Dimensionless quantity

In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is applicable. It is thus a bare number, and is therefore also known as a quantity of dimension one.^[1] Dimensionless quantities are widely used in many fields, such as mathematics, physics, engineering, and economics. Numerous well-known quantities, such as $π$ , $e$ , and $φ$ , are dimensionless. By contrast, examples of quantities with dimensions are length, time, and speed, which are measured in dimensional units, such as meter, second and meter/second.

Dimensionless quantities are often obtained as products or ratios of quantities that are not dimensionless, but whose dimensions cancel in the mathematical operation. This is the case, for instance, with the engineering strain, a measure of deformation. It is defined as change in length, divided by initial length, but because these quantities both have dimensions L (length), the result is a dimensionless quantity.

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), angle units (degrees, radians, grad), dalton and mole. 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 one 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 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:

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

Proportional occurrences, e.g. 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.
Radian measure of 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, light-years, etc.), as long as the same unit is used for both.
Relative density
Relative atomic mass - measured in daltons
Amount of substance - measured in moles as ratio between a given number of particles and Avogadro number or as ratio of mass and molar mass.
Reynolds number is commonly used in fluid mechanics to characterize flow, incorporating both properties of the fluid and the flow. It is interpreted as the ratio of inertial forces to viscous forces and can indicate flow regime as well as correlate to frictional heating in application to flow in pipes
Cost of transport is the efficiency in moving from one place to another

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 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;
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	$V = \frac{ n_d - 1 }{ n_F - n_C }$	optics (dispersion in optical materials)
Activity coefficient	$\gamma$	$\gamma= \frac {{a}}{{x}}$	chemistry (Proportion of "active" molecules or atoms)
Albedo	$\alpha$	$\alpha= (1-D) \bar \alpha(\theta_i) + D \bar{ \bar \alpha}$	climatology, astronomy (reflectivity of surfaces or bodies)
Archimedes number	Ar	$\mathrm{Ar} = \frac{g L^3 \rho_\ell (\rho - \rho_\ell)}{\mu^2}$	fluid mechanics (motion of fluids due to density differences)
Arrhenius number	$\alpha$	$\alpha = \frac{E_a}{RT}$	chemistry (ratio of activation energy to thermal energy)^[5]
Atomic weight	M		chemistry (mass of atom over one atomic mass unit, u, where carbon-12 is exactly 12 u)
Atwood number	A	$\mathrm{A} = \frac{\rho_1 - \rho_2} {\rho_1 + \rho_2}$	fluid mechanics (onset of instabilities in fluid mixtures due to density differences)
Bagnold number	Ba	$\mathrm{Ba} = \frac{\rho d^2 \lambda^{1/2} \gamma}{\mu}$	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	$\mathrm{Be} = \frac{\Delta P L^2} {\mu \alpha}$	fluid mechanics (dimensionless pressure drop along a channel)^[7]
