Dimensionless
In dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension. It is thus a "pure" number, and as such always has a dimension of 1.^{[1]} Dimensionless quantities are widely used in mathematics, physics, engineering, economics, and in everyday life (such as in counting). Numerous wellknown quantities, such as π, e, and φ, are dimensionless. By contrast, nondimensionless quantities are measured in units of length, area, time, etc.
Dimensionless quantities are often defined as products or ratios of quantities that are not dimensionless, but whose dimensions cancel out when their 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.
Contents
Properties
 Even though a dimensionless quantity has no physical dimension associated with it, it can still have dimensionless units. To show the quantity being measured (for example 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 years over meters). This may be the case when calculating slopes 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^{−6}), ppb (= 10^{−9}), ppt (= 10^{−12}) and angle units (degrees, radians, grad). Units of number such as the dozen and the gross are also dimensionless.
 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 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 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.
Buckingham π theorem
Another consequence of the Buckingham π theorem of dimensional analysis is that the functional dependence between a certain number (say, n) of variables can be reduced by the number (say, k) of independent dimensions occurring in those variables to give a set of p = n − k independent, dimensionless quantities. For the purposes of the experimenter, different systems that share the same description by dimensionless quantity are equivalent.
Example
The 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.
Those n = 5 variables are built up from k = 3 dimensions:
 Length: L (m)
 Time: T (s)
 Mass: M (kg)
According to the πtheorem, 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:
 Reynolds number (a dimensionless number describing the fluid flow regime)
 Power number (describing the stirrer and also involves the density of the fluid)
Standards efforts
The International Committee for Weights and Measures contemplated defining the unit of 1 as the 'uno', but the idea was dropped.^{[2]}^{[3]}^{[4]}
Examples
 Consider this example: Sarah says, "Out of every 10 apples I gather, 1 is rotten." The rottentogathered ratio is (1 apple) / (10 apples) = 0.1 = 10%, which is a dimensionless quantity.
 Plane angles – 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 radians as the unit, the length that is compared is the length of the radius of the circle. When using degree as the units, the arc's length is compared to 1/360 of the circumference of the circle.
 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, lightyears, etc.), as long as the same unit is used for both.
Dimensionless physical constants
Certain fundamental physical constants, 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, charge, and temperature are chosen. The resulting system of units is known as the natural units. However, not all physical constants 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:
 α ≈ 1/137.036, the fine structure constant which is the coupling constant for the electromagnetic interaction;
