Wednesday 13 August 2014

Dimensionless quantity

From Wikipedia, the chargeless encyclopedia
In dimensional analysis, a dimensionless abundance or abundance of ambit one is a abundance after an associated concrete dimension. It is appropriately a "pure" number, and as such consistently has a ambit of 1.[1] Dimensionless quantities are broadly acclimated in mathematics, physics, engineering, economics, and in accustomed activity (such as in counting). Numerous acclaimed quantities, such as ПЂ, e, and П†, are dimensionless. By contrast, non-dimensionless quantities are abstinent in units of length, area, time, etc.
Dimensionless quantities are generally authentic as articles or ratios of quantities that are not dimensionless, but whose ambit abolish out if their admiral are multiplied. This is the case, for instance, with the engineering strain, a admeasurement of deformation. It is authentic as change in length, disconnected by antecedent length, but back these quantities both accept ambit L (length), the aftereffect is a dimensionless quantity.
Contents [hide]
1 Properties
2 Buckingham ПЂ theorem
2.1 Example
3 Standards efforts
4 Examples
5 Dimensionless concrete constants
6 List of dimensionless quantities
7 See also
8 References
9 External links

Properties

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Even admitting a dimensionless abundance has no concrete ambit associated with it, it can still accept dimensionless units. To appearance the abundance getting abstinent (for archetype accumulation atom or birthmark fraction), it is sometimes accessible to use the aforementioned units in both the numerator and denominator (kg/kg or mol/mol). The abundance may aswell be accustomed as a arrangement of two altered units that accept the aforementioned ambit (for instance, ablaze years over meters). This may be the case if artful slopes in graphs, or if authoritative assemblage conversions. Such characters does not announce the attendance of concrete dimensions, and is absolutely a notational convention. Other accepted dimensionless units are % (= 0.01), ‰ (= 0.001), ppm (= 10−6), ppb (= 10−9), ppt (= 10−12) and bend units (degrees, radians, grad). Units of amount such as the dozen and the gross are aswell dimensionless.
The arrangement of two quantities with the aforementioned ambit is dimensionless, and has the aforementioned amount behindhand of the units acclimated to account them. For instance, if physique A exerts a force of consequence F on physique B, and B exerts a force of consequence f on A, again the arrangement F/f is consistently according to 1, behindhand of the absolute units acclimated to admeasurement F and f. This is a axiological acreage of dimensionless ACCOMMODATION ANDfollows from the acceptance that the laws of physics are absolute of the arrangement of units acclimated in their expression. In this case, if the arrangement F/f was not consistently according to 1, but afflicted if one switched from SI to CGS, that would beggarly that Newton's Third Law's accuracy or canard would depend on the arrangement of units used, which would belie this axiological hypothesis. This acceptance that the laws of physics are not accidental aloft a specific assemblage arrangement is the base for the Buckingham ПЂ theorem. A account of this assumption is that any concrete law can be bidding as an character involving alone dimensionless combinations (ratios or products) of the variables affiliated by the law (e. g., burden and aggregate are affiliated by Boyle's Law – they are inversely proportional). If the dimensionless combinations' ethics afflicted with the systems of units, again the blueprint would not be an identity, and Buckingham's assumption would not hold.

Buckingham π theorem

Another aftereffect of the Buckingham ПЂ assumption of dimensional assay is that the anatomic assurance amid a assertive amount (say, n) of variables can be bargain by the amount (say, k) of absolute ambit occurring in those variables to accord a set of p = n в€’ k independent, dimensionless quantities. For the purposes of the experimenter, altered systems that allotment the aforementioned description by dimensionless abundance are equivalent.
Example[edit]
The ability burning of a stirrer with a accustomed appearance is a action of the body and the bendability of the aqueous to be stirred, the admeasurement of the stirrer accustomed by its diameter, and the acceleration of the stirrer. Therefore, we accept n = 5 variables apery our example.
Those n = 5 variables are congenital up from k = 3 dimensions:
Length: L (m)
Time: T (s)
Mass: M (kg)
According to the ПЂ-theorem, the n = 5 variables can be bargain by the k = 3 ambit to anatomy p = n в€’ k = 5 в€’ 3 = 2 absolute dimensionless numbers, which are, in case of the stirrer:
Reynolds amount (a dimensionless amount anecdotic the aqueous breeze regime)
Power amount (describing the stirrer and aswell involves the body of the fluid)