Bejan number (thermodynamics)	Be	$\mathrm{Be} = \frac{\dot S'_{\mathrm{gen},\, \Delta T}}{\dot S'_{\mathrm{gen},\, \Delta T}+ \dot S'_{\mathrm{gen},\, \Delta p}}$	thermodynamics (ratio of heat transfer irreversibility to total irreversibility due to heat transfer and fluid friction)^[8]
Bingham number	Bm	$\mathrm{Bm} = \frac{ \tau_y L }{ \mu V }$	fluid mechanics, rheology (ratio of yield stress to viscous stress)^[5]
Biot number	Bi	$\mathrm{Bi} = \frac{h L_C}{k_b}$	heat transfer (surface vs. volume conductivity of solids)
Blake number	Bl or B	$\mathrm{B} = \frac{u \rho}{\mu (1 - \epsilon) D}$	geology, fluid mechanics, porous media (inertial over viscous forces in fluid flow through porous media)
Bodenstein number	Bo or Bd	$\mathrm{Bo} = vL/\mathcal{D} = \mathrm{Re}\, \mathrm{Sc}$	chemistry (residence-time distribution; similar to the axial mass transfer Peclet number)^[9]
Bond number	Bo	$\mathrm{Bo} = \frac{\rho a L^2}{\gamma}$	geology, fluid mechanics, porous media (buoyant versus capilary forces, similar to the Eötvös number) ^[10]
Brinkman number	Br	$\mathrm{Br} = \frac {\mu U^2}{\kappa (T_w - T_0)}$	heat transfer, fluid mechanics (conduction from a wall to a viscous fluid)
Brownell–Katz number	N_BK	$\mathrm{N}_\mathrm{BK} = \frac{u \mu}{k_\mathrm{rw}\sigma}$	fluid mechanics (combination of capillary number and Bond number) ^[11]
Capillary number	Ca	$\mathrm{Ca} = \frac{\mu V}{\gamma}$	porous media, fluid mechanics (viscous forces versus surface tension)
Chandrasekhar number	Q	$\mathrm{Q} = \frac{{B_0}^2 d^2}{\mu_0 \rho \nu \lambda}$	magnetohydrodynamics (ratio of the Lorentz force to the viscosity in magnetic convection)
Colburn J factors	J_M, J_H, J_D		turbulence; heat, mass, and momentum transfer (dimensionless transfer coefficients)
Coefficient of kinetic friction	$\mu_k$		mechanics (friction of solid bodies in translational motion)
Coefficient of static friction	$\mu_s$		mechanics (friction of solid bodies at rest)
Coefficient of determination	$R^2$		statistics (proportion of variance explained by a statistical model)
Coefficient of variation	$\frac{\sigma}{\mu}$	$\frac{\sigma}{\mu}$	statistics (ratio of standard deviation to expectation)
Correlation	ρ or r	$\frac{{\mathbb E}[(X-\mu_X)(Y-\mu_Y)]}{\sigma_X \sigma_Y}$ or $\frac{\sum_{k=1}^n (x_k-\bar x)(y_k-\bar y)}{\sqrt{\sum_{k=1}^n (x_k-\bar x)^2 \sum_{k=1}^n (y_k-\bar y)^2}}$ where $\bar x = \sum_{k=1}^n x_k/n$ and similarly for $\bar y$	statistics (measure of linear dependence)
Courant–Friedrich–Levy number	C or 𝜈	$C = \frac {u\,\Delta t} {\Delta x}$	mathematics (numerical solutions of hyperbolic PDEs)^[12]
Damkohler number	Da	$\mathrm{Da} = k \tau$	chemistry (reaction time scales vs. residence time)
Damping ratio	$\zeta$	$\zeta = \frac{c}{2 \sqrt{km}}$	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	$\mathrm{Da} = \frac{K}{d^2}$	porous media (ratio of permeability to cross-sectional area)
Dean number	D	$\mathrm{D} = \frac{\rho V d}{\mu} \left( \frac{d}{2 R} \right)^{1/2}$	turbulent flow (vortices in curved ducts)
Deborah number	De	$\mathrm{De} = \frac{t_\mathrm{c}}{t_\mathrm{p}}$	rheology (viscoelastic fluids)
Decibel	dB		acoustics, electronics, control theory (ratio of two intensities or powers of a wave)
Drag coefficient	c_d	$c_\mathrm{d} = \dfrac{2 F_\mathrm{d}}{\rho v^2 A}\, ,$	aeronautics, fluid dynamics (resistance to fluid motion)