 β (or μ) ≈ 1836, the protontoelectron 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;
 α_{s}, the coupling constant for the strong force;
 α_{G} ≈ 1.75×10^{−45}, the gravitational coupling constant.
List of dimensionless quantities
All numbers are dimensionless quantities. Certain dimensionless quantities of some importance are given below:
Name  Standard symbol  Definition  Field of application 

Abbe number  V  $V\; =\; \backslash frac\{\; n\_d\; \; 1\; \}\{\; n\_F\; \; n\_C\; \}$  optics (dispersion in optical materials) 
Activity coefficient  $\backslash gamma$  $\backslash gamma=\; \backslash frac$Template:A  chemistry (Proportion of "active" molecules or atoms) 
Albedo  $\backslash alpha$  $\{\backslash alpha\}=\; (1D)\; \backslash bar\; \backslash alpha(\backslash theta\_i)\; +\; D\; \backslash bar\{\; \backslash bar\; \backslash alpha\}$  climatology, astronomy (reflectivity of surfaces or bodies) 
Archimedes number  Ar  $\backslash mathrm\{Ar\}\; =\; \backslash frac\{g\; L^3\; \backslash rho\_\backslash ell\; (\backslash rho\; \; \backslash rho\_\backslash ell)\}\{\backslash mu^2\}$  fluid mechanics (motion of fluids due to density differences) 
Arrhenius number  $\backslash alpha$  $\backslash alpha\; =\; \backslash frac\{E\_a\}\{RT\}$  chemistry (ratio of activation energy to thermal energy)^{[5]} 
Atomic weight  M  chemistry (mass of atom over one atomic mass unit, u, where carbon12 is exactly 12 u)  
Atwood number  A  $\backslash mathrm\{A\}\; =\; \backslash frac\{\backslash rho\_1\; \; \backslash rho\_2\}\; \{\backslash rho\_1\; +\; \backslash rho\_2\}$  fluid mechanics (onset of instabilities in fluid mixtures due to density differences) 
Bagnold number  Ba  $\backslash mathrm\{Ba\}\; =\; \backslash frac\{\backslash rho\; d^2\; \backslash lambda^\{1/2\}\; \backslash gamma\}\{\backslash mu\}$  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  $\backslash mathrm\{Be\}\; =\; \backslash frac\{\backslash Delta\; P\; L^2\}\; \{\backslash mu\; \backslash alpha\}$  fluid mechanics (dimensionless pressure drop along a channel)^{[7]} 
Bejan number (thermodynamics) 
Be  $\backslash mathrm\{Be\}\; =\; \backslash frac\{\backslash dot\; S\text{'}\_\{\backslash mathrm\{gen\},\backslash ,\; \backslash Delta\; T\}\}\{\backslash dot\; S\text{'}\_\{\backslash mathrm\{gen\},\backslash ,\; \backslash Delta\; T\}+\; \backslash dot\; S\text{'}\_\{\backslash mathrm\{gen\},\backslash ,\; \backslash Delta\; p\}\}$  thermodynamics (ratio of heat transfer irreversibility to total irreversibility due to heat transfer and fluid friction)^{[8]} 
Bingham number  Bm  $\backslash mathrm\{Bm\}\; =\; \backslash frac\{\; \backslash tau\_y\; L\; \}\{\; \backslash mu\; V\; \}$  fluid mechanics, rheology (ratio of yield stress to viscous stress)^{[5]} 
Biot number  Bi  $\backslash mathrm\{Bi\}\; =\; \backslash frac\{h\; L\_C\}\{k\_b\}$  heat transfer (surface vs. volume conductivity of solids) 
Blake number  Bl or B  $\backslash mathrm\{B\}\; =\; \backslash frac\{u\; \backslash rho\}\{\backslash mu\; (1\; \; \backslash epsilon)\; D\}$  geology, fluid mechanics, porous media (inertial over viscous forces in fluid flow through porous media) 
Bodenstein number  Bo or Bd  $\backslash mathrm\{Bo\}\; =\; vL/\backslash mathcal\{D\}\; =\; \backslash mathrm\{Re\}\backslash ,\; \backslash mathrm\{Sc\}$  chemistry (residencetime distribution; similar to the axial mass transfer Peclet number)^{[9]} 
