List of dimensionless quantities

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

Name	Standard symbol	Definition	Field of application
Abbe number	V	${\displaystyle V={\frac {n_{d}-1}{n_{F}-n_{C}}}}$	optics (dispersion in optical materials)
Activity coefficient	${\displaystyle \gamma }$	${\displaystyle \gamma ={\frac {a}{x}}}$	chemistry (Proportion of "active" molecules or atoms)
Albedo	${\displaystyle \alpha }$	${\displaystyle \alpha =(1-D){\bar {\alpha }}(\theta _{i})+D{\bar {\bar {\alpha }}}}$	climatology, astronomy (reflectivity of surfaces or bodies)
Archimedes number	Ar	${\displaystyle \mathrm {Ar} ={\frac {gL^{3}\rho _{\ell }(\rho -\rho _{\ell })}{\mu ^{2}}}}$	fluid mechanics (motion of fluids due to density differences)
Arrhenius number	${\displaystyle \alpha }$	${\displaystyle \alpha ={\frac {E_{a}}{RT}}}$	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)
Atwood number	A	${\displaystyle \mathrm {A} ={\frac {\rho _{1}-\rho _{2}}{\rho _{1}+\rho _{2}}}}$	fluid mechanics (onset of instabilities in fluid mixtures due to density differences)
Bagnold number	Ba	${\displaystyle \mathrm {Ba} ={\frac {\rho d^{2}\lambda ^{1/2}\gamma }{\mu }}}$	fluid mechanics, geology (ratio of grain collision stresses to viscous fluid stresses in flow of a granular material such as grain and sand)[2]
Basic reproduction number	${\displaystyle R_{0}}$	number of infections caused on average by an infectious individual over entire infectious period	epidemiology
Bejan number (fluid mechanics)	Be	${\displaystyle \mathrm {Be} ={\frac {\Delta PL^{2}}{\mu \alpha }}}$	fluid mechanics (dimensionless pressure drop along a channel)[3]
Bejan number (thermodynamics)	Be	${\displaystyle \mathrm {Be} ={\frac {{\dot {S}}'_{\mathrm {gen} ,\,\Delta T}}{{\dot {S}}'_{\mathrm {gen} ,\,\Delta T}+{\dot {S}}'_{\mathrm {gen} ,\,\Delta p}}}}$	thermodynamics (ratio of heat transfer irreversibility to total irreversibility due to heat transfer and fluid friction)[4]
Bingham number	Bm	${\displaystyle \mathrm {Bm} ={\frac {\tau _{y}L}{\mu V}}}$	fluid mechanics, rheology (ratio of yield stress to viscous stress)[1]
Biot number	Bi	${\displaystyle \mathrm {Bi} ={\frac {hL_{C}}{k_{b}}}}$	heat transfer (surface vs. volume conductivity of solids)
Blake number	Bl or B	${\displaystyle \mathrm {B} ={\frac {u\rho }{\mu (1-\epsilon )D}}}$	geology, fluid mechanics, porous media (inertial over viscous forces in fluid flow through porous media)
Blondeau number	${\displaystyle B_{\kappa }}$	${\displaystyle \mathrm {B_{\kappa }} ={\frac {t_{g}v_{f}}{l_{mf}}}}$	sport science, team sports[5]
Bodenstein number	Bo or Bd	${\displaystyle \mathrm {Bo} =vL/{\mathcal {D}}=\mathrm {Re} \,\mathrm {Sc} }$	chemistry (residence-time distribution; similar to the axial mass transfer Peclet number)[6]
Bond number	Bo	${\displaystyle \mathrm {Bo} ={\frac {\rho aL^{2}}{\gamma }}}$	geology, fluid mechanics, porous media (buoyant versus capillary forces, similar to the Eötvös number) [7]
