List of dimensionless quantities

From HandWiki

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

Biology and medicine

Name Standard symbol Definition Field of application
Basic reproduction number [math]\displaystyle{ R_0 }[/math] number of infections caused on average by an infectious individual over entire infectious period epidemiology
Body fat percentage total mass of fat divided by total body mass, multiplied by 100 biology
Kt/V Kt/V medicine (hemodialysis and peritoneal dialysis treatment; dimensionless time)
Waist–hip ratio waist circumference divided by hip circumference biology
Waist-to-chest ratio waist circumference divided by chest circumference biology
Waist-to-height ratio waist circumference divided by height biology

Chemistry

Name Standard symbol Definition Field of application
Activity coefficient [math]\displaystyle{ \gamma }[/math] [math]\displaystyle{ \gamma= \frac {{a}}{{x}} }[/math] chemistry (Proportion of "active" molecules or atoms)
Arrhenius number [math]\displaystyle{ \alpha }[/math] [math]\displaystyle{ \alpha = \frac{E_a}{RT} }[/math] chemistry (ratio of activation energy to thermal energy)[1]
Atomic weight M chemistry (mass of atom over one atomic mass unit, u, where carbon-12 is exactly 12 u)
Bodenstein number Bo or Bd [math]\displaystyle{ \mathrm{Bo} = vL/\mathcal{D} = \mathrm{Re}\, \mathrm{Sc} }[/math] chemistry (residence-time distribution; similar to the axial mass transfer Peclet number)[2]
Damkohler number Da [math]\displaystyle{ \mathrm{Da} = k \tau }[/math] chemistry (reaction time scales vs. residence time)
Hatta number Ha [math]\displaystyle{ \mathrm{Ha} = \frac{N_{\mathrm{A}0}}{N_{\mathrm{A}0}^{\mathrm{phys}}} }[/math] chemical engineering (adsorption enhancement due to chemical reaction)
Jakob number Ja [math]\displaystyle{ \mathrm{Ja} = \frac{c_p (T_\mathrm{s} - T_\mathrm{sat}) }{\Delta H_{\mathrm{f}} } }[/math] chemistry (ratio of sensible to latent energy absorbed during liquid-vapor phase change)[3]
pH [math]\displaystyle{ \mathrm{pH} }[/math] [math]\displaystyle{ \mathrm{pH} = - \log_{10}(a_{\textrm{H}^+}) }[/math] chemistry (the measure of the acidity or basicity of an aqueous solution)
van 't Hoff factor i [math]\displaystyle{ i = 1 + \alpha (n - 1) }[/math] quantitative analysis (Kf and Kb)
Wagner number Wa [math]\displaystyle{ \mathrm{Wa} = \frac{\kappa}{l} \frac{\mathrm{d}\eta}{\mathrm{d}i} }[/math] electrochemistry (ratio of kinetic polarization resistance to solution ohmic resistance in an electrochemical cell)[4]
Weaver flame speed number Wea [math]\displaystyle{ \mathrm{Wea} = \frac{w}{w_\mathrm{H}} 100 }[/math] combustion (laminar burning velocity relative to hydrogen gas)[5]

