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

chemistry (Proportion of "active" molecules or atoms)

Albedo

\alpha

\alpha= (1D) \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 carbon12 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 (residencetime 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{\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 crosssectional 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}}

optics, photography (ratio of focal length to diameter of aperture)

Föppl–von Kármán number

\gamma

\gamma = \frac{Y r^2}{\kappa}

virology, solid mechanics (thinshell 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 selfsimilar 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 liquidvapor 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 (countercurrent twophase 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 momentumtransport)

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}}

twophase 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}

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\,\frac\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)

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]}