Examples

Consider this example: Sarah says, "Out of every 10 apples I gather, 1 is rotten." The rotten-to-gathered arrangement is (1 apple) / (10 apples) = 0.1 = 10%, which is a dimensionless quantity.
Plane angles – An bend is abstinent as the arrangement of the breadth of a circle's arc subtended by an bend whose acme is the centre of the amphitheater to some added length. The ratio—i.e., breadth disconnected by length—is dimensionless. When application radians as the unit, the breadth that is compared is the breadth of the ambit of the circle. When application amount as the units, the arc's breadth is compared to 1/360 of the ambit of the circle.
In the case of the dimensionless abundance ПЂ, getting the arrangement of a circle's ambit to its diameter, the amount would be connected behindhand of what assemblage is acclimated to admeasurement a circle's ambit and bore (e.g., centimetres, miles, light-years, etc.), as continued as the aforementioned assemblage is acclimated for both.

Dimensionless physical constants

Certain axiological concrete constants, such as the acceleration of ablaze in a vacuum, the accepted gravitational constant, Planck's connected and Boltzmann's connected can be normalized to 1 if adapted units for time, length, mass, charge, and temperature are chosen. The consistent arrangement of units is accepted as the accustomed units. However, not all concrete constants can be normalized in this fashion. For example, the ethics of the afterward constants are absolute of the arrangement of units and have to be bent experimentally:
α ≈ 1/137.036, the accomplished anatomy connected which is the coupling connected for the electromagnetic interaction;
β (or μ) ≈ 1836, the proton-to-electron accumulation ratio. This arrangement is the blow accumulation of the proton disconnected by that of the electron. An akin arrangement can be authentic for any elementary particle;
О±s, the coupling connected for the able force;
αG ≈ 1.75×10−45, the gravitational coupling constant.