Physics

Physical constants

Fluids and heat transfer

Main page: Physics:Dimensionless numbers in fluid mechanics
Name Standard symbol Definition Field of application
Archimedes number Ar [math]\displaystyle{ \mathrm{Ar} = \frac{g L^3 \rho_\ell (\rho - \rho_\ell)}{\mu^2} }[/math] fluid mechanics (motion of fluids due to density differences)
Asakuma number As [math]\displaystyle{ \mathrm{As} = \frac{W} {\alpha \rho d_p H } }[/math] heat transfer (ratio of heat generation of microwave dielectric heating to thermal diffusion[disambiguation needed] )[6]
Atwood number A [math]\displaystyle{ \mathrm{A} = \frac{\rho_1 - \rho_2} {\rho_1 + \rho_2} }[/math] fluid mechanics (onset of instabilities in fluid mixtures due to density differences)
Bagnold number Ba [math]\displaystyle{ \mathrm{Ba} = \frac{\rho d^2 \lambda^{1/2} \dot{\gamma} }{\mu} }[/math] fluid mechanics, geology (ratio of grain collision stresses to viscous fluid stresses in flow of a granular material such as grain and sand)[7]
Bejan number
(fluid mechanics)		 Be [math]\displaystyle{ \mathrm{Be} = \frac{\Delta P L^2} {\mu \alpha} }[/math] fluid mechanics (dimensionless pressure drop along a channel)[8]
Bejan number
(thermodynamics)		 Be [math]\displaystyle{ \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 irreversibility to total irreversibility due to heat transfer and fluid friction)[9]
Bingham number Bm [math]\displaystyle{ \mathrm{Bm} = \frac{ \tau_y L }{ \mu V } }[/math] fluid mechanics, rheology (ratio of yield stress to viscous stress)[1]
Biot number Bi [math]\displaystyle{ \mathrm{Bi} = \frac{h L_C}{k_b} }[/math] heat transfer (surface vs. volume conductivity of solids)
Blake number Bl or B [math]\displaystyle{ \mathrm{B} = \frac{u \rho}{\mu (1 - \epsilon) D} }[/math] geology, fluid mechanics, porous media (inertial over viscous forces in fluid flow through porous media)
Bond number Bo [math]\displaystyle{ \mathrm{Bo} = \frac{\rho a L^2}{\gamma} }[/math] geology, fluid mechanics, porous media (buoyant versus capillary forces, similar to the Eötvös number) [10]
Brinkman number Br [math]\displaystyle{ \mathrm{Br} = \frac {\mu U^2}{\kappa (T_w - T_0)} }[/math] heat transfer, fluid mechanics (conduction from a wall to a viscous fluid)
Brownell–Katz number NBK [math]\displaystyle{ \mathrm{N}_\mathrm{BK} = \frac{u \mu}{k_\mathrm{rw}\sigma} }[/math] fluid mechanics (combination of capillary number and Bond number) [11]
Capillary number Ca [math]\displaystyle{ \mathrm{Ca} = \frac{\mu V}{\gamma} }[/math] porous media, fluid mechanics (viscous forces versus surface tension)
Chandrasekhar number Q [math]\displaystyle{ \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)
Colburn J factors JM, JH, JD turbulence; heat, mass, and momentum transfer (dimensionless transfer coefficients)
Darcy friction factor Cf or fD fluid mechanics (fraction of pressure losses due to friction in a pipe; four times the Fanning friction factor)
Dean number D [math]\displaystyle{ \mathrm{D} = \frac{\rho V d}{\mu} \left( \frac{d}{2 R} \right)^{1/2} }[/math] turbulent flow (vortices in curved ducts)
Deborah number De [math]\displaystyle{ \mathrm{De} = \frac{t_\mathrm{c}}{t_\mathrm{p}} }[/math] rheology (viscoelastic fluids)
Drag coefficient cd [math]\displaystyle{ c_\mathrm{d} = \dfrac{2 F_\mathrm{d}}{\rho v^2 A}\, , }[/math] aeronautics, fluid dynamics (resistance to fluid motion)
Eckert number Ec [math]\displaystyle{ \mathrm{Ec} = \frac{V^2}{c_p\Delta T} }[/math] convective heat transfer (characterizes dissipation of energy; ratio of kinetic energy to enthalpy)
Ekman number Ek [math]\displaystyle{ \mathrm{Ek} = \frac{\nu}{2D^2\Omega\sin\varphi} }[/math] geophysics (viscous versus Coriolis forces)
Eötvös number Eo [math]\displaystyle{ \mathrm{Eo}=\frac{\Delta\rho \,g \,L^2}{\sigma} }[/math] fluid mechanics (shape of bubbles or drops)
Ericksen number Er [math]\displaystyle{ \mathrm{Er}=\frac{\mu v L}{K} }[/math] fluid dynamics (liquid crystal flow behavior; viscous over elastic forces)
Euler number Eu [math]\displaystyle{ \mathrm{Eu}=\frac{\Delta{}p}{\rho V^2} }[/math] hydrodynamics (stream pressure versus inertia forces)