Bond number  Bo  $\backslash mathrm\{Bo\}\; =\; \backslash frac\{\backslash rho\; a\; L^2\}\{\backslash gamma\}$  geology, fluid mechanics, porous media (buoyant versus capilary forces, similar to the Eötvös number) ^{[10]} 
Brinkman number  Br  $\backslash mathrm\{Br\}\; =\; \backslash frac\; \{\backslash mu\; U^2\}\{\backslash kappa\; (T\_w\; \; T\_0)\}$  heat transfer, fluid mechanics (conduction from a wall to a viscous fluid) 
Brownell–Katz number  N_{BK}  $\backslash mathrm\{N\}\_\backslash mathrm\{BK\}\; =\; \backslash frac\{u\; \backslash mu\}\{k\_\backslash mathrm\{rw\}\backslash sigma\}$  fluid mechanics (combination of capillary number and Bond number) ^{[11]} 
Capillary number  Ca  $\backslash mathrm\{Ca\}\; =\; \backslash frac\{\backslash mu\; V\}\{\backslash gamma\}$  porous media (viscous forces versus surface tension) 
Chandrasekhar number  Q  $\backslash mathrm\{Q\}\; =\; \backslash frac$  mechanics (the level of damping in a system) 
Darcy friction factor  C_{f} or f_{D}  fluid mechanics (fraction of pressure losses due to friction in a pipe; four times the Fanning friction factor)  
Darcy number  Da  $\backslash mathrm\{Da\}\; =\; \backslash frac\{K\}\{d^2\}$  porous media (ratio of permeability to crosssectional area) 
Dean number  D  $\backslash mathrm\{D\}\; =\; \backslash frac\{\backslash rho\; V\; d\}\{\backslash mu\}\; \backslash left(\; \backslash frac\{d\}\{2\; R\}\; \backslash right)^\{1/2\}$  turbulent flow (vortices in curved ducts) 
Deborah number  De  $\backslash mathrm\{De\}\; =\; \backslash frac\{t\_\backslash mathrm\{c\}\}\{t\_\backslash mathrm\{p\}\}$  rheology (viscoelastic fluids) 
Decibel  dB  acoustics, electronics, control theory (ratio of two intensities or powers of a wave)  
Drag coefficient  c_{d}  $c\_\backslash mathrm\{d\}\; =\; \backslash dfrac\{2\; F\_\backslash mathrm\{d\}\}\{\backslash rho\; v^2\; A\}\backslash ,\; ,$  aeronautics, fluid dynamics (resistance to fluid motion) 
Dukhin number  Du  $\backslash mathrm\{Du\}\; =\; \backslash frac\{\backslash kappa^\{\backslash sigma\}\}\{L\; \backslash lambda\}$  optics (slit diffraction)^{[17]} 
Froude number  Fr  $\backslash mathrm\{Fr\}\; =\; \backslash frac\{v\}\{\backslash sqrt\{g\backslash ell\}\}$  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)^{[18]}  
Galilei number  Ga  $\backslash mathrm\{Ga\}\; =\; \backslash frac\{g\backslash ,\; L^3\}\{\backslash nu^2\}$  fluid mechanics (gravitational over viscous forces) 
Golden ratio  $\backslash varphi$  $\backslash varphi\; =\; \backslash frac\{1+\backslash sqrt\{5\}\}\{2\}\; \backslash approx\; 1.61803$  mathematics, aesthetics (long side length of selfsimilar rectangle) 
Görtler number  G  $\backslash mathrm\{G\}\; =\; \backslash frac\{U\_e\; \backslash theta\}\{\backslash nu\}\; \backslash left(\; \backslash frac\{\backslash theta\}\{R\}\; \backslash right)^\{1/2\}$  fluid dynamics (boundary layer flow along a concave wall) 
Graetz number  Gz  $\backslash mathrm\{Gz\}\; =\; \{D\_H\; \backslash over\; L\}\; \backslash mathrm\{Re\}\backslash ,\; \backslash mathrm\{Pr\}$  heat transfer, fluid mechanics (laminar flow through a conduit; also used in mass transfer) 
Grashof number  Gr  $\backslash mathrm\{Gr\}\_L\; =\; \backslash frac\{g\; \backslash beta\; (T\_s\; \; T\_\backslash infty\; )\; L^3\}\{\backslash nu\; ^2\}$  heat transfer, natural convection (ratio of the buoyancy to viscous force) 