List of dimensionless quantities

All numbers are dimensionless quantities. Certain dimensionless quantities of some accent are accustomed 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, astrochemistry (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 physique differences)
Arrhenius number \alpha \alpha = \frac{E_a}{RT} chemistry (ratio of activation activity to thermal energy)[5]
Atomic weight M chemistry (mass of atom over one diminutive accumulation unit, u, breadth carbon-12 is absolutely 12 u)
Atwood number A \mathrm{A} = \frac{\rho_1 - \rho_2} {\rho_1 + \rho_2} fluid mechanics (onset of instabilities in aqueous mixtures due to physique differences)
Bagnold number Ba \mathrm{Ba} = \frac{\rho d^2 \lambda^{1/2} \gamma}{\mu} fluid mechanics, cartography (ratio of atom blow stresses to adhesive aqueous stresses in breeze of a diminutive actual such as atom and sand)[6]
Bejan number
(fluid mechanics) Be \mathrm{Be} = \frac{\Delta P L^2} {\mu \alpha} fluid mechanics (dimensionless burden bead forth 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 calefaction alteration irreversibility to absolute irreversibility due to calefaction alteration and aqueous friction)[8]
Bingham number Bm \mathrm{Bm} = \frac{ \tau_y L }{ \mu V } fluid mechanics, rheology (ratio of crop accent to adhesive stress)[5]
Biot number Bi \mathrm{Bi} = \frac{h L_C}{k_b} heat alteration (surface vs. aggregate application of solids)
Blake number Bl or B \mathrm{B} = \frac{u \rho}{\mu (1 - \epsilon) D} geology, aqueous mechanics, absorptive media (inertial over adhesive armament in aqueous breeze through absorptive media)
Bodenstein number Bo or Bd \mathrm{Bo} = vL/\mathcal{D} = \mathrm{Re}\, \mathrm{Sc} chemistry (residence-time distribution; agnate to the axial accumulation alteration Peclet number)[9]
Bond number Bo \mathrm{Bo} = \frac{\rho a L^2}{\gamma} geology, aqueous mechanics, absorptive media (buoyant against capilary forces, agnate to the Eötvös number) [10]
Brinkman number Br \mathrm{Br} = \frac {\mu U^2}{\kappa (T_w - T_0)} heat transfer, aqueous mechanics (conduction from a bank to a adhesive fluid)
Brownell–Katz number NBK \mathrm{N}_\mathrm{BK} = \frac{u \mu}{k_\mathrm{rw}\sigma} fluid mechanics (combination of capillary amount and Bond number) [11]
Capillary number Ca \mathrm{Ca} = \frac{\mu V}{\gamma} porous media, aqueous mechanics (viscous armament against credible 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 bendability in alluring convection)
Colburn J factors JM, JH, JD turbulence; heat, mass, and drive alteration (dimensionless alteration coefficients)
Coefficient of active friction \mu_k mechanics (friction of solid bodies in translational motion)
Coefficient of changeless friction \mu_s mechanics (friction of solid bodies at rest)
Coefficient of determination R^2 statistics (proportion of about-face explained by a statistical model)
Coefficient of variation \frac{\sigma}{\mu} \frac{\sigma}{\mu} statistics (ratio of accepted aberration to expectation)
Correlation ПЃ 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}} breadth \bar x = \sum_{k=1}^n x_k/n and analogously for \bar y statistics (measure of beeline dependence)
Courant–Friedrich–Levy number C or 𝜈 C = \frac {u\,\Delta t} {\Delta x} mathematics (numerical solutions of abstract PDEs)[12]
Damkohler number Da \mathrm{Da} = k \tau chemistry (reaction time scales vs. abode time)
Damping ratio \zeta \zeta = \frac{c}{2 \sqrt{km}} mechanics (the akin of damping in a system)
Darcy abrasion factor Cf or fD fluid mechanics (fraction of burden losses due to abrasion in a pipe; four times the Fanning abrasion 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 breeze (vortices in arced ducts)
Deborah number De \mathrm{De} = \frac{t_\mathrm{c}}{t_\mathrm{p}} rheology (viscoelastic fluids)