Dukhin number	Du	$\mathrm{Du} = \frac{\kappa^{\sigma}}{{\Kappa_m} a}$	colloid science (ratio of electric surface conductivity to the electric bulk conductivity in heterogeneous systems)
Eckert number	Ec	$\mathrm{Ec} = \frac{V^2}{c_p\Delta T}$	convective heat transfer (characterizes dissipation of energy; ratio of kinetic energy to enthalpy)
Ekman number	Ek	$\mathrm{Ek} = \frac{\nu}{2D^2\Omega\sin\varphi}$	geophysics (viscous versus Coriolis forces)
Elasticity (economics)	E	$E_{x,y} = \frac{\partial \ln(x)}{\partial \ln(y)} = \frac{\partial x}{\partial y}\frac{y}{x}$	economics (response of demand or supply to price changes)
Eötvös number	Eo	$\mathrm{Eo}=\frac{\Delta\rho \,g \,L^2}{\sigma}$	fluid mechanics (shape of bubbles or drops)
Ericksen number	Er	$\mathrm{Er}=\frac{\mu v L}{K}$	fluid dynamics (liquid crystal flow behavior; viscous over elastic forces)
Euler number	Eu	$\mathrm{Eu}=\frac{\Delta{}p}{\rho V^2}$	hydrodynamics (stream pressure versus inertia forces)
Euler's number	e	$e = \displaystyle\sum\limits_{n = 0}^{ \infty} \dfrac{1}{n!} \approx 2.71828$	mathematics (base of the natural logarithm)
Excess temperature coefficient	$\Theta_r$	$\Theta_r = \frac{c_p (T-T_e)}{U_e^2/2}$	heat transfer, fluid dynamics (change in internal energy versus kinetic energy)^[13]
Fanning friction factor	f		fluid mechanics (fraction of pressure losses due to friction in a pipe; 1/4th the Darcy friction factor)^[14]
Feigenbaum constants	$\alpha$ , $\delta$	$\alpha \approx 2.50290,$ $\ \delta \approx 4.66920$	chaos theory (period doubling)^[15]
Fine structure constant	$\alpha$	$\alpha = \frac{e^2}{2\varepsilon_0 hc}$	quantum electrodynamics (QED) (coupling constant characterizing the strength of the electromagnetic interaction)
f-number	f	$f = \frac {{\ell}}{{D}}$	optics, photography (ratio of focal length to diameter of aperture)
Föppl–von Kármán number	$\gamma$	$\gamma = \frac{Y r^2}{\kappa}$	virology, solid mechanics (thin-shell buckling)
Fourier number	Fo	$\mathrm{Fo} = \frac{\alpha t}{L^2}$	heat transfer, mass transfer (ratio of diffusive rate versus storage rate)
Fresnel number	F	$\mathit{F} = \frac{a^{2}}{L \lambda}$	optics (slit diffraction)^[16]
Froude number	Fr	$\mathrm{Fr} = \frac{v}{\sqrt{g\ell}}$	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)^[17]
Galilei number	Ga	$\mathrm{Ga} = \frac{g\, L^3}{\nu^2}$	fluid mechanics (gravitational over viscous forces)
Golden ratio	$\varphi$	$\varphi = \frac{1+\sqrt{5}}{2} \approx 1.61803$	mathematics, aesthetics (long side length of self-similar rectangle)
Görtler number	G	$\mathrm{G} = \frac{U_e \theta}{\nu} \left( \frac{\theta}{R} \right)^{1/2}$	fluid dynamics (boundary layer flow along a concave wall)
Graetz number	Gz	$\mathrm{Gz} = {D_H \over L} \mathrm{Re}\, \mathrm{Pr}$	heat transfer, fluid mechanics (laminar flow through a conduit; also used in mass transfer)
Grashof number	Gr	$\mathrm{Gr}_L = \frac{g \beta (T_s - T_\infty ) L^3}{\nu ^2}$	heat transfer, natural convection (ratio of the buoyancy to viscous force)
Gravitational coupling constant	$\alpha_G$	$\alpha_G=\frac{Gm_e^2}{\hbar c}$	gravitation (attraction between two massy elementary particles; analogous to the Fine structure constant)
Hatta number	Ha	$\mathrm{Ha} = \frac{N_{\mathrm{A}0}}{N_{\mathrm{A}0}^{\mathrm{phys}}}$	chemical engineering (adsorption enhancement due to chemical reaction)