Brinkman number	Br	${\displaystyle \mathrm {Br} ={\frac {\mu U^{2}}{\kappa (T_{w}-T_{0})}}}$	heat transfer, fluid mechanics (conduction from a wall to a viscous fluid)
Brownell–Katz number	NBK	${\displaystyle \mathrm {N} _{\mathrm {BK} }={\frac {u\mu }{k_{\mathrm {rw} }\sigma }}}$	fluid mechanics (combination of capillary number and Bond number) [8]
Capillary number	Ca	${\displaystyle \mathrm {Ca} ={\frac {\mu V}{\gamma }}}$	porous media, fluid mechanics (viscous forces versus surface tension)
Chandrasekhar number	Q	${\displaystyle \mathrm {Q} ={\frac {{B_{0}}^{2}d^{2}}{\mu _{0}\rho \nu \lambda }}}$	magnetohydrodynamics (ratio of the Lorentz force to the viscosity in magnetic convection)
Colburn J factors	JM, JH, JD		turbulence; heat, mass, and momentum transfer (dimensionless transfer coefficients)
Coefficient of kinetic friction	${\displaystyle \mu _{k}}$		mechanics (friction of solid bodies in translational motion)
Coefficient of static friction	${\displaystyle \mu _{s}}$		mechanics (friction of solid bodies at rest)
Coefficient of determination	${\displaystyle R^{2}}$		statistics (proportion of variance explained by a statistical model)
Coefficient of variation	${\displaystyle {\frac {\sigma }{\mu }}}$	${\displaystyle {\frac {\sigma }{\mu }}}$	statistics (ratio of standard deviation to expectation)
Cohesion number	Coh	${\displaystyle Coh={\frac {1}{\rho g}}\left({\frac {\Gamma ^{5}}{{E^{*}}^{2}{R^{*}}^{8}}}\right)^{\frac {1}{3}}}$	Chemical engineering, material science, mechanics (A scale to show the energy needed for detaching two solid particles)[9][10]
Correlation	ρ or r	${\displaystyle {\frac {\operatorname {E} [(X-\mu _{X})(Y-\mu _{Y})]}{\sigma _{X}\sigma _{Y}}}}$	statistics (measure of linear dependence)
Cost of transport	COT	${\displaystyle \mathrm {COT} ={\frac {E}{mgd}}}$	energy efficiency, economics (ratio of energy input to kinetic motion)
Courant–Friedrich–Levy number	C or 𝜈	${\displaystyle C={\frac {u\,\Delta t}{\Delta x}}}$	mathematics (numerical solutions of hyperbolic PDEs)[11]
Damkohler number	Da	${\displaystyle \mathrm {Da} =k\tau }$	chemistry (reaction time scales vs. residence time)
Damping ratio	${\displaystyle \zeta }$	${\displaystyle \zeta ={\frac {c}{2{\sqrt {km}}}}}$	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	${\displaystyle \mathrm {Da} ={\frac {K}{d^{2}}}}$	porous media (ratio of permeability to cross-sectional area)
Dean number	D	${\displaystyle \mathrm {D} ={\frac {\rho Vd}{\mu }}\left({\frac {d}{2R}}\right)^{1/2}}$	turbulent flow (vortices in curved ducts)
Deborah number	De	${\displaystyle \mathrm {De} ={\frac {t_{\mathrm {c} }}{t_{\mathrm {p} }}}}$	rheology (viscoelastic fluids)
Decibel	dB		acoustics, electronics, control theory (ratio of two intensities or powers of a wave)
Drag coefficient	cd	${\displaystyle c_{\mathrm {d} }={\dfrac {2F_{\mathrm {d} }}{\rho v^{2}A}}\,,}$	aeronautics, fluid dynamics (resistance to fluid motion)
Dukhin number	Du	${\displaystyle \mathrm {Du} ={\frac {\kappa ^{\sigma }}{{\mathrm {K} _{m}}a}}}$	colloid science (ratio of electric surface conductivity to the electric bulk conductivity in heterogeneous systems)