Decibel dB acoustics, electronics, ascendancy approach (ratio of two intensities or admiral of a wave)
Drag coefficient cd c_\mathrm{d} = \dfrac{2 F_\mathrm{d}}{\rho v^2 A}\, , aeronautics, aqueous dynamics (resistance to aqueous motion)
Dukhin number Du \mathrm{Du} = \frac{\kappa^{\sigma}}{{\Kappa_m} a} colloid science (ratio of electric credible application to the electric aggregate application in amalgamate systems)
Eckert number Ec \mathrm{Ec} = \frac{V^2}{c_p\Delta T} convective calefaction alteration (characterizes amusement of energy; arrangement of active activity to enthalpy)
Ekman number Ek \mathrm{Ek} = \frac{\nu}{2D^2\Omega\sin\varphi} geophysics (viscous against 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 appeal or accumulation to amount 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 clear breeze behavior; adhesive over adaptable forces)
Euler number Eu \mathrm{Eu}=\frac{\Delta{}p}{\rho V^2} hydrodynamics (stream burden against apathy forces)
Euler's number e e = \displaystyle\sum\limits_{n = 0}^{ \infty} \dfrac{1}{n!} = \approx 2.71828 mathematics (base of the accustomed logarithm)
Excess temperature coefficient \Theta_r \Theta_r = \frac{c_p (T-T_e)}{U_e^2/2} heat transfer, aqueous dynamics (change in centralized activity against active energy)[13]
Fanning abrasion factor f fluid mechanics (fraction of burden losses due to abrasion in a pipe; 1/4th the Darcy abrasion factor)[14]
Feigenbaum constants \alpha, \delta \alpha \approx 2.50290,
\ \delta \approx 4.66920 chaos approach (period doubling)[15]
Fine anatomy constant \alpha \alpha = \frac{e^2}{2\varepsilon_0 hc} quantum electrodynamics (QED) (coupling connected anecdotic the backbone of the electromagnetic interaction)
f-number f f = \frac {{\ell}}{{D}} optics, photography (ratio of focal breadth to bore 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, accumulation alteration (ratio of deviating amount against accumulator 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 credible behaviour; arrangement of a body's apathy to gravitational forces)
Gain – electronics (signal achievement to arresting input)
Gain ratio – bicycling (system of apery gearing; breadth catholic over breadth pedaled)[17]
Galilei number Ga \mathrm{Ga} = \frac{g\, L^3}{\nu^2} fluid mechanics (gravitational over adhesive forces)
Golden ratio \varphi \varphi = \frac{1+\sqrt{5}}{2} \approx 1.61803 mathematics, aesthetics (long ancillary breadth of self-similar rectangle)
Görtler number G \mathrm{G} = \frac{U_e \theta}{\nu} \left( \frac{\theta}{R} \right)^{1/2} fluid dynamics (boundary band breeze forth a biconcave wall)
Graetz number Gz \mathrm{Gz} = {D_H \over L} \mathrm{Re}\, \mathrm{Pr} heat transfer, aqueous mechanics (laminar breeze through a conduit; aswell acclimated in accumulation transfer)
Grashof number Gr \mathrm{Gr}_L = \frac{g \beta (T_s - T_\infty ) L^3}{\nu ^2} heat transfer, accustomed alteration (ratio of the airiness to adhesive force)
Gravitational coupling constant \alpha_G \alpha_G=\frac{Gm_e^2}{\hbar c} gravitation (attraction amid two massy elementary particles; akin to the Fine anatomy constant)
Hatta number Ha \mathrm{Ha} = \frac{N_{\mathrm{A}0}}{N_{\mathrm{A}0}^{\mathrm{phys}}} chemical engineering (adsorption accessory due to actinic reaction)
Hagen number Hg \mathrm{Hg} = -\frac{1}{\rho}\frac{\mathrm{d} p}{\mathrm{d} x}\frac{L^3}{\nu^2} heat alteration (ratio of the airiness to adhesive force in affected convection)
Hydraulic gradient i i = \frac{\mathrm{d}h}{\mathrm{d}l} = \frac{h_2 - h_1}{\mathrm{length}} fluid mechanics, groundwater breeze (pressure arch over distance)
Iribarren number Ir \mathrm{Ir} = \frac{\tan \alpha}{\sqrt{H/L_0}} wave mechanics (breaking credible force after-effects on a slope)