Gravitational coupling constant  $\backslash alpha\_G$  $\backslash alpha\_G=\backslash frac\{Gm\_e^2\}\{\backslash hbar\; c\}$  gravitation (attraction between two massy elementary particles; analogous to the Fine structure constant) 
Hatta number  Ha  $\backslash mathrm\{Ha\}\; =\; \backslash frac\{N\_\{\backslash mathrm\{A\}0\}\}\{N\_\{\backslash mathrm\{A\}0\}^\{\backslash mathrm\{phys\}\}\}$  chemical engineering (adsorption enhancement due to chemical reaction) 
Hagen number  Hg  $\backslash mathrm\{Hg\}\; =\; \backslash frac\{1\}\{\backslash rho\}\backslash frac\{\backslash mathrm\{d\}\; p\}\{\backslash mathrm\{d\}\; x\}\backslash frac\{L^3\}\{\backslash nu^2\}$  heat transfer (ratio of the buoyancy to viscous force in forced convection) 
Hydraulic gradient  i  $i\; =\; \backslash frac\{\backslash mathrm\{d\}h\}\{\backslash mathrm\{d\}l\}\; =\; \backslash frac\{h\_2\; \; h\_1\}\{\backslash mathrm\{length\}\}$  fluid mechanics, groundwater flow (pressure head over distance) 
Iribarren number  Ir  $\backslash mathrm\{Ir\}\; =\; \backslash frac\{\backslash tan\; \backslash alpha\}\{\backslash sqrt\{H/L\_0\}\}$  wave mechanics (breaking surface gravity waves on a slope) 
Jakob Number  Ja  $\backslash mathrm\{Ja\}\; =\; \backslash frac\{c\_p\; (T\_\backslash mathrm\{s\}\; \; T\_\backslash mathrm\{sat\})\; \}\{\backslash Delta\; H\_\{\backslash mathrm\{f\}\}\; \}$  chemistry (ratio of sensible to latent energy absorbed during liquidvapor phase change)^{[19]} 
Karlovitz number  Ka  $\backslash mathrm\{Ka\}\; =\; k\; t\_c$  turbulent combustion (characteristic flow time times flame stretch rate) 
Keulegan–Carpenter number  K_{C}  $\backslash mathrm\{K\_C\}\; =\; \backslash frac\{V\backslash ,T\}\{L\}$  fluid dynamics (ratio of drag force to inertia for a bluff object in oscillatory fluid flow) 
Knudsen number  Kn  $\backslash mathrm\{Kn\}\; =\; \backslash frac\; \{\backslash lambda\}\{L\}$  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  $\backslash mathrm\{Ku\}\; =\; \backslash frac\{U\_h\; \backslash rho\_g^\{1/2\}\}\{\backslash left(\{\backslash sigma\; g\; (\backslash rho\_l\; \; \backslash rho\_g)\}\backslash right)^\{1/4\}\}$  fluid mechanics (countercurrent twophase flow)^{[20]} 
Laplace number  La  $\backslash mathrm\{La\}\; =\; \backslash frac\{\backslash sigma\; \backslash rho\; L\}\{\backslash mu^2\}$  fluid dynamics (free convection within immiscible fluids; ratio of surface tension to momentumtransport) 
Lewis number  Le  $\backslash mathrm\{Le\}\; =\; \backslash frac\{\backslash alpha\}\{D\}\; =\; \backslash frac\{\backslash mathrm\{Sc\}\}\{\backslash mathrm\{Pr\}\}$  heat and mass transfer (ratio of thermal to mass diffusivity) 
Lift coefficient  C_{L}  $C\_\backslash mathrm\{L\}\; =\; \backslash frac\{L\}\{q\backslash ,S\}$  aerodynamics (lift available from an airfoil at a given angle of attack) 
Lockhart–Martinelli parameter  $\backslash chi$  $\backslash chi\; =\; \backslash frac\{m\_\backslash ell\}\{m\_g\}\; \backslash sqrt\{\backslash frac\{\backslash rho\_g\}\{\backslash rho\_\backslash ell\}\}$  twophase flow (flow of wet gases; liquid fraction)^{[21]} 
Love numbers  h, k, l  geophysics (solidity of earth and other planets)  
Lundquist number  S  $S\; =\; \backslash frac\{\backslash mu\_0LV\_A\}\{\backslash eta\}$  plasma physics (ratio of a resistive time to an Alfvén wave crossing time in a plasma) 