Excess temperature coefficient [math]\displaystyle{ \Theta_r }[/math] [math]\displaystyle{ \Theta_r = \frac{c_p (T-T_e)}{U_e^2/2} }[/math] heat transfer, fluid dynamics (change in internal energy versus kinetic energy)[12]
Fanning friction factor f fluid mechanics (fraction of pressure losses due to friction in a pipe; 1/4th the Darcy friction factor)[13]
Fourier number Fo [math]\displaystyle{ \mathrm{Fo} = \frac{\alpha t}{L^2} }[/math] heat transfer, mass transfer (ratio of diffusive rate versus storage rate)
Froude number Fr [math]\displaystyle{ \mathrm{Fr} = \frac{v}{\sqrt{g\ell}} }[/math] fluid mechanics (wave and surface behaviour; ratio of a body's inertia to gravitational forces)
Galilei number Ga [math]\displaystyle{ \mathrm{Ga} = \frac{g\, L^3}{\nu^2} }[/math] fluid mechanics (gravitational over viscous forces)
Görtler number G [math]\displaystyle{ \mathrm{G} = \frac{U_e \theta}{\nu} \left( \frac{\theta}{R} \right)^{1/2} }[/math] fluid dynamics (boundary layer flow along a concave wall)
Graetz number Gz [math]\displaystyle{ \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)
Grashof number Gr [math]\displaystyle{ \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)
Hagen number Hg [math]\displaystyle{ \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)
Hydraulic gradient i [math]\displaystyle{ i = \frac{\mathrm{d}h}{\mathrm{d}l} = \frac{h_2 - h_1}{\mathrm{length}} }[/math] fluid mechanics, groundwater flow (pressure head over distance)
Karlovitz number Ka [math]\displaystyle{ \mathrm{Ka} = \frac{t_F}{t_\eta} }[/math] turbulent combustion (characteristic chemical time scale to Kolmogorov time scale)
Keulegan–Carpenter number KC [math]\displaystyle{ \mathrm{K_C} = \frac{V\,T}{L} }[/math] fluid dynamics (ratio of drag force to inertia for a bluff object in oscillatory fluid flow)
Knudsen number Kn [math]\displaystyle{ \mathrm{Kn} = \frac {\lambda}{L} }[/math] gas dynamics (ratio of the molecular mean free path length to a representative physical length scale)
Kutateladze number Ku [math]\displaystyle{ \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)[14]
Laplace number La [math]\displaystyle{ \mathrm{La} = \frac{\sigma \rho L}{\mu^2} }[/math] fluid dynamics (free convection within immiscible fluids; ratio of surface tension to momentum-transport)
Lewis number Le [math]\displaystyle{ \mathrm{Le} = \frac{\alpha}{D} = \frac{\mathrm{Sc}}{\mathrm{Pr}} }[/math] heat and mass transfer (ratio of thermal to mass diffusivity)
Lift coefficient CL [math]\displaystyle{ C_\mathrm{L} = \frac{L}{q\,S} }[/math] aerodynamics (lift available from an airfoil at a given angle of attack)
Lockhart–Martinelli parameter [math]\displaystyle{ \chi }[/math] [math]\displaystyle{ \chi = \frac{m_\ell}{m_g} \sqrt{\frac{\rho_g}{\rho_\ell}} }[/math] two-phase flow (flow of wet gases; liquid fraction)[15]
Mach number M or Ma [math]\displaystyle{ \mathrm{M} = \frac{{v}}{{v_\mathrm{sound}}} }[/math] gas dynamics (compressible flow; dimensionless velocity)
Magnetic Reynolds number Rm [math]\displaystyle{ \mathrm{R}_\mathrm{m} = \frac{U L}{\eta} }[/math] magnetohydrodynamics (ratio of magnetic advection to magnetic diffusion)
Manning roughness coefficient n open channel flow (flow driven by gravity)[16]
Marangoni number Mg [math]\displaystyle{ \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 viscous forces)
Markstein number [math]\displaystyle{ \mathcal{M} }[/math] [math]\displaystyle{ \mathcal{M} = \frac{\mathcal{L}_b}{\delta_L} }[/math] fluid dynamics, combustion (turbulent combustion flames)
Morton number Mo [math]\displaystyle{ \mathrm{Mo} = \frac{g \mu_c^4 \, \Delta \rho}{\rho_c^2 \sigma^3} }[/math] fluid dynamics (determination of bubble/drop shape)
Nusselt number Nu [math]\displaystyle{ \mathrm{Nu}_d =\frac{hd}{k} }[/math] heat transfer (forced convection; ratio of convective to conductive heat transfer)
Ohnesorge number Oh [math]\displaystyle{ \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]\displaystyle{ \mathrm{Pe}_d = \frac{du\rho c_p}{k} = \mathrm{Re}_d\, \mathrm{Pr} }[/math] heat transfer (advection–diffusion problems; total momentum transfer to molecular heat transfer)
