. 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
|
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
|
NBK
|
\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
|
Cf or fD
|
|
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
|
cd
|
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 (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
|
KC
|
\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
|
CL
|
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}
|
gas dynamics (compressible flow; dimensionless velocity)
|
Magnetic Reynolds number
|
Rm
|
\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
|
NP
|
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
|
Np
|
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
|
CP
|
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
|
Crr
|
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 Sk
|
\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 (Kf and Kb)
|
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]
|
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