Mach number  M or Ma  $\backslash mathrm\{M\}\; =\; \backslash frac$}  gas dynamics (compressible flow; dimensionless velocity) 
Magnetic Reynolds number  R_{m}  $\backslash mathrm\{R\}\_\backslash mathrm\{m\}\; =\; \backslash frac\{U\; L\}\{\backslash eta\}$  magnetohydrodynamics (ratio of magnetic advection to magnetic diffusion) 
Manning roughness coefficient  n  open channel flow (flow driven by gravity)^{[22]}  
Marangoni number  Mg  $\backslash mathrm\{Mg\}\; =\; \; \{\backslash frac\{\backslash mathrm\{d\}\backslash sigma\}\{\backslash mathrm\{d\}T\}\}\backslash frac\{L\; \backslash Delta\; T\}\{\backslash eta\; \backslash alpha\}$  fluid mechanics (Marangoni flow; thermal surface tension forces over viscous forces) 
Morton number  Mo  $\backslash mathrm\{Mo\}\; =\; \backslash frac\{g\; \backslash mu\_c^4\; \backslash ,\; \backslash Delta\; \backslash rho\}\{\backslash rho\_c^2\; \backslash sigma^3\}$  fluid dynamics (determination of bubble/drop shape) 
Nusselt number  Nu  $\backslash mathrm\{Nu\}\; =\backslash frac\{hd\}\{k\}$  heat transfer (forced convection; ratio of convective to conductive heat transfer) 
Ohnesorge number  Oh  $\backslash mathrm\{Oh\}\; =\; \backslash frac\{\; \backslash mu\}\{\; \backslash sqrt\{\backslash rho\; \backslash sigma\; L\; \}\}\; =\; \backslash frac\{\backslash sqrt\{\backslash mathrm\{We\}\}\}\{\backslash mathrm\{Re\}\}$  fluid dynamics (atomization of liquids, Marangoni flow) 
Péclet number  Pe  $\backslash mathrm\{Pe\}\; =\; \backslash frac\{du\backslash rho\; c\_p\}\{k\}\; =\; \backslash mathrm\{Re\}\backslash ,\; \backslash mathrm\{Pr\}$  heat transfer (advection–diffusion problems; total momentum transfer to molecular heat transfer) 
Peel number  N_{P}  $N\_\backslash mathrm\{P\}\; =\; \backslash frac\{\backslash text\{Restoring\; force\}\}\{\backslash text\{Adhesive\; force\}\}$  coating (adhesion of microstructures with substrate)^{[23]} 
Perveance  K  $\{K\}\; =\; \backslash frac$Template:I 0\,\fracTemplate:2{\mathrm{d}\varepsilon_\mathrm{axial}}  elasticity (load in transverse and longitudinal direction) 
Porosity  $\backslash phi$  $\backslash phi\; =\; \backslash frac\{V\_\backslash mathrm\{V\}\}\{V\_\backslash mathrm\{T\}\}$  geology, porous media (void fraction of the medium) 
Power factor  P/S  electronics (real power to apparent power)  
Power number  N_{p}  $N\_\{p\}\; =\; \{P\backslash over\; \backslash rho\; n^\{3\}\; d^\{5\}\}$  electronics (power consumption by agitators; resistance force versus inertia force) 
Prandtl number  Pr  $\backslash mathrm\{Pr\}\; =\; \backslash frac\{\backslash nu\}\{\backslash alpha\}\; =\; \backslash frac\{c\_p\; \backslash mu\}\{k\}$  heat transfer (ratio of viscous diffusion rate over thermal diffusion rate) 
Prater number  β  $\backslash beta\; =\; \backslash frac\{\backslash Delta\; H\_r\; D\_\{TA\}^e\; C\_\{AS\}\}\{\backslash lambda^e\; T\_s\}$  reaction engineering (ratio of heat evolution to heat conduction within a catalyst pellet)^{[24]} 
Pressure coefficient  C_{P}  $C\_p\; =\; \{p\; \; p\_\backslash infty\; \backslash over\; \backslash frac\{1\}\{2\}\; \backslash rho\_\backslash infty\; V\_\backslash infty^2\}$  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  $\backslash text\{arc\; length\}/\backslash text\{radius\}$  mathematics (measurement of planar angles, 1 radian = 180/π degrees) 