Eckert number	Ec	${\displaystyle \mathrm {Ec} ={\frac {V^{2}}{c_{p}\Delta T}}}$	convective heat transfer (characterizes dissipation of energy; ratio of kinetic energy to enthalpy)
Ekman number	Ek	${\displaystyle \mathrm {Ek} ={\frac {\nu }{2D^{2}\Omega \sin \varphi }}}$	geophysics (viscous versus Coriolis forces)
Elasticity (economics)	E	${\displaystyle E_{x,y}={\frac {\partial \ln(x)}{\partial \ln(y)}}={\frac {\partial x}{\partial y}}{\frac {y}{x}}}$	economics (response of demand or supply to price changes)
Eötvös number	Eo	${\displaystyle \mathrm {Eo} ={\frac {\Delta \rho \,g\,L^{2}}{\sigma }}}$	fluid mechanics (shape of bubbles or drops)
Ericksen number	Er	${\displaystyle \mathrm {Er} ={\frac {\mu vL}{K}}}$	fluid dynamics (liquid crystal flow behavior; viscous over elastic forces)
Euler number	Eu	${\displaystyle \mathrm {Eu} ={\frac {\Delta {}p}{\rho V^{2}}}}$	hydrodynamics (stream pressure versus inertia forces)
Euler's number	e	${\displaystyle e=\displaystyle \sum \limits _{n=0}^{\infty }{\dfrac {1}{n!}}\approx 2.71828}$	mathematics (base of the natural logarithm)
Excess temperature coefficient	${\displaystyle \Theta _{r}}$	${\displaystyle \Theta _{r}={\frac {c_{p}(T-T_{e})}{U_{e}^{2}/2}}}$	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]
Feigenbaum constants	${\displaystyle \alpha }$, ${\displaystyle \delta }$	${\displaystyle \alpha \approx 2.50290,}$ ${\displaystyle \ \delta \approx 4.66920}$	chaos theory (period doubling)[14]
Fine-structure constant	${\displaystyle \alpha }$	${\displaystyle \alpha ={\frac {e^{2}}{4\pi \varepsilon _{0}\hbar c}}}$	quantum electrodynamics (QED) (coupling constant characterizing the strength of the electromagnetic interaction)
f-number	f	${\displaystyle f={\frac {\ell }{D}}}$	optics, photography (ratio of focal length to diameter of aperture)
Föppl–von Kármán number	${\displaystyle \gamma }$	${\displaystyle \gamma ={\frac {Yr^{2}}{\kappa }}}$	virology, solid mechanics (thin-shell buckling)
Fourier number	Fo	${\displaystyle \mathrm {Fo} ={\frac {\alpha t}{L^{2}}}}$	heat transfer, mass transfer (ratio of diffusive rate versus storage rate)
Fresnel number	F	${\displaystyle {\mathit {F}}={\frac {a^{2}}{L\lambda }}}$	optics (slit diffraction)[15]
Froude number	Fr	${\displaystyle \mathrm {Fr} ={\frac {v}{\sqrt {g\ell }}}}$	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)[16]
Galilei number	Ga	${\displaystyle \mathrm {Ga} ={\frac {g\,L^{3}}{\nu ^{2}}}}$	fluid mechanics (gravitational over viscous forces)
Golden ratio	${\displaystyle \varphi }$	${\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.61803}$	mathematics, aesthetics (long side length of self-similar rectangle)
Görtler number	G	${\displaystyle \mathrm {G} ={\frac {U_{e}\theta }{\nu }}\left({\frac {\theta }{R}}\right)^{1/2}}$	fluid dynamics (boundary layer flow along a concave wall)