Péclet number Pe [math]\displaystyle{ \mathrm{Pe}_d = \frac{du}{D} = \mathrm{Re}_d\, \mathrm{Sc} }[/math] mass transfer (advection–diffusion problems; total momentum transfer to diffusive mass transfer)
Prandtl number Pr [math]\displaystyle{ \mathrm{Pr} = \frac{\nu}{\alpha} = \frac{c_p \mu}{k} }[/math] heat transfer (ratio of viscous diffusion rate over thermal diffusion rate)
Pressure coefficient CP [math]\displaystyle{ 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)
Rayleigh number Ra [math]\displaystyle{ \mathrm{Ra}_{x} = \frac{g \beta} {\nu \alpha} (T_s - T_\infin) x^3 }[/math] heat transfer (buoyancy versus viscous forces in free convection)
Reynolds number Re [math]\displaystyle{ \mathrm{Re}_L = \frac{vL\rho}{\mu} }[/math] fluid mechanics (ratio of fluid inertial and viscous forces)[1]
Richardson number Ri [math]\displaystyle{ \mathrm{Ri} = \frac{gh}{u^2} = \frac{1}{\mathrm{Fr}^2} }[/math] fluid dynamics (effect of buoyancy on flow stability; ratio of potential over kinetic energy)[17]
Roshko number Ro [math]\displaystyle{ \mathrm{Ro} = {f L^{2}\over \nu} =\mathrm{St}\,\mathrm{Re} }[/math] fluid dynamics (oscillating flow, vortex shedding)
Schmidt number Sc [math]\displaystyle{ \mathrm{Sc}_D = \frac{\nu}{D} }[/math] mass transfer (viscous over molecular diffusion rate)[18]
Shape factor H [math]\displaystyle{ H = \frac {\delta^*}{\theta} }[/math] boundary layer flow (ratio of displacement thickness to momentum thickness)
Sherwood number Sh [math]\displaystyle{ \mathrm{Sh}_D = \frac{K L}{D} }[/math] mass transfer (forced convection; ratio of convective to diffusive mass transport)
Sommerfeld number S [math]\displaystyle{ \mathrm{S} = \left( \frac{r}{c} \right)^2 \frac {\mu N}{P} }[/math] hydrodynamic lubrication (boundary lubrication)[19]
Stanton number St [math]\displaystyle{ \mathrm{St} = \frac{h}{c_p \rho V} = \frac{\mathrm{Nu}}{\mathrm{Re}\,\mathrm{Pr}} }[/math] heat transfer and fluid dynamics (forced convection)
Stokes number Stk or Sk [math]\displaystyle{ \mathrm{Stk} = \frac{\tau U_o}{d_c} }[/math] particles suspensions (ratio of characteristic time of particle to time of flow)
Strouhal number St or Sr [math]\displaystyle{ \mathrm{St} = {\omega L\over v} }[/math] fluid dynamics (continuous and pulsating flow; nondimensional frequency)[20]
Stuart number N [math]\displaystyle{ \mathrm{N} = \frac {B^2 L_{c} \sigma}{\rho U} = \frac{\mathrm{Ha}^2}{\mathrm{Re}} }[/math] magnetohydrodynamics (ratio of electromagnetic to inertial forces)
Taylor number Ta [math]\displaystyle{ \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 viscous forces)
Ursell number U [math]\displaystyle{ \mathrm{U} = \frac{H\, \lambda^2}{h^3} }[/math] wave mechanics (nonlinearity of surface gravity waves on a shallow fluid layer)
Vadasz number Va [math]\displaystyle{ \mathrm{Va} = \frac{\phi\, \mathrm{Pr}}{\mathrm{Da}} }[/math] porous media (governs the effects of porosity [math]\displaystyle{ \phi }[/math], the Prandtl number and the Darcy number on flow in a porous medium) [21]
Wallis parameter j* [math]\displaystyle{ j^* = R \left( \frac{\omega \rho}{\mu} \right)^\frac{1}{2} }[/math] multiphase flows (nondimensional superficial velocity)[22]
Weber number We [math]\displaystyle{ \mathrm{We} = \frac{\rho v^2 l}{\sigma} }[/math] multiphase flow (strongly curved surfaces; ratio of inertia to surface tension)
Weissenberg number Wi [math]\displaystyle{ \mathrm{Wi} = \dot{\gamma} \lambda }[/math] viscoelastic flows (shear rate times the relaxation time)[23]
Womersley number [math]\displaystyle{ \alpha }[/math] [math]\displaystyle{ \alpha = R \left( \frac{\omega \rho}{\mu} \right)^\frac{1}{2} }[/math] biofluid mechanics (continuous and pulsating flows; ratio of pulsatile flow frequency to viscous effects)[24]
Zel'dovich number [math]\displaystyle{ \beta }[/math] [math]\displaystyle{ \beta = \frac{E}{RT_f} \frac{T_f-T_o}{T_f} }[/math] fluid dynamics, Combustion (Measure of activation energy)