Hagen number	Hg	$\mathrm{Hg} = -\frac{1}{\rho}\frac{\mathrm{d} p}{\mathrm{d} x}\frac{L^3}{\nu^2}$	heat transfer (ratio of the buoyancy to viscous force in forced convection)
Hydraulic gradient	i	$i = \frac{\mathrm{d}h}{\mathrm{d}l} = \frac{h_2 - h_1}{\mathrm{length}}$	fluid mechanics, groundwater flow (pressure head over distance)
Iribarren number	Ir	$\mathrm{Ir} = \frac{\tan \alpha}{\sqrt{H/L_0}}$	wave mechanics (breaking surface gravity waves on a slope)
Jakob number	Ja	$\mathrm{Ja} = \frac{c_p (T_\mathrm{s} - T_\mathrm{sat}) }{\Delta H_{\mathrm{f}} }$	chemistry (ratio of sensible to latent energy absorbed during liquid-vapor phase change)^[18]
Karlovitz number	Ka	$\mathrm{Ka} = k t_c$	turbulent combustion (characteristic flow time times flame stretch rate)
Keulegan–Carpenter number	K_C	$\mathrm{K_C} = \frac{V\,T}{L}$	fluid dynamics (ratio of drag force to inertia for a bluff object in oscillatory fluid flow)
Knudsen number	Kn	$\mathrm{Kn} = \frac {\lambda}{L}$	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	$\mathrm{Ku} = \frac{U_h \rho_g^{1/2}}{\left({\sigma g (\rho_l - \rho_g)}\right)^{1/4}}$	fluid mechanics (counter-current two-phase flow)^[19]
Laplace number	La	$\mathrm{La} = \frac{\sigma \rho L}{\mu^2}$	fluid dynamics (free convection within immiscible fluids; ratio of surface tension to momentum-transport)
Lewis number	Le	$\mathrm{Le} = \frac{\alpha}{D} = \frac{\mathrm{Sc}}{\mathrm{Pr}}$	heat and mass transfer (ratio of thermal to mass diffusivity)
Lift coefficient	C_L	$C_\mathrm{L} = \frac{L}{q\,S}$	aerodynamics (lift available from an airfoil at a given angle of attack)
Lockhart–Martinelli parameter	$\chi$	$\chi = \frac{m_\ell}{m_g} \sqrt{\frac{\rho_g}{\rho_\ell}}$	two-phase flow (flow of wet gases; liquid fraction)^[20]
Love numbers	h, k, l		geophysics (solidity of earth and other planets)
Lundquist number	S	$S = \frac{\mu_0LV_A}{\eta}$	plasma physics (ratio of a resistive time to an Alfvén wave crossing time in a plasma)
Mach number	M or Ma	$\mathrm{M} = \frac{{v}}{{v_\mathrm{sound}}}$	gas dynamics (compressible flow; dimensionless velocity)
Magnetic Reynolds number	R_m	$\mathrm{R}_\mathrm{m} = \frac{U L}{\eta}$	magnetohydrodynamics (ratio of magnetic advection to magnetic diffusion)
Manning roughness coefficient	n		open channel flow (flow driven by gravity)^[21]
Marangoni number	Mg	$\mathrm{Mg} = - {\frac{\mathrm{d}\sigma}{\mathrm{d}T}}\frac{L \Delta T}{\eta \alpha}$	fluid mechanics (Marangoni flow; thermal surface tension forces over viscous forces)
Morton number	Mo	$\mathrm{Mo} = \frac{g \mu_c^4 \, \Delta \rho}{\rho_c^2 \sigma^3}$	fluid dynamics (determination of bubble/drop shape)
Nusselt number	Nu	$\mathrm{Nu} =\frac{hd}{k}$	heat transfer (forced convection; ratio of convective to conductive heat transfer)
Ohnesorge number	Oh	$\mathrm{Oh} = \frac{ \mu}{ \sqrt{\rho \sigma L }} = \frac{\sqrt{\mathrm{We}}}{\mathrm{Re}}$	fluid dynamics (atomization of liquids, Marangoni flow)
Péclet number	Pe	$\mathrm{Pe} = \frac{du\rho c_p}{k} = \mathrm{Re}\, \mathrm{Pr}$	heat transfer (advection–diffusion problems; total momentum transfer to molecular heat transfer)
Peel number	N_P	$N_\mathrm{P} = \frac{\text{Restoring force}}{\text{Adhesive force}}$	coating (adhesion of microstructures with substrate)^[22]