Graetz number	Gz	${\displaystyle \mathrm {Gz} ={D_{H} \over L}\mathrm {Re} \,\mathrm {Pr} }$	heat transfer, fluid mechanics (laminar flow through a conduit; also used in mass transfer)
Grashof number	Gr	${\displaystyle \mathrm {Gr} _{L}={\frac {g\beta (T_{s}-T_{\infty })L^{3}}{\nu ^{2}}}}$	heat transfer, natural convection (ratio of the buoyancy to viscous force)
Hatta number	Ha	${\displaystyle \mathrm {Ha} ={\frac {N_{\mathrm {A} 0}}{N_{\mathrm {A} 0}^{\mathrm {phys} }}}}$	chemical engineering (adsorption enhancement due to chemical reaction)
Hagen number	Hg	${\displaystyle \mathrm {Hg} =-{\frac {1}{\rho }}{\frac {\mathrm {d} p}{\mathrm {d} x}}{\frac {L^{3}}{\nu ^{2}}}}$	heat transfer (ratio of the buoyancy to viscous force in forced convection)
Havnes parameter	${\displaystyle P_{H}}$	${\displaystyle P_{H}={\frac {Z_{d}n_{d}}{n_{i}}}}$	In Dusty plasma physics, ratio of the total charge ${\displaystyle Z_{d}}$ carried by the dust particles ${\displaystyle d}$ to the charge carried by the ions ${\displaystyle i}$, with ${\displaystyle n}$ the number density of particles
Helmholtz number	${\displaystyle He}$	${\displaystyle He={\frac {wa}{c_{0}}}=k_{0}a}$	The most important parameter in duct acoustics. If ${\displaystyle \omega }$ is the dimensional frequency, then ${\displaystyle k_{0}}$ is the corresponding free field wavenumber and ${\displaystyle He}$ is the corresponding dimensionless frequency [17]
Hydraulic gradient	i	${\displaystyle i={\frac {\mathrm {d} h}{\mathrm {d} l}}={\frac {h_{2}-h_{1}}{\mathrm {length} }}}$	fluid mechanics, groundwater flow (pressure head over distance)
Iribarren number	Ir	${\displaystyle \mathrm {Ir} ={\frac {\tan \alpha }{\sqrt {H/L_{0}}}}}$	wave mechanics (breaking surface gravity waves on a slope)
Jakob number	Ja	${\displaystyle \mathrm {Ja} ={\frac {c_{p}(T_{\mathrm {s} }-T_{\mathrm {sat} })}{\Delta H_{\mathrm {f} }}}}$	chemistry (ratio of sensible to latent energy absorbed during liquid-vapor phase change)[18]
Karlovitz number	Ka	${\displaystyle \mathrm {Ka} ={\frac {t_{F}}{t_{\eta }}}}$	turbulent combustion (characteristic chemical time scale to Kolmogorov time scale)
Keulegan–Carpenter number	KC	${\displaystyle \mathrm {K_{C}} ={\frac {V\,T}{L}}}$	fluid dynamics (ratio of drag force to inertia for a bluff object in oscillatory fluid flow)
Knudsen number	Kn	${\displaystyle \mathrm {Kn} ={\frac {\lambda }{L}}}$	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	${\displaystyle \mathrm {Ku} ={\frac {U_{h}\rho _{g}^{1/2}}{\left({\sigma g(\rho _{l}-\rho _{g})}\right)^{1/4}}}}$	fluid mechanics (counter-current two-phase flow)[19]
Laplace number	La	${\displaystyle \mathrm {La} ={\frac {\sigma \rho L}{\mu ^{2}}}}$	fluid dynamics (free convection within immiscible fluids; ratio of surface tension to momentum-transport)
Lewis number	Le	${\displaystyle \mathrm {Le} ={\frac {\alpha }{D}}={\frac {\mathrm {Sc} }{\mathrm {Pr} }}}$	heat and mass transfer (ratio of thermal to mass diffusivity)
Lift coefficient	CL	${\displaystyle C_{\mathrm {L} }={\frac {L}{q\,S}}}$	aerodynamics (lift available from an airfoil at a given angle of attack)