Solids

Name Standard symbol Definition Field of application
Coefficient of kinetic friction [math]\displaystyle{ \mu_k }[/math] mechanics (friction of solid bodies in translational motion)
Coefficient of static friction [math]\displaystyle{ \mu_s }[/math] mechanics (friction of solid bodies at rest)
Dieterich-Ruina-Rice number [math]\displaystyle{ \mathrm{R_u} }[/math] [math]\displaystyle{ \mathrm{R_u} = \frac{W}{L}\frac{(b-a)\bar{\sigma}}{G} }[/math] mechanics, friction, rheology, geophysics (stiffness ratio for frictional contacts)[25]
Föppl–von Kármán number [math]\displaystyle{ \gamma }[/math] [math]\displaystyle{ \gamma = \frac{Y r^2}{\kappa} }[/math] virology, solid mechanics (thin-shell buckling)
Rockwell scale – mechanical hardness (indentation hardness of a material)
Rolling resistance coefficient Crr [math]\displaystyle{ C_{rr} = \frac{F}{N_f} }[/math] vehicle dynamics (ratio of force needed for motion of a wheel over the normal force)

Optics

Name Standard symbol Definition Field of application
Abbe number V [math]\displaystyle{ V = \frac{ n_d - 1 }{ n_F - n_C } }[/math] optics (dispersion in optical materials)
f-number N [math]\displaystyle{ N = \frac{f}{D} }[/math] optics, photography (ratio of focal length to diameter of aperture)
Fresnel number F [math]\displaystyle{ \mathit{F} = \frac{a^{2}}{L \lambda} }[/math] optics (slit diffraction)[26]
Refractive index n [math]\displaystyle{ n=\frac{c}{v} }[/math] electromagnetism, optics (speed of light in vacuum over speed of light in a material)
Transmittance T [math]\displaystyle{ T = \frac{I}{I_0} }[/math] optics, spectroscopy (the ratio of the intensities of radiation exiting through and incident on a sample)

Mathematics and statistics

Name Standard symbol Definition Field of application
Coefficient of determination [math]\displaystyle{ R^2 }[/math] statistics (proportion of variance explained by a statistical model)
Coefficient of variation [math]\displaystyle{ \frac{\sigma}{\mu} }[/math] [math]\displaystyle{ \frac{\sigma}{\mu} }[/math] statistics (ratio of standard deviation to expectation)
Correlation ρ or r [math]\displaystyle{ \frac{\operatorname{E}[(X-\mu_X)(Y-\mu_Y)]}{\sigma_X \sigma_Y} }[/math] statistics (measure of linear dependence)
Courant–Friedrich–Levy number C or 𝜈 [math]\displaystyle{ C = \frac {u\,\Delta t} {\Delta x} }[/math] mathematics (numerical solutions of hyperbolic PDEs)[27]
Euler's number e [math]\displaystyle{ e = \displaystyle\sum\limits_{n = 0}^{ \infty} \dfrac{1}{n!} \approx 2.71828 }[/math] mathematics (base of the natural logarithm)
Feigenbaum constants [math]\displaystyle{ \alpha }[/math], [math]\displaystyle{ \delta }[/math] [math]\displaystyle{ \alpha \approx 2.50290, }[/math]
[math]\displaystyle{ \ \delta \approx 4.66920 }[/math]		 chaos theory (period doubling)[28]
Golden ratio [math]\displaystyle{ \varphi }[/math] [math]\displaystyle{ \varphi = \frac{1+\sqrt{5}}{2} \approx 1.61803 }[/math] mathematics, aesthetics (long side length of self-similar rectangle)
Pi [math]\displaystyle{ \pi }[/math] [math]\displaystyle{ \pi = \frac{C}{D} \approx 3.14159 }[/math] mathematics (ratio of a circle's circumference to its diameter)
Radian measure rad [math]\displaystyle{ \text{arc length}/\text{radius} }[/math] mathematics (measurement of planar angles, 1 radian = 180/π degrees)
Steradian measure sr measurement of solid angles