Perveance	K	${K} = \frac{{I}}{{I_0}}\,\frac{{2}}{{\beta}^3{\gamma}^3} (1-\gamma^2f_e)$	charged particle transport (measure of the strength of space charge in a charged particle beam)
pH	$\mathrm{pH}$	$\mathrm{pH} = - \log_{10}(a_{\textrm{H}^+}) = \log_{10}\left(\frac{1}{a_{\textrm{H}^+}}\right)$	chemistry (the measure of the acidity or basicity of an aqueous solution)
Pi	$\pi$	$\pi = \frac{C}{d} \approx 3.14159$	mathematics (ratio of a circle's circumference to its diameter)
Pixel	px		digital imaging (smallest addressable unit)
Poisson's ratio	$\nu$	$\nu = -\frac{\mathrm{d}\varepsilon_\mathrm{trans}}{\mathrm{d}\varepsilon_\mathrm{axial}}$	elasticity (load in transverse and longitudinal direction)
Porosity	$\phi$	$\phi = \frac{V_\mathrm{V}}{V_\mathrm{T}}$	geology, porous media (void fraction of the medium)
Power factor	P/S		electronics (real power to apparent power)
Power number	N_p	$N_p = {P\over \rho n^3 d^5}$	electronics (power consumption by agitators; resistance force versus inertia force)
Prandtl number	Pr	$\mathrm{Pr} = \frac{\nu}{\alpha} = \frac{c_p \mu}{k}$	heat transfer (ratio of viscous diffusion rate over thermal diffusion rate)
Prater number	β	$\beta = \frac{-\Delta H_r D_{TA}^e C_{AS}}{\lambda^e T_s}$	reaction engineering (ratio of heat evolution to heat conduction within a catalyst pellet)^[23]
Pressure coefficient	C_P	$C_p = {p - p_\infty \over \frac{1}{2} \rho_\infty V_\infty^2}$	aerodynamics, hydrodynamics (pressure experienced at a point on an airfoil; dimensionless pressure variable)
Q factor	Q	$Q = 2 \pi f_r \frac{\text{Energy Stored}}{\text{Power Loss}}$	physics, engineering (damping of oscillator or resonator; energy stored versus energy lost)
Radian measure	rad	$\text{arc length}/\text{radius}$	mathematics (measurement of planar angles, 1 radian = 180/π degrees)
Rayleigh number	Ra	$\mathrm{Ra}_{x} = \frac{g \beta} {\nu \alpha} (T_s - T_\infin) x^3$	heat transfer (buoyancy versus viscous forces in free convection)
Refractive index	n	$n=\frac{c}{v}$	electromagnetism, optics (speed of light in a vacuum over speed of light in a material)
Relative density	RD	$RD = \frac{\rho_\mathrm{substance}}{\rho_\mathrm{reference}}$	hydrometers, material comparisons (ratio of density of a material to a reference material—usually water)
Relative permeability	$\mu_r$	$\mu_r = \frac{\mu}{\mu_0}$	magnetostatics (ratio of the permeability of a specific medium to free space)
Relative permittivity	$\varepsilon_r$	$\varepsilon_{r} = \frac{C_{x}} {C_{0}}$	electrostatics (ratio of capacitance of test capacitor with dielectric material versus vacuum)
Reynolds number	Re	$\mathrm{Re} = \frac{vL\rho}{\mu}$	fluid mechanics (ratio of fluid inertial and viscous forces)^[5]
Richardson number	Ri	$\mathrm{Ri} = \frac{gh}{u^2} = \frac{1}{\mathrm{Fr}^2}$	fluid dynamics (effect of buoyancy on flow stability; ratio of potential over kinetic energy)^[24]
Rockwell scale	–		mechanical hardness (indentation hardness of a material)
Rolling resistance coefficient	C_rr	$C_{rr} = \frac{F}{N_f}$	vehicle dynamics (ratio of force needed for motion of a wheel over the normal force)
Roshko number	Ro	$\mathrm{Ro} = {f L^{2}\over \nu} =\mathrm{St}\,\mathrm{Re}$	fluid dynamics (oscillating flow, vortex shedding)
Rossby number	Ro	$\mathrm{Ro}=\frac{U}{Lf}$	geophysics (ratio of inertial to Coriolis force)
Rouse number	P or Z	$\mathrm{P} = \frac{w_s}{\kappa u_*}$	sediment transport (ratio of the sediment fall velocity and the upwards velocity of grain)