Rayleigh number  Ra  $\backslash mathrm\{Ra\}\_\{x\}\; =\; \backslash frac\{g\; \backslash beta\}\; \{\backslash nu\; \backslash alpha\}\; (T\_s\; \; T\_\backslash infin)\; x^3$  heat transfer (buoyancy versus viscous forces in free convection) 
Refractive index  n  $n=\backslash frac\{c\}\{v\}$  electromagnetism, optics (speed of light in a vacuum over speed of light in a material) 
Relative density  RD  $RD\; =\; \backslash frac\{\backslash rho\_\backslash mathrm\{substance\}\}\{\backslash rho\_\backslash mathrm\{reference\}\}$  hydrometers, material comparisons (ratio of density of a material to a reference material—usually water) 
Relative permeability  $\backslash mu\_r$  $\backslash mu\_r\; =\; \backslash frac\{\backslash mu\}\{\backslash mu\_0\}$  magnetostatics (ratio of the permeability of a specific medium to free space) 
Relative permittivity  $\backslash varepsilon\_r$  $\backslash varepsilon\_\{r\}\; =\; \backslash frac\{C\_\{x\}\}\; \{C\_\{0\}\}$  electrostatics (ratio of capacitance of test capacitor with dielectric material versus vacuum) 
Reynolds number  Re  $\backslash mathrm\{Re\}\; =\; \backslash frac\{vL\backslash rho\}\{\backslash mu\}$  fluid mechanics (ratio of fluid inertial and viscous forces)^{[5]} 
Richardson number  Ri  $\backslash mathrm\{Ri\}\; =\; \backslash frac\{gh\}\{u^2\}\; =\; \backslash frac\{1\}\{\backslash mathrm\{Fr\}^2\}$  fluid dynamics (effect of buoyancy on flow stability; ratio of potential over kinetic energy)^{[25]} 
Rockwell scale  –  mechanical hardness (indentation hardness of a material)  
Rolling resistance coefficient  C_{rr}  $C\_\{rr\}\; =\; \backslash frac\{F\}\{N\_f\}$  vehicle dynamics (ratio of force needed for motion of a wheel over the normal force) 
Roshko number  Ro  $\backslash mathrm\{Ro\}\; =\; \{f\; L^\{2\}\backslash over\; \backslash nu\}\; =\backslash mathrm\{St\}\backslash ,\backslash mathrm\{Re\}$  fluid dynamics (oscillating flow, vortex shedding) 
Rossby number  Ro  $\backslash mathrm\{Ro\}=\backslash frac\{U\}\{Lf\}$  geophysics (ratio of inertial to Coriolis force) 
Rouse number  P or Z  $\backslash mathrm\{P\}\; =\; \backslash frac\{w\_s\}\{\backslash kappa\; u\_*\}$  sediment transport (ratio of the sediment fall velocity and the upwards velocity of grain) 
Schmidt number  Sc  $\backslash mathrm\{Sc\}\; =\; \backslash frac\{\backslash nu\}\{D\}$  mass transfer (viscous over molecular diffusion rate)^{[26]} 
Shape factor  H  $H\; =\; \backslash frac\; \{\backslash delta^*\}\{\backslash theta\}$  boundary layer flow (ratio of displacement thickness to momentum thickness) 
Sherwood number  Sh  $\backslash mathrm\{Sh\}\; =\; \backslash frac\{K\; L\}\{D\}$  mass transfer (forced convection; ratio of convective to diffusive mass transport) 
Shields parameter  $\backslash tau\_*$ or $\backslash theta$  $\backslash tau\_\{\backslash ast\}\; =\; \backslash frac\{\backslash tau\}\{(\backslash rho\_s\; \; \backslash rho)\; g\; D\}$  sediment transport (threshold of sediment movement due to fluid motion; dimensionless shear stress) 
Sommerfeld number  S  $\backslash mathrm\{S\}\; =\; \backslash left(\; \backslash frac\{r\}\{c\}\; \backslash right)^2\; \backslash frac\; \{\backslash mu\; N\}\{P\}$  hydrodynamic lubrication (boundary lubrication)^{[27]} 
Specific gravity  SG  (same as Relative density)  
Stanton number  St  $\backslash mathrm\{St\}\; =\; \backslash frac\{h\}\{c\_p\; \backslash rho\; V\}\; =\; \backslash frac\{\backslash mathrm\{Nu\}\}\{\backslash mathrm\{Re\}\backslash ,\backslash mathrm\{Pr\}\}$  heat transfer and fluid dynamics (forced convection) 