Geography, geology and geophysics

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

Sport

Name Standard symbol Definition Field of application
Blondeau number [math]\displaystyle{ B_\kappa }[/math] [math]\displaystyle{ \mathrm{B_\kappa} = \frac{t_g v_f}{l_{mf}} }[/math] sport science, team sports[29]
Gain ratio – bicycling (system of representing gearing; length traveled over length pedaled)[30]
Goal average – [math]\displaystyle{ \text{Goal average }=\frac{\text{goals scored}}{\text{goals conceded}} }[/math] Association football[31]
Runs Per Wicket Ratio RpW ratio [math]\displaystyle{ \text{RpW ratio }=\frac{\text{runs scored}}{\text{wickets lost}} \div \frac{\text{runs conceded}}{\text{wickets taken}} }[/math] cricket[32]
Winning percentage – Various, e.g. [math]\displaystyle{ \frac{\text{Games won}}{\text{Games played}} }[/math] or [math]\displaystyle{ \frac{\text{Points won}}{\text{Points contested}} }[/math] Various sports

Other fields

Name Standard symbol Definition Field of application
Capacity factor [math]\displaystyle{ \frac{\text{actual electrical energy output}}{\text{maximum possible electrical energy output}} }[/math] energy
Cohesion number Coh [math]\displaystyle{ Coh=\frac{1}{\rho g}\left ( \frac{\Gamma^5}{{E^*}^2{R^*}^8} \right )^{\frac{1}{3}} }[/math] Chemical engineering, material science, mechanics (A scale to show the energy needed for detaching two solid particles)[33][34]
Cost of transport COT [math]\displaystyle{ \mathrm{COT} = \frac{E}{mgd} }[/math] energy efficiency, economics (ratio of energy input to kinetic motion)
Damping ratio [math]\displaystyle{ \zeta }[/math] [math]\displaystyle{ \zeta = \frac{c}{2 \sqrt{km}} }[/math] mechanics, electrical engineering (the level of damping in a system)
Darcy number Da [math]\displaystyle{ \mathrm{Da} = \frac{K}{d^2} }[/math] porous media (ratio of permeability to cross-sectional area)
Decibel dB acoustics, electronics, control theory (ratio of two intensities or powers of a wave)
Dukhin number Du [math]\displaystyle{ \mathrm{Du} = \frac{\kappa^{\sigma}}{{\Kappa_m} a} }[/math] colloid science (ratio of electric surface conductivity to the electric bulk conductivity in heterogeneous systems)
Elasticity
(economics)		 E [math]\displaystyle{ E_{x,y} = \frac{\partial \ln(x)}{\partial \ln(y)} = \frac{\partial x}{\partial y}\frac{y}{x} }[/math] economics (response of demand or supply to price changes)
Fine-structure constant [math]\displaystyle{ \alpha }[/math] [math]\displaystyle{ \alpha = \frac{e^2}{4\pi\varepsilon_0 \hbar c} }[/math] quantum electrodynamics (QED) (coupling constant characterizing the strength of the electromagnetic interaction)
Gain – electronics (signal output to signal input)
Havnes parameter [math]\displaystyle{ P_H }[/math] [math]\displaystyle{ P_H = \frac{Z_d n_d}{n_i} }[/math] In Dusty plasma physics, ratio of the total charge [math]\displaystyle{ Z_d }[/math] carried by the dust particles [math]\displaystyle{ d }[/math] to the charge carried by the ions [math]\displaystyle{ i }[/math], with [math]\displaystyle{ n }[/math] the number density of particles
Helmholtz number [math]\displaystyle{ He }[/math] [math]\displaystyle{ He = \frac{wa}{c_0} = k_0a }[/math] The most important parameter in duct acoustics. If [math]\displaystyle{ \omega }[/math] is the dimensional frequency, then [math]\displaystyle{ k_0 }[/math] is the corresponding free field wavenumber and [math]\displaystyle{ He }[/math] is the corresponding dimensionless frequency [35]
Iribarren number Ir [math]\displaystyle{ \mathrm{Ir} = \frac{\tan \alpha}{\sqrt{H/L_0}} }[/math] wave mechanics (breaking surface gravity waves on a slope)