Jakob number Ja \mathrm{Ja} = \frac{c_p (T_\mathrm{s} - T_\mathrm{sat}) }{\Delta H_{\mathrm{f}} } chemistry (ratio of alive to abeyant activity captivated during liquid-vapor appearance change)[18]
Karlovitz number Ka \mathrm{Ka} = k t_c turbulent agitation (characteristic breeze time times blaze amplitude rate)
Keulegan–Carpenter number KC \mathrm{K_C} = \frac{V\,T}{L} fluid dynamics (ratio of annoyance force to apathy for a barefaced article in oscillatory aqueous flow)
Knudsen number Kn \mathrm{Kn} = \frac {\lambda}{L} gas dynamics (ratio of the atomic beggarly chargeless aisle breadth to a adumbrative concrete breadth 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 alteration aural immiscible fluids; arrangement of credible astriction to momentum-transport)
Lewis number Le \mathrm{Le} = \frac{\alpha}{D} = \frac{\mathrm{Sc}}{\mathrm{Pr}} heat and accumulation alteration (ratio of thermal to accumulation diffusivity)
Lift coefficient CL C_\mathrm{L} = \frac{L}{q\,S} aerodynamics (lift accessible from an airfoil at a accustomed bend of attack)
Lockhart–Martinelli parameter \chi \chi = \frac{m_\ell}{m_g} \sqrt{\frac{\rho_g}{\rho_\ell}} two-phase breeze (flow of wet gases; aqueous fraction)[20]
Love numbers h, k, l geophysics (solidity of apple and added planets)
Lundquist number S S = \frac{\mu_0LV_A}{\eta} plasma physics (ratio of a arresting time to an AlfvГ©n beachcomber bridge 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 Rm \mathrm{R}_\mathrm{m} = \frac{U L}{\eta} magnetohydrodynamics (ratio of alluring advection to alluring diffusion)
Manning acerbity coefficient n open approach breeze (flow apprenticed 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 credible astriction armament over adhesive 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 alteration (forced convection; arrangement of convective to conductive calefaction 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 alteration (advection–diffusion problems; absolute drive alteration to atomic calefaction 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{{I}}{{I_0}}\,\frac{{2}}{{\beta}^3{\gamma}^3} (1-\gamma^2f_e) charged atom carriage (measure of the backbone of amplitude allegation in a answerable atom beam)
pH \mathrm{pH} \mathrm{pH} = - \log_{10}(a_{\textrm{H}^+}) = \log_{10}\left(\frac{1}{a_{\textrm{H}^+}}\right) the admeasurement of the acidity or basicity of an aqueous solution
Pi \pi \pi = \frac{C}{d} \approx 3.14159 mathematics (ratio of a circle's ambit 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 axle and longitudinal direction)
Porosity \phi \phi = \frac{V_\mathrm{V}}{V_\mathrm{T}} geology, absorptive media (void atom of the medium)
Power factor P/S electronics (real ability to credible power)
Power number Np N_p = {P\over \rho n^3 d^5} electronics (power afire by agitators; attrition force against apathy force)
Prandtl number Pr \mathrm{Pr} = \frac{\nu}{\alpha} = \frac{c_p \mu}{k} heat alteration (ratio of adhesive circulation amount over thermal circulation rate)
Prater number ОІ \beta = \frac{-\Delta H_r D_{TA}^e C_{AS}}{\lambda^e T_s} reaction engineering (ratio of calefaction change to calefaction advice aural a agitator pellet)[23]
Pressure coefficient CP C_p = {p - p_\infty \over \frac{1}{2} \rho_\infty V_\infty^2} aerodynamics, hydrodynamics (pressure accomplished at a point on an airfoil; dimensionless burden variable)
Q factor Q physics, engineering (damping of oscillator or resonator; activity stored against activity lost)
Radian measure rad \text{arc length}/\text{radius} mathematics (measurement of collapsed angles, 1 radian = 180/ПЂ degrees)
Rayleigh number Ra \mathrm{Ra}_{x} = \frac{g \beta} {\nu \alpha} (T_s - T_\infin) x^3 heat alteration (buoyancy against adhesive armament in chargeless convection)
Refractive index n n=\frac{c}{v} electromagnetism, eyes (speed of ablaze in a exhaustion over acceleration of ablaze in a material)
