho_\mathrm{reference}}</math> || [[hydrometer]]s, material comparisons (ratio of [[density]] of a material to a reference material—usually [[water]])
| |
| |-
| |
| | [[Permeability (electromagnetism)|Relative permeability]] || <math>\mu_r</math> || <math>\mu_r = \frac{\mu}{\mu_0}</math> || [[magnetostatics]] (ratio of the permeability of a specific medium to free space)
| |
| |-
| |
| | [[Relative permittivity]] || <math>\varepsilon_r</math> || <math>\varepsilon_{r} = \frac{C_{x}} {C_{0}}</math> || [[electrostatics]] (ratio of [[capacitance]] of test [[capacitor]] with [[dielectric]] material versus [[vacuum]])
| |
| |-
| |
| | [[Reynolds number]] || Re || <math>\mathrm{Re} = \frac{vL\rho}{\mu}</math> || [[fluid mechanics]] (ratio of fluid [[inertia]]l and [[viscosity|viscous]] forces)<ref name="berkley">{{Cite web|title=Table of Dimensionless Numbers |format=PDF |url=http://www.cchem.berkeley.edu/gsac/grad_info/prelims/binders/dimensionless_numbers.pdf|accessdate=2009-11-05}}</ref>
| |
| |-
| |
| | [[Richardson number]] || Ri || <math> \mathrm{Ri} = \frac{gh}{u^2} = \frac{1}{\mathrm{Fr}^2} </math> || [[fluid dynamics]] (effect of [[buoyancy]] on flow stability; ratio of [[Potential Energy|potential]] over [[kinetic energy]])<ref>[http://apollo.lsc.vsc.edu/classes/met455/notes/section4/2.html Richardson number]</ref>
| |
| |-
| |
| | [[Rockwell scale]] || – || || [[hardness|mechanical hardness]] ([[indentation hardness]] of a material)
| |
| |-
| |
| | [[Rolling resistance#Rolling resistance coefficient examples|Rolling resistance coefficient]] || ''C<sub>rr</sub>'' ||<math>C_{rr} = \frac{F}{N_f} </math>|| [[vehicle dynamics]] (ratio of [[force]] needed for motion of a [[wheel]] over the [[normal force]])
| |
| |-
| |
| | [[Roshko number]] || Ro || <math> \mathrm{Ro} = {f L^{2}\over \nu} =\mathrm{St}\,\mathrm{Re} </math> || [[fluid dynamics]] (oscillating flow, [[vortex]] [[vortex shedding|shedding]])
| |
| |-
| |
| | [[Rossby number]] || Ro || <math>\mathrm{Ro}=\frac{U}{Lf}</math> || [[geophysics]] (ratio of [[inertia]]l to [[Coriolis force]])
| |
| |-
| |
| | [[Rouse number]] || P or Z || <math>\mathrm{P} = \frac{w_s}{\kappa u_*}</math> || [[sediment transport]] (ratio of the sediment [[terminal velocity|fall velocity]] and the upwards velocity of grain)
| |
| |-
| |
| | [[Schmidt number]] || Sc || <math>\mathrm{Sc} = \frac{\nu}{D}</math> || [[mass transfer]] ([[viscosity|viscous]] over molecular [[diffusion]] rate)<ref>[http://www.ent.ohiou.edu/~hbwang/fluidynamics.htm Schmidt number]</ref>
| |
| |-
| |
| | [[Shape factor (boundary layer flow)|Shape factor]] || ''H'' || <math>H = \frac {\delta^*}{\theta}</math> || [[boundary layer flow]] (ratio of displacement thickness to momentum thickness)
| |
| |-
| |
| | [[Sherwood number]] || Sh || <math>\mathrm{Sh} = \frac{K L}{D} </math> || [[mass transfer]] ([[forced convection]]; ratio of [[convection|convective]] to [[diffusion|diffusive]] mass transport)
| |
| |-
| |
| | [[Shields parameter]] || <math>\tau_*</math> or <math>\theta</math> || <math>\tau_{\ast} = \frac{\tau}{(\rho_s - \rho) g D}</math> || [[sediment transport]] (threshold of [[sediment transport|sediment movement]] due to fluid motion; dimensionless [[shear stress]])
| |
| |-
| |
| | [[Sommerfeld number]] || S || <math> \mathrm{S} = \left( \frac{r}{c} \right)^2 \frac {\mu N}{P}</math> || [[hydrodynamic lubrication]] (boundary [[lubrication]])<ref>[http://epubl.luth.se/avslutade/0348-8373/41/ Sommerfeld number]</ref>
| |
| |-
| |
| | [[Specific gravity]] || ''SG'' || || (same as [[Relative density]])
| |
| |-
| |
| | [[Stanton number]] || St || <math>\mathrm{St} = \frac{h}{c_p \rho V} = \frac{\mathrm{Nu}}{\mathrm{Re}\,\mathrm{Pr}} </math> || [[heat transfer]] and [[fluid dynamics]] (forced [[convection]])
| |
| |-
| |
| | [[Stefan number]] || Ste ||<math>\mathrm{Ste} = \frac{c_p \Delta T}{L}</math>|| [[phase transition|phase change]], [[thermodynamics]] (ratio of [[sensible heat]] to [[latent heat]])
| |
| |-
| |
| | [[Stokes number]] || Stk or S<sub>k</sub> ||<math>\mathrm{Stk} = \frac{\tau U_o}{d_c}</math>|| [[Suspension (chemistry)|particles suspensions]] (ratio of characteristic [[time]] of particle to time of flow)
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| |-
| |
| | [[Strain (materials science)|Strain]] || <math>\epsilon</math> ||<math>\epsilon = \cfrac{\partial{F}}{\partial{X}} - 1</math>|| [[materials science]], [[elasticity (physics)|elasticity]] (displacement between particles in the body relative to a reference length)