Lockhart–Martinelli parameter	${\displaystyle \chi }$	${\displaystyle \chi ={\frac {m_{\ell }}{m_{g}}}{\sqrt {\frac {\rho _{g}}{\rho _{\ell }}}}}$	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	${\displaystyle S={\frac {\mu _{0}LV_{A}}{\eta }}}$	plasma physics (ratio of a resistive time to an Alfvén wave crossing time in a plasma)
Mach number	M or Ma	${\displaystyle \mathrm {M} ={\frac {v}{v_{\mathrm {sound} }}}}$	gas dynamics (compressible flow; dimensionless velocity)
Magnetic Reynolds number	Rm	${\displaystyle \mathrm {R} _{\mathrm {m} }={\frac {UL}{\eta }}}$	magnetohydrodynamics (ratio of magnetic advection to magnetic diffusion)
Manning roughness coefficient	n		open channel flow (flow driven by gravity)[21]
Marangoni number	Mg	${\displaystyle \mathrm {Mg} =-{\frac {\mathrm {d} \sigma }{\mathrm {d} T}}{\frac {L\Delta T}{\eta \alpha }}}$	fluid mechanics (Marangoni flow; thermal surface tension forces over viscous forces)
Markstein number	${\displaystyle {\mathcal {M}}}$	${\displaystyle {\mathcal {M}}={\frac {{\mathcal {L}}_{b}}{\delta _{L}}}}$	fluid dynamics, combustion (turbulent combustion flames)
Morton number	Mo	${\displaystyle \mathrm {Mo} ={\frac {g\mu _{c}^{4}\,\Delta \rho }{\rho _{c}^{2}\sigma ^{3}}}}$	fluid dynamics (determination of bubble/drop shape)
Nusselt number	Nu	${\displaystyle \mathrm {Nu} ={\frac {hd}{k}}}$	heat transfer (forced convection; ratio of convective to conductive heat transfer)
Ohnesorge number	Oh	${\displaystyle \mathrm {Oh} ={\frac {\mu }{\sqrt {\rho \sigma L}}}={\frac {\sqrt {\mathrm {We} }}{\mathrm {Re} }}}$	fluid dynamics (atomization of liquids, Marangoni flow)
Péclet number	Pe	${\displaystyle \mathrm {Pe} ={\frac {du\rho c_{p}}{k}}=\mathrm {Re} \,\mathrm {Pr} }$	heat transfer (advection–diffusion problems; total momentum transfer to molecular heat transfer)
Peel number	NP	${\displaystyle N_{\mathrm {P} }={\frac {\text{Restoring force}}{\text{Adhesive force}}}}$	coating (adhesion of microstructures with substrate)[22]
Perveance	K	${\displaystyle {K}={\frac {I}{I_{0}}}\,{\frac {2}{{\beta }^{3}{\gamma }^{3}}}(1-\gamma ^{2}f_{e})}$	charged particle transport (measure of the strength of space charge in a charged particle beam)
pH	${\displaystyle \mathrm {pH} }$	${\displaystyle \mathrm {pH} =-\log _{10}(a_{{\textrm {H}}^{+}})}$	chemistry (the measure of the acidity or basicity of an aqueous solution)
Pi	${\displaystyle \pi }$	${\displaystyle \pi ={\frac {C}{d}}\approx 3.14159}$	mathematics (ratio of a circle's circumference to its diameter)
Pierce parameter	${\displaystyle C}$	${\displaystyle C^{3}={\frac {Z_{c}I_{K}}{4V_{K}}}}$	Traveling wave tube
Pixel	px		digital imaging (smallest addressable unit)
Beta (plasma physics)	${\displaystyle \beta }$	${\displaystyle \beta ={\frac {nk_{B}T}{B^{2}/2\mu _{0}}}}$	Plasma (physics) and Fusion power. Ratio of plasma thermal pressure to magnetic pressure, controlling the level of turbulence in a magnetised plasma.