Schmidt number	Sc	$\mathrm{Sc} = \frac{\nu}{D}$	mass transfer (viscous over molecular diffusion rate)^[25]
Shape factor	H	$H = \frac {\delta^*}{\theta}$	boundary layer flow (ratio of displacement thickness to momentum thickness)
Sherwood number	Sh	$\mathrm{Sh} = \frac{K L}{D}$	mass transfer (forced convection; ratio of convective to diffusive mass transport)
Shields parameter	$\tau_*$ or $\theta$	$\tau_{\ast} = \frac{\tau}{(\rho_s - \rho) g D}$	sediment transport (threshold of sediment movement due to fluid motion; dimensionless shear stress)
Sommerfeld number	S	$\mathrm{S} = \left( \frac{r}{c} \right)^2 \frac {\mu N}{P}$	hydrodynamic lubrication (boundary lubrication)^[26]
Specific gravity	SG		(same as Relative density)
Stanton number	St	$\mathrm{St} = \frac{h}{c_p \rho V} = \frac{\mathrm{Nu}}{\mathrm{Re}\,\mathrm{Pr}}$	heat transfer and fluid dynamics (forced convection)
Stefan number	Ste	$\mathrm{Ste} = \frac{c_p \Delta T}{L}$	phase change, thermodynamics (ratio of sensible heat to latent heat)
Stokes number	Stk or S_k	$\mathrm{Stk} = \frac{\tau U_o}{d_c}$	particles suspensions (ratio of characteristic time of particle to time of flow)
Strain	$\epsilon$	$\epsilon = \cfrac{\partial{F}}{\partial{X}} - 1$	materials science, elasticity (displacement between particles in the body relative to a reference length)
Strouhal number	St or Sr	$\mathrm{St} = {\omega L\over v}$	fluid dynamics (continuous and pulsating flow; nondimensional frequency)^[27]
Stuart number	N	$\mathrm{N} = \frac {B^2 L_{c} \sigma}{\rho U} = \frac{\mathrm{Ha}^2}{\mathrm{Re}}$	magnetohydrodynamics (ratio of electromagnetic to inertial forces)
Taylor number	Ta	$\mathrm{Ta} = \frac{4\Omega^2 R^4}{\nu^2}$	fluid dynamics (rotating fluid flows; inertial forces due to rotation of a fluid versus viscous forces)
Transmittance	T	$T = \frac{I}{I_0}$	optics, spectroscopy (the ratio of the intensities of radiation exiting through and incident on a sample)
Ursell number	U	$\mathrm{U} = \frac{H\, \lambda^2}{h^3}$	wave mechanics (nonlinearity of surface gravity waves on a shallow fluid layer)
Vadasz number	Va	$\mathrm{Va} = \frac{\phi\, \mathrm{Pr}}{\mathrm{Da}}$	porous media (governs the effects of porosity $\phi$ , the Prandtl number and the Darcy number on flow in a porous medium) ^[28]
van 't Hoff factor	i	$i = 1 + \alpha (n - 1)$	quantitative analysis (K_f and K_b)
Wallis parameter	j^*	$j^* = R \left( \frac{\omega \rho}{\mu} \right)^\frac{1}{2}$	multiphase flows (nondimensional superficial velocity)^[29]
Weaver flame speed number	Wea	$\mathrm{Wea} = \frac{w}{w_\mathrm{H}} 100$	combustion (laminar burning velocity relative to hydrogen gas)^[30]
Weber number	We	$\mathrm{We} = \frac{\rho v^2 l}{\sigma}$	multiphase flow (strongly curved surfaces; ratio of inertia to surface tension)
Weissenberg number	Wi	$\mathrm{Wi} = \dot{\gamma} \lambda$	viscoelastic flows (shear rate times the relaxation time)^[31]
Womersley number	$\alpha$	$\alpha = R \left( \frac{\omega \rho}{\mu} \right)^\frac{1}{2}$	biofluid mechanics (continuous and pulsating flows; ratio of pulsatile flow frequency to viscous effects)^[32]

References

↑ "1.8 (1.6) quantity of dimension one dimensionless quantity". International vocabulary of metrology — Basic and general concepts and associated terms (VIM). ISO. 2008. Retrieved 2011-03-22.