Load factor [math]\displaystyle{ \frac{\text{average load}}{\text{peak load}} }[/math] energy
Lundquist number S [math]\displaystyle{ S = \frac{\mu_0LV_A}{\eta} }[/math] plasma physics (ratio of a resistive time to an Alfvén wave crossing time in a plasma)
Peel number NP [math]\displaystyle{ N_\mathrm{P} = \frac{\text{Restoring force}}{\text{Adhesive force}} }[/math] coating (adhesion of microstructures with substrate)[36]
Perveance K [math]\displaystyle{ {K} = \frac{{I}}{{I_0}}\,\frac{{2}}{{\beta}^3{\gamma}^3} (1-\gamma^2f_e) }[/math] charged particle transport (measure of the strength of space charge in a charged particle beam)
Pierce parameter [math]\displaystyle{ C }[/math] [math]\displaystyle{ C^3=\frac{Z_c I_K}{4 V_K} }[/math] Traveling wave tube
Pixel px digital imaging (smallest addressable unit)
Beta (plasma physics) [math]\displaystyle{ \beta }[/math] [math]\displaystyle{ \beta = \frac{n k_B T}{B^2/2\mu_0} }[/math] Plasma and Fusion power. Ratio of plasma thermal pressure to magnetic pressure, controlling the level of turbulence in a magnetised plasma.
Poisson's ratio [math]\displaystyle{ \nu }[/math] [math]\displaystyle{ \nu = -\frac{\mathrm{d}\varepsilon_\mathrm{trans}}{\mathrm{d}\varepsilon_\mathrm{axial}} }[/math] elasticity (strain in transverse and longitudinal direction)
Power factor pf [math]\displaystyle{ pf = \frac{P}{S} }[/math] electrical (real power to apparent power)
Power number Np [math]\displaystyle{ N_p = {P\over \rho n^3 d^5} }[/math] fluid mechanics, power consumption by rotary agitators; resistance force versus inertia force)
Prater number β [math]\displaystyle{ \beta = \frac{-\Delta H_r D_{TA}^e C_{AS}}{\lambda^e T_s} }[/math] reaction engineering (ratio of heat evolution to heat conduction within a catalyst pellet)[37]
Q factor Q [math]\displaystyle{ Q = 2 \pi f_r \frac{\text{Energy Stored}}{\text{Power Loss}} }[/math] physics, engineering (Damping ratio of oscillator or resonator; energy stored versus energy lost)
Relative density RD [math]\displaystyle{ RD = \frac{\rho_\mathrm{substance}}{\rho_\mathrm{reference}} }[/math] hydrometers, material comparisons (ratio of density of a material to a reference material—usually water)
Relative permeability [math]\displaystyle{ \mu_r }[/math] [math]\displaystyle{ \mu_r = \frac{\mu}{\mu_0} }[/math] magnetostatics (ratio of the permeability of a specific medium to free space)
Relative permittivity [math]\displaystyle{ \varepsilon_r }[/math] [math]\displaystyle{ \varepsilon_{r} = \frac{C_{x}} {C_{0}} }[/math] electrostatics (ratio of capacitance of test capacitor with dielectric material versus vacuum)
Rouse number P or Z [math]\displaystyle{ \mathrm{P} = \frac{w_s}{\kappa u_*} }[/math] sediment transport (ratio of the sediment fall velocity and the upwards velocity of grain)
Shields parameter [math]\displaystyle{ \tau_* }[/math] or [math]\displaystyle{ \theta }[/math] [math]\displaystyle{ \tau_{\ast} = \frac{\tau}{(\rho_s - \rho) g D} }[/math] sediment transport (threshold of sediment movement due to fluid motion; dimensionless shear stress)
Specific gravity SG (same as Relative density)
Stefan number Ste [math]\displaystyle{ \mathrm{Ste} = \frac{c_p \Delta T}{L} }[/math] phase change, thermodynamics (ratio of sensible heat to latent heat)
Strain [math]\displaystyle{ \epsilon }[/math] [math]\displaystyle{ \epsilon = \cfrac{\partial{F}}{\partial{X}} - 1 }[/math] materials science, elasticity (displacement between particles in the body relative to a reference length)