Relative density RD RD = \frac{\rho_\mathrm{substance}}{\rho_\mathrm{reference}} hydrometers, actual comparisons (ratio of physique of a actual to a advertence material—usually water)
Relative permeability \mu_r \mu_r = \frac{\mu}{\mu_0} magnetostatics (ratio of the permeability of a specific average to chargeless space)
Relative permittivity \varepsilon_r \varepsilon_{r} = \frac{C_{x}} {C_{0}} electrostatics (ratio of capacitance of assay capacitor with dielectric actual against vacuum)
Reynolds number Re \mathrm{Re} = \frac{vL\rho}{\mu} fluid mechanics (ratio of aqueous inertial and adhesive forces)[5]
Richardson number Ri \mathrm{Ri} = \frac{gh}{u^2} = \frac{1}{\mathrm{Fr}^2} fluid dynamics (effect of airiness on breeze stability; arrangement of abeyant over active energy)[24]
Rockwell scale – mechanical acerbity (indentation acerbity of a material)
Rolling attrition coefficient Crr C_{rr} = \frac{F}{N_f} vehicle dynamics (ratio of force bare for motion of a caster over the accustomed force)
Roshko number Ro \mathrm{Ro} = {f L^{2}\over \nu} =\mathrm{St}\,\mathrm{Re} fluid dynamics (oscillating flow, amphitheater 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 carriage (ratio of the debris abatement acceleration and the upwards acceleration of grain)
Schmidt number Sc \mathrm{Sc} = \frac{\nu}{D} mass alteration (viscous over atomic circulation rate)[25]
Shape factor H H = \frac {\delta^*}{\theta} boundary band breeze (ratio of displacement array to drive thickness)
Sherwood number Sh \mathrm{Sh} = \frac{K L}{D} mass alteration (forced convection; arrangement of convective to deviating accumulation transport)
Shields parameter \tau_* or \theta \tau_{\ast} = \frac{\tau}{(\rho_s - \rho) g D} sediment carriage (threshold of debris movement due to aqueous motion; dimensionless microburst 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 About density)
Stanton number St \mathrm{St} = \frac{h}{c_p \rho V} = \frac{\mathrm{Nu}}{\mathrm{Re}\,\mathrm{Pr}} heat alteration and aqueous dynamics (forced convection)
Stefan number Ste \mathrm{Ste} = \frac{c_p \Delta T}{L} phase change, thermodynamics (ratio of alive calefaction to abeyant heat)
Stokes number Stk or Sk \mathrm{Stk} = \frac{\tau U_o}{d_c} particles suspensions (ratio of appropriate time of atom to time of flow)
Strain \epsilon \epsilon = \cfrac{\partial{F}}{\partial{X}} - 1 materials science, animation (displacement amid particles in the physique about to a advertence 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 aqueous flows; inertial armament due to circling of a aqueous against adhesive forces)
Ursell number U \mathrm{U} = \frac{H\, \lambda^2}{h^3} wave mechanics (nonlinearity of credible force after-effects on a bank aqueous layer)
Vadasz number Va \mathrm{Va} = \frac{\phi\, \mathrm{Pr}}{\mathrm{Da}} porous media (governs the furnishings of porosity \phi, the Prandtl amount and the Darcy amount on breeze in a absorptive medium) [28]
van 't Hoff factor i i = 1 + \alpha (n - 1) quantitative assay (Kf and Kb)
Wallis parameter j* j^* = R \left( \frac{\omega \rho}{\mu} \right)^\frac{1}{2} multiphase flows (nondimensional apparent velocity)[29]
Weaver blaze acceleration number Wea \mathrm{Wea} = \frac{w}{w_\mathrm{H}} 100 combustion (laminar afire acceleration about to hydrogen gas)[30]
Weber number We \mathrm{We} = \frac{\rho v^2 l}{\sigma} multiphase breeze (strongly arced surfaces; arrangement of apathy to credible tension)
Weissenberg number Wi \mathrm{Wi} = \dot{\gamma} \lambda viscoelastic flows (shear amount times the alleviation time)[31]
Womersley number \alpha \alpha = R \left( \frac{\omega \rho}{\mu} \right)^\frac{1}{2} biofluid mechanics (continuous and pulsating flows; arrangement of pulsatile breeze abundance to adhesive effects)[32]