Poisson's ratio	${\displaystyle \nu }$	${\displaystyle \nu =-{\frac {\mathrm {d} \varepsilon _{\mathrm {trans} }}{\mathrm {d} \varepsilon _{\mathrm {axial} }}}}$	elasticity (strain in transverse and longitudinal direction)
Porosity	${\displaystyle \phi }$	${\displaystyle \phi ={\frac {V_{\mathrm {V} }}{V_{\mathrm {T} }}}}$	geology, porous media (void fraction of the medium)
Power factor	pf	${\displaystyle pf={\frac {P}{S}}}$	electrical (real power to apparent power)
Power number	Np	${\displaystyle N_{p}={P \over \rho n^{3}d^{5}}}$	electronics (power consumption by agitators; resistance force versus inertia force)
Prandtl number	Pr	${\displaystyle \mathrm {Pr} ={\frac {\nu }{\alpha }}={\frac {c_{p}\mu }{k}}}$	heat transfer (ratio of viscous diffusion rate over thermal diffusion rate)
Prater number	β	${\displaystyle \beta ={\frac {-\Delta H_{r}D_{TA}^{e}C_{AS}}{\lambda ^{e}T_{s}}}}$	reaction engineering (ratio of heat evolution to heat conduction within a catalyst pellet)[23]
Pressure coefficient	CP	${\displaystyle C_{p}={p-p_{\infty } \over {\frac {1}{2}}\rho _{\infty }V_{\infty }^{2}}}$	aerodynamics, hydrodynamics (pressure experienced at a point on an airfoil; dimensionless pressure variable)
Q factor	Q	${\displaystyle Q=2\pi f_{r}{\frac {\text{Energy Stored}}{\text{Power Loss}}}}$	physics, engineering (damping of oscillator or resonator; energy stored versus energy lost)
Radian measure	rad	${\displaystyle {\text{arc length}}/{\text{radius}}}$	mathematics (measurement of planar angles, 1 radian = 180/π degrees)
Rayleigh number	Ra	${\displaystyle \mathrm {Ra} _{x}={\frac {g\beta }{\nu \alpha }}(T_{s}-T_{\infty })x^{3}}$	heat transfer (buoyancy versus viscous forces in free convection)
Refractive index	n	${\displaystyle n={\frac {c}{v}}}$	electromagnetism, optics (speed of light in a vacuum over speed of light in a material)
Relative density	RD	${\displaystyle RD={\frac {\rho _{\mathrm {substance} }}{\rho _{\mathrm {reference} }}}}$	hydrometers, material comparisons (ratio of density of a material to a reference material—usually water)
Relative permeability	${\displaystyle \mu _{r}}$	${\displaystyle \mu _{r}={\frac {\mu }{\mu _{0}}}}$	magnetostatics (ratio of the permeability of a specific medium to free space)
Relative permittivity	${\displaystyle \varepsilon _{r}}$	${\displaystyle \varepsilon _{r}={\frac {C_{x}}{C_{0}}}}$	electrostatics (ratio of capacitance of test capacitor with dielectric material versus vacuum)
Reynolds number	Re	${\displaystyle \mathrm {Re} ={\frac {vL\rho }{\mu }}}$	fluid mechanics (ratio of fluid inertial and viscous forces)[1]
Richardson number	Ri	${\displaystyle \mathrm {Ri} ={\frac {gh}{u^{2}}}={\frac {1}{\mathrm {Fr} ^{2}}}}$	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	${\displaystyle C_{rr}={\frac {F}{N_{f}}}}$	vehicle dynamics (ratio of force needed for motion of a wheel over the normal force)
Roshko number	Ro	${\displaystyle \mathrm {Ro} ={fL^{2} \over \nu }=\mathrm {St} \,\mathrm {Re} }$	fluid dynamics (oscillating flow, vortex shedding)
Rossby number	Ro	${\displaystyle \mathrm {Ro} ={\frac {U}{Lf}}}$	geophysics (ratio of inertial to Coriolis force)
Rouse number	P or Z	${\displaystyle \mathrm {P} ={\frac {w_{s}}{\kappa u_{*}}}}$	sediment transport (ratio of the sediment fall velocity and the upwards velocity of grain)
Schmidt number	Sc	${\displaystyle \mathrm {Sc} ={\frac {\nu }{D}}}$	mass transfer (viscous over molecular diffusion rate)[25]
Shape factor	H	${\displaystyle H={\frac {\delta ^{*}}{\theta }}}$	boundary layer flow (ratio of displacement thickness to momentum thickness)
Sherwood number	Sh	${\displaystyle \mathrm {Sh} ={\frac {KL}{D}}}$	mass transfer (forced convection; ratio of convective to diffusive mass transport)