| |
| |-
| |
| | [[Strouhal number]] || St or Sr ||<math>\mathrm{St} = {\omega L\over v} </math>|| [[fluid dynamics]] (continuous and pulsating flow; nondimensional [[frequency]])<ref>[http://www.engineeringtoolbox.com/strouhal-number-d_582.html Strouhal number], Engineering Toolbox</ref>
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| |-
| |
| | [[Stuart number]] || N || <math> \mathrm{N} = \frac {B^2 L_{c} \sigma}{\rho U} = \frac{\mathrm{Ha}^2}{\mathrm{Re}} </math> || [[magnetohydrodynamics]] (ratio of [[electromagnetic force|electromagnetic]] to inertial forces)
| |
| |-
| |
| | [[Taylor number]] || Ta ||<math> \mathrm{Ta} = \frac{4\Omega^2 R^4}{\nu^2}</math>|| [[fluid dynamics]] (rotating fluid flows; inertial forces due to [[rotation]] of a [[fluid]] versus [[viscosity|viscous forces]])
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| |-
| |
| | [[Ursell number]] || U ||<math>\mathrm{U} = \frac{H\, \lambda^2}{h^3}</math>|| [[wave|wave mechanics]] (nonlinearity of [[ocean surface wave|surface gravity waves]] on a shallow fluid layer)
| |
| |-
| |
| | [[Vadasz number]] || Va ||<math>\mathrm{Va} = \frac{\phi\, \mathrm{Pr}}{\mathrm{Da}}</math>|| [[porous medium|porous media]] (governs the effects of [[porosity]] <math>\phi</math>, the [[Prandtl number]] and the [[Darcy number]] on flow in a [[porous medium]]) <ref>{{Cite doi|10.1098/rspa.2000.0657}}</ref>
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| |-
| |
| | [[van 't Hoff factor]] || ''i'' ||<math> i = 1 + \alpha (n - 1)</math>|| [[quantitative analysis (chemistry)|quantitative analysis]] ([[Freezing-point depression|''K''<sub>f</sub>]] and [[Boiling point elevation|''K''<sub>b</sub>]])
| |
| |-
| |
| | [[Wallis parameter]] || ''j''<sup>*</sup> ||<math>j^* = R \left( \frac{\omega \rho}{\mu} \right)^\frac{1}{2}</math>|| [[multiphase flow]]s (nondimensional [[superficial velocity]])<ref>{{cite doi|10.1016/S0029-5493(99)00005-9}}</ref>
| |
| |-
| |
| | [[Weaver flame speed number]] || Wea ||<math>\mathrm{Wea} = \frac{w}{w_\mathrm{H}} 100</math> || [[combustion]] ([[laminar flow|laminar]] burning [[velocity]] relative to [[hydrogen]] gas)<ref>{{Cite doi|10.1016/B978-0-12-416013-2.00008-7}}</ref>
| |
| |-
| |
| | [[Weber number]] || We ||<math>\mathrm{We} = \frac{\rho v^2 l}{\sigma}</math>|| [[multiphase flow]] (strongly curved surfaces; ratio of [[inertia]] to [[surface tension]])
| |
| |-
| |
| | [[Weissenberg number]] || Wi ||<math>\mathrm{Wi} = \dot{\gamma} \lambda </math>|| [[viscoelastic]] flows ([[shear rate]] times the relaxation time)<ref>[http://physics.ucsd.edu/~des/Shear1999.pdf Weissenberg number]</ref>
| |
| |-
| |
| | [[Womersley number]] || <math>\alpha</math> ||<math>\alpha = R \left( \frac{\omega \rho}{\mu} \right)^\frac{1}{2}</math>|| [[biofluid mechanics]] (continuous and pulsating flows; ratio of [[pulsatile flow]] [[frequency]] to [[viscosity|viscous effects]])<ref>[http://www.seas.upenn.edu/courses/belab/LabProjects/2001/be310s01m2.doc Womersley number]</ref>
| |
| |}
| |
| | |
| ==See also==
| |
| * [[Similitude (model)]]
| |
| * [[Orders of magnitude (numbers)]]
| |
| * [[Dimensional analysis]]
| |
| * [[Dimensionless physical constant]]
| |
| * [[Normalization (statistics)]] and [[standardized moment]], the analogous concepts in [[statistics]]
| |
| * [[Buckingham π theorem]]
| |
| | |
| ==References==
| |
| {{Reflist|2}}
| |
| | |
| ==External links==
| |
| * [[John Baez]], "[http://math.ucr.edu/home/baez/constants.html How Many Fundamental Constants Are There?]"
| |
| * Huba, J. D., 2007, ''[http://www.ipp.mpg.de/~dpc/nrl/ NRL Plasma Formulary: Dimensionless Numbers of Fluid Mechanics.]'' [[United States Naval Research Laboratory|Naval Research Laboratory]]. p. [http://www.ipp.mpg.de/~dpc/nrl/23.html 23], [http://www.ipp.mpg.de/~dpc/nrl/24.html 24], [http://www.ipp.mpg.de/~dpc/nrl/25.html 25]
| |
| * Sheppard, Mike, 2007, "[http://www.mit.edu/~mi22295/constants/constants.html Systematic Search for Expressions of Dimensionless Constants using the NIST database of Physical Constants.]"
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| {{NonDimFluMech}}
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| {{DEFAULTSORT:Dimensionless Quantity}}
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| [[Category:Physical constants]]
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| [[Category:Dimensionless numbers| ]]
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