↑ "BIPM Consultative Committee for Units (CCU), 15th Meeting" (PDF). 17–18 April 2003. Retrieved 2010-01-22.
↑ "BIPM Consultative Committee for Units (CCU), 16th Meeting" (PDF). Retrieved 2010-01-22.
↑ Dybkaer, René (2004). "An ontology on property for physical, chemical, and biological systems". APMIS Suppl. (117): 1–210. PMID 15588029.
↑ 5.0 5.1 5.2 "Table of Dimensionless Numbers" (PDF). Retrieved 2009-11-05.
↑ Bagnold number
↑ Bhattacharjee S., Grosshandler W.L. (1988). "The formation of wall jet near a high temperature wall under microgravity environment". ASME MTD 96: 711–6.
↑ Paoletti S., Rispoli F., Sciubba E. (1989). "Calculation of exergetic losses in compact heat exchanger passager". ASME AES 10 (2): 21–9.
↑ Becker, A.; Hüttinger, K. J. (1998). "Chemistry and kinetics of chemical vapor deposition of pyrocarbon—II pyrocarbon deposition from ethylene, acetylene and 1,3-butadiene in the low temperature regime". Carbon 36 (3): 177. doi:10.1016/S0008-6223(97)00175-9.
↑ Bond number
↑ http://www.onepetro.org/mslib/servlet/onepetropreview?id=00020506
↑ Courant–Friedrich–Levy number
↑ Schetz, Joseph A. (1993). Boundary Layer Analysis. Englewood Cliffs, NJ: Prentice-Hall, Inc. pp. 132–134. ISBN 0-13-086885-X.
↑ Fanning friction factor
↑ Feigenbaum constants
↑ Fresnel number
↑ Gain Ratio - Sheldon Brown
↑ Incropera, Frank P. (2007). Fundamentals of heat and mass transfer. John Wiley & Sons, Inc. p. 376.
↑ Tan, R. B. H.; Sundar, R. (2001). "On the froth–spray transition at multiple orifices". Chemical Engineering Science 56 (21–22): 6337. doi:10.1016/S0009-2509(01)00247-0.
↑ Lockhart–Martinelli parameter
↑ Manning coefficient PDF (109 KB)
↑ Van Spengen, W. M.; Puers, R.; De Wolf, I. (2003). "The prediction of stiction failures in MEMS". IEEE Transactions on Device and Materials Reliability 3 (4): 167. doi:10.1109/TDMR.2003.820295.
↑ Davis, Mark E.; Davis, Robert J. (2012). Fundamentals of Chemical Reaction Engineering. Dover. p. 215. ISBN 978-0486488554.
↑ Richardson number
↑ Schmidt number
↑ Sommerfeld number
↑ Strouhal number, Engineering Toolbox
↑ Straughan, B. (2001). "A sharp nonlinear stability threshold in rotating porous convection". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 457 (2005): 87–88. Bibcode:2001RSPSA.457...87S. doi:10.1098/rspa.2000.0657.
↑ Petritsch, G.; Mewes, D. (1999). "Experimental investigations of the flow patterns in the hot leg of a pressurized water reactor". Nuclear Engineering and Design 188: 75. doi:10.1016/S0029-5493(99)00005-9.
↑ Kuneš, J. (2012). "Technology and Mechanical Engineering". Dimensionless Physical Quantities in Science and Engineering. pp. 353–390. doi:10.1016/B978-0-12-416013-2.00008-7. ISBN 9780124160132.
↑ Weissenberg number
↑ Womersley number

External links

John Baez, "How Many Fundamental Constants Are There?"
Huba, J. D., 2007, NRL Plasma Formulary: Dimensionless Numbers of Fluid Mechanics. Naval Research Laboratory. p. 23, 24, 25
Sheppard, Mike, 2007, "Systematic Search for Expressions of Dimensionless Constants using the NIST database of Physical Constants."

Dimensionless numbers in fluid dynamics