Shields parameter	${\displaystyle \tau _{*}}$ or ${\displaystyle \theta }$	${\displaystyle \tau _{\ast }={\frac {\tau }{(\rho _{s}-\rho )gD}}}$	sediment transport (threshold of sediment movement due to fluid motion; dimensionless shear stress)
Sommerfeld number	S	${\displaystyle \mathrm {S} =\left({\frac {r}{c}}\right)^{2}{\frac {\mu N}{P}}}$	hydrodynamic lubrication (boundary lubrication)[26]
Specific gravity	SG		(same as Relative density)
Stanton number	St	${\displaystyle \mathrm {St} ={\frac {h}{c_{p}\rho V}}={\frac {\mathrm {Nu} }{\mathrm {Re} \,\mathrm {Pr} }}}$	heat transfer and fluid dynamics (forced convection)
Stefan number	Ste	${\displaystyle \mathrm {Ste} ={\frac {c_{p}\Delta T}{L}}}$	phase change, thermodynamics (ratio of sensible heat to latent heat)
Stokes number	Stk or Sk	${\displaystyle \mathrm {Stk} ={\frac {\tau U_{o}}{d_{c}}}}$	particles suspensions (ratio of characteristic time of particle to time of flow)
Strain	${\displaystyle \epsilon }$	${\displaystyle \epsilon ={\cfrac {\partial {F}}{\partial {X}}}-1}$	materials science, elasticity (displacement between particles in the body relative to a reference length)
Strouhal number	St or Sr	${\displaystyle \mathrm {St} ={\omega L \over v}}$	fluid dynamics (continuous and pulsating flow; nondimensional frequency)[27]
Stuart number	N	${\displaystyle \mathrm {N} ={\frac {B^{2}L_{c}\sigma }{\rho U}}={\frac {\mathrm {Ha} ^{2}}{\mathrm {Re} }}}$	magnetohydrodynamics (ratio of electromagnetic to inertial forces)
Taylor number	Ta	${\displaystyle \mathrm {Ta} ={\frac {4\Omega ^{2}R^{4}}{\nu ^{2}}}}$	fluid dynamics (rotating fluid flows; inertial forces due to rotation of a fluid versus viscous forces)
Transmittance	T	${\displaystyle T={\frac {I}{I_{0}}}}$	optics, spectroscopy (the ratio of the intensities of radiation exiting through and incident on a sample)
Ursell number	U	${\displaystyle \mathrm {U} ={\frac {H\,\lambda ^{2}}{h^{3}}}}$	wave mechanics (nonlinearity of surface gravity waves on a shallow fluid layer)
Vadasz number	Va	${\displaystyle \mathrm {Va} ={\frac {\phi \,\mathrm {Pr} }{\mathrm {Da} }}}$	porous media (governs the effects of porosity ${\displaystyle \phi }$, the Prandtl number and the Darcy number on flow in a porous medium) [28]
van 't Hoff factor	i	${\displaystyle i=1+\alpha (n-1)}$	quantitative analysis (Kf and Kb)
Wallis parameter	j*	${\displaystyle j^{*}=R\left({\frac {\omega \rho }{\mu }}\right)^{\frac {1}{2}}}$	multiphase flows (nondimensional superficial velocity)[29]
Wagner number	Wa	${\displaystyle \mathrm {Wa} ={\frac {\kappa }{l}}{\frac {\mathrm {d} \eta }{\mathrm {d} i}}}$	electrochemistry (ratio of kinetic polarization resistance to solution ohmic resistance in an electrochemical cell)[30]
Weaver flame speed number	Wea	${\displaystyle \mathrm {Wea} ={\frac {w}{w_{\mathrm {H} }}}100}$	combustion (laminar burning velocity relative to hydrogen gas)[31]
Weber number	We	${\displaystyle \mathrm {We} ={\frac {\rho v^{2}l}{\sigma }}}$	multiphase flow (strongly curved surfaces; ratio of inertia to surface tension)
Weissenberg number	Wi	${\displaystyle \mathrm {Wi} ={\dot {\gamma }}\lambda }$	viscoelastic flows (shear rate times the relaxation time)[32]
Womersley number	${\displaystyle \alpha }$	${\displaystyle \alpha =R\left({\frac {\omega \rho }{\mu }}\right)^{\frac {1}{2}}}$	biofluid mechanics (continuous and pulsating flows; ratio of pulsatile flow frequency to viscous effects)[33]
Zel'dovich number	${\displaystyle \beta }$	${\displaystyle \beta ={\frac {E}{RT_{f}}}{\frac {T_{f}-T_{o}}{T_{f}}}}$	fluid dynamics, Combustion (Measure of activation energy)