Stefan number  Ste  $\backslash mathrm\{Ste\}\; =\; \backslash frac\{c\_p\; \backslash Delta\; T\}\{L\}$  phase change, thermodynamics (ratio of sensible heat to latent heat) 
Stokes number  Stk or S_{k}  $\backslash mathrm\{Stk\}\; =\; \backslash frac\{\backslash tau\; U\_o\}\{d\_c\}$  particles suspensions (ratio of characteristic time of particle to time of flow) 
Strain  $\backslash epsilon$  $\backslash epsilon\; =\; \backslash cfrac\{\backslash partial\{F\}\}\{\backslash partial\{X\}\}\; \; 1$  materials science, elasticity (displacement between particles in the body relative to a reference length) 
Strouhal number  St or Sr  $\backslash mathrm\{St\}\; =\; \{\backslash omega\; L\backslash over\; v\}$  fluid dynamics (continuous and pulsating flow; nondimensional frequency)^{[28]} 
Stuart number  N  $\backslash mathrm\{N\}\; =\; \backslash frac\; \{B^2\; L\_\{c\}\; \backslash sigma\}\{\backslash rho\; U\}\; =\; \backslash frac\{\backslash mathrm\{Ha\}^2\}\{\backslash mathrm\{Re\}\}$  magnetohydrodynamics (ratio of electromagnetic to inertial forces) 
Taylor number  Ta  $\backslash mathrm\{Ta\}\; =\; \backslash frac\{4\backslash Omega^2\; R^4\}\{\backslash nu^2\}$  fluid dynamics (rotating fluid flows; inertial forces due to rotation of a fluid versus viscous forces) 
Ursell number  U  $\backslash mathrm\{U\}\; =\; \backslash frac\{H\backslash ,\; \backslash lambda^2\}\{h^3\}$  wave mechanics (nonlinearity of surface gravity waves on a shallow fluid layer) 
Vadasz number  Va  $\backslash mathrm\{Va\}\; =\; \backslash frac\{\backslash phi\backslash ,\; \backslash mathrm\{Pr\}\}\{\backslash mathrm\{Da\}\}$  porous media (governs the effects of porosity $\backslash phi$, the Prandtl number and the Darcy number on flow in a porous medium) ^{[29]} 
van 't Hoff factor  i  $i\; =\; 1\; +\; \backslash alpha\; (n\; \; 1)$  quantitative analysis (K_{f} and K_{b}) 
Wallis parameter  j^{*}  $j^*\; =\; R\; \backslash left(\; \backslash frac\{\backslash omega\; \backslash rho\}\{\backslash mu\}\; \backslash right)^\backslash frac\{1\}\{2\}$  multiphase flows (nondimensional superficial velocity)^{[30]} 
Weaver flame speed number  Wea  $\backslash mathrm\{Wea\}\; =\; \backslash frac\{w\}\{w\_\backslash mathrm\{H\}\}\; 100$  combustion (laminar burning velocity relative to hydrogen gas)^{[31]} 
Weber number  We  $\backslash mathrm\{We\}\; =\; \backslash frac\{\backslash rho\; v^2\; l\}\{\backslash sigma\}$  multiphase flow (strongly curved surfaces; ratio of inertia to surface tension) 
Weissenberg number  Wi  $\backslash mathrm\{Wi\}\; =\; \backslash dot\{\backslash gamma\}\; \backslash lambda$  viscoelastic flows (shear rate times the relaxation time)^{[32]} 
Womersley number  $\backslash alpha$  $\backslash alpha\; =\; R\; \backslash left(\; \backslash frac\{\backslash omega\; \backslash rho\}\{\backslash mu\}\; \backslash right)^\backslash frac\{1\}\{2\}$  biofluid mechanics (continuous and pulsating flows; ratio of pulsatile flow frequency to viscous effects)^{[33]} 
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
External links
 How Many Fundamental Constants Are There?"
 Huba, J. D., 2007, 25
 Sheppard, Mike, 2007, "Systematic Search for Expressions of Dimensionless Constants using the NIST database of Physical Constants."