References

  1. ↑ 1.0 1.1 1.2 "Table of Dimensionless Numbers". http://www.cchem.berkeley.edu/gsac/grad_info/prelims/binders/dimensionless_numbers.pdf. 
  2. ↑ 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. 
  3. ↑ Incropera, Frank P. (2007). Fundamentals of heat and mass transfer. John Wiley & Sons, Inc. p. 376. ISBN 9780470055540. https://archive.org/details/fundamentalsheat00incr_869. 
  4. ↑ Popov, Konstantin I.; Djokić, Stojan S.; Grgur, Branimir N. (2002). Fundamental Aspects of Electrometallurgy. Boston, MA: Springer. pp. 101–102. ISBN 978-0-306-47564-1. 
  5. ↑ Kuneš, J. (2012). "Technology and Mechanical Engineering". Dimensionless Physical Quantities in Science and Engineering. pp. 353–390. doi:10.1016/B978-0-12-416013-2.00008-7. ISBN 978-0-12-416013-2. 
  6. ↑ Asakuma, Y. (2020). "A dimensionless number for microwave non-equilibrium local heating through surfactant desorption". Colloids and Surfaces A: Physicochemical and Engineering Aspects. 591. pp. 124560. 
  7. ↑ Bagnold number
  8. ↑ Bhattacharjee S.; Grosshandler W.L. (1988). "The formation of wall jet near a high temperature wall under microgravity environment". ASME MTD 96: 711–6. Bibcode: 1988nht.....1..711B. 
  9. ↑ Paoletti S.; Rispoli F.; Sciubba E. (1989). "Calculation of exergetic losses in compact heat exchanger passager". ASME AES 10 (2): 21–9. 
  10. ↑ Bond number
  11. ↑ "Home". OnePetro. 2015-05-04. http://www.onepetro.org/mslib/servlet/onepetropreview?id=00020506. 
  12. ↑ Schetz, Joseph A. (1993). Boundary Layer Analysis. Englewood Cliffs, NJ: Prentice-Hall, Inc.. pp. 132–134. ISBN 0-13-086885-X. https://archive.org/details/boundarylayerana00sche. 
  13. ↑ "Fanning friction factor". http://www.engineering.uiowa.edu/~cee081/Exams/Final/Final.htm. 
  14. ↑ 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. Bibcode: 2001ChEnS..56.6337T. 
  15. ↑ Lockhart–Martinelli parameter
  16. ↑ "Manning coefficient". 10 June 2013. http://www.epa.gov/ORD/NRMRL/pubs/600r01043/600R01043chap2.pdf.  (109 KB)
  17. ↑ Richardson number
  18. ↑ Schmidt number
  19. ↑ Sommerfeld number
  20. ↑ Strouhal number, Engineering Toolbox
  21. ↑ 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. doi:10.1098/rspa.2000.0657. Bibcode: 2001RSPSA.457...87S. 
  22. ↑ Petritsch, G.; Mewes, D. (1999). "Experimental investigations of the flow patterns in the hot leg of a pressurized water reactor". Nuclear Engineering and Design 188: 75–84. doi:10.1016/S0029-5493(99)00005-9. 
  23. ↑ Weissenberg number
  24. ↑ Womersley number
  25. ↑ Barbot, S. (2019). "Slow-slip, slow earthquakes, period-two cycles, full and partial ruptures, and deterministic chaos in a single asperity fault". Tectonophysics 768: 228171. doi:10.1016/j.tecto.2019.228171. Bibcode: 2019Tectp.76828171B. 
  26. ↑ Fresnel number
  27. ↑ Courant–Friedrich–Levy number
  28. ↑ Feigenbaum constants
  29. ↑ Blondeau, J. (2021). "The influence of field size, goal size and number of players on the average number of goals scored per game in variants of football and hockey: the Pi-theorem applied to team sports". Journal of Quantitative Analysis in Sports 17 (2): 145–154. doi:10.1515/jqas-2020-0009. https://doi.org/10.1515/jqas-2020-0009. 
  30. ↑ Gain Ratio – Sheldon Brown
  31. ↑ "goal average". https://dictionary.cambridge.org/dictionary/english/goal-average. 
  32. ↑ "World Test Championship Playing Conditions: What's different?". https://icc-static-files.s3.amazonaws.com/ICC/document/2019/07/31/6b4241d8-1b33-44b5-8a83-579380989fb9/Changes-to-Test-PCs-for-WTC.pdf. 
  33. ↑ Behjani, Mohammadreza Alizadeh; Rahmanian, Nejat; Ghani, Nur Fardina bt Abdul; Hassanpour, Ali (2017). "An investigation on process of seeded granulation in a continuous drum granulator using DEM". Advanced Powder Technology 28 (10): 2456–2464. doi:10.1016/j.apt.2017.02.011. http://eprints.whiterose.ac.uk/113300/3/APT_Seeded_Granulation-accepted%20manuscript%20Feb%202017.pdf. 
  34. ↑ Alizadeh Behjani, Mohammadreza; Hassanpour, Ali; Ghadiri, Mojtaba; Bayly, Andrew (2017). "Numerical Analysis of the Effect of Particle Shape and Adhesion on the Segregation of Powder Mixtures" (in en). EPJ Web of Conferences 140: 06024. doi:10.1051/epjconf/201714006024. ISSN 2100-014X. Bibcode: 2017EPJWC.14006024A. 
  35. ↑ S.W. RIENSTRA, 2015, Fundamentals of Duct Acoustics, Von Karman Institute Lecture Notes
  36. ↑ 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. 
  37. ↑ Davis, Mark E.; Davis, Robert J. (2012). Fundamentals of Chemical Reaction Engineering. Dover. p. 215. ISBN 978-0-486-48855-4. 




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