Dimensionless numbers in fluid mechanics

Dimensionless numbers (or characteristic numbers) have an important role in analyzing the behavior of fluids and their flow as well as in other transport phenomena. ^[1] They include the Reynolds and the Mach numbers, which describe as ratios the relative magnitude of fluid and physical system characteristics, such as density, viscosity, speed of sound, and flow speed. To compare a real situation (e.g. an aircraft) with a small-scale model it is necessary to keep the important characteristic numbers the same. Names and formulation of these numbers were standardized in ISO 31-12 and in ISO 80000-11.

Diffusive numbers in transport phenomena

Dimensionless numbers in transport phenomena

vs.	Inertial	Viscous	Thermal	Mass
Inertial	v d	Re	Pe	PeAB
Viscous	Re −1	μ/ρ, ν	Pr	Sc
Thermal	Pe −1	Pr −1	α	Le
Mass	PeAB −1	Sc −1	Le −1	D

As a general example of how dimensionless numbers arise in fluid mechanics, the classical numbers in transport phenomena of mass, momentum, and energy are principally analyzed by the ratio of effective diffusivities in each transport mechanism. The six dimensionless numbers give the relative strengths of the different phenomena of inertia, viscosity, conductive heat transport, and diffusive mass transport. (In the table, the diagonals give common symbols for the quantities, and the given dimensionless number is the ratio of the left column quantity over top row quantity; e.g. Re = inertial force/viscous force = vd/ν.) These same quantities may alternatively be expressed as ratios of characteristic time, length, or energy scales. Such forms are less commonly used in practice, but can provide insight into particular applications.

Droplet formation

Dimensionless numbers in droplet formation

vs.	Momentum	Viscosity	Surface tension	Gravity	Kinetic energy
Momentum	ρ v d	Re		Fr
Viscosity	Re −1	ρ ν, μ	Oh, Ca, La −1	Ga −1
Surface tension		Oh −1, Ca −1, La	σ	Je	We −1
Gravity	Fr −1	Ga	Bo	g
Kinetic energy			We		ρ v 2 d

Droplet formation mostly depends on momentum, viscosity and surface tension. ^[2] In inkjet printing for example, an ink with a too high Ohnesorge number would not jet properly, and an ink with a too low Ohnesorge number would be jetted with many satellite drops. ^[3] Not all of the quantity ratios are explicitly named, though each of the unnamed ratios could be expressed as a product of two other named dimensionless numbers.

List

All numbers are dimensionless quantities. See other article for extensive list of dimensionless quantities. Certain dimensionless quantities of some importance to fluid mechanics are given below:

Name	Standard symbol	Definition	Named after	Field of application
Archimedes number	Ar	${\displaystyle \mathrm {Ar} ={\frac {gL^{3}\rho _{\ell }(\rho -\rho _{\ell })}{\mu ^{2}}}}$	Archimedes	fluid mechanics (motion of fluids due to density differences)
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	${\displaystyle \mathrm {Ba} ={\frac {\rho d^{2}\lambda ^{1/2}{\dot {\gamma }}}{\mu }}}$	Ralph Bagnold	Granular flow (grain collision stresses to viscous fluid stresses)
Bejan number	Be	${\displaystyle \mathrm {Be} ={\frac {\Delta PL^{2}}{\mu \alpha }}}$	Adrian Bejan	fluid mechanics (dimensionless pressure drop along a channel) [4]
Bingham number	Bm	${\displaystyle \mathrm {Bm} ={\frac {\tau _{y}L}{\mu V}}}$	Eugene C. Bingham	fluid mechanics, rheology (ratio of yield stress to viscous stress) [5]
Biot number	Bi	${\displaystyle \mathrm {Bi} ={\frac {hL_{C}}{k_{b}}}}$	Jean-Baptiste Biot	heat transfer (surface vs. volume conductivity of solids)
Blake number	Bl or B	${\displaystyle \mathrm {B} ={\frac {u\rho }{\mu (1-\epsilon )D}}}$	Frank C. Blake (1892–1926)	geology, fluid mechanics, porous media (inertial over viscous forces in fluid flow through porous media)
Bond number	Bo	${\displaystyle \mathrm {Bo} ={\frac {\rho aL^{2}}{\gamma }}}$	Wilfrid Noel Bond	geology, fluid mechanics, porous media (buoyant versus capillary forces, similar to the Eötvös number) [6]
Brinkman number	Br	${\displaystyle \mathrm {Br} ={\frac {\mu U^{2}}{\kappa (T_{w}-T_{0})}}}$	Henri Brinkman	heat transfer, fluid mechanics (conduction from a wall to a viscous fluid)
Burger number	Bu	${\displaystyle \mathrm {Bu} =\left({\dfrac {\mathrm {Ro} }{\mathrm {Fr} }}\right)^{2}}$	Alewyn P. Burger (1927–2003)	meteorology, oceanography (density stratification versus Earth's rotation)
Brownell–Katz number	NBK	${\displaystyle \mathrm {N} _{\mathrm {BK} }={\frac {u\mu }{k_{\mathrm {rw} }\sigma }}}$	Lloyd E. Brownell and Donald L. Katz	fluid mechanics (combination of capillary number and Bond number) [7]
Capillary number	Ca	${\displaystyle \mathrm {Ca} ={\frac {\mu V}{\gamma }}}$		porous media, fluid mechanics (viscous forces versus surface tension)
Cauchy number	Ca	${\displaystyle \mathrm {Ca} ={\frac {\rho u^{2}}{K}}}$	Augustin-Louis Cauchy	compressible flows (inertia forces versus compressibility force)
Cavitation number	Ca	${\displaystyle \mathrm {Ca} ={\frac {p-p_{\mathrm {v} }}{{\frac {1}{2}}\rho v^{2}}}}$		multiphase flow (hydrodynamic cavitation, pressure over dynamic pressure)
Chandrasekhar number	C	${\displaystyle \mathrm {C} ={\frac {B^{2}L^{2}}{\mu _{o}\mu D_{M}}}}$	Subrahmanyan Chandrasekhar	hydromagnetics (Lorentz force versus viscosity)
Colburn J factors	JM, JH, JD		Allan Philip Colburn (1904–1955)	turbulence; heat, mass, and momentum transfer (dimensionless transfer coefficients)
Damkohler number	Da	${\displaystyle \mathrm {Da} =k\tau }$	Gerhard Damköhler	chemistry (reaction time scales vs. residence time)
Darcy friction factor	Cf or fD		Henry Darcy	fluid mechanics (fraction of pressure losses due to friction in a pipe; four times the Fanning friction factor)
Darcy number	Da	${\displaystyle \mathrm {Da} ={\frac {k}{d^{2}}}}$	Henry Darcy	Fluid dynamics (permeability of the medium versus its cross-sectional area in porous media)
Dean number	D	${\displaystyle \mathrm {D} ={\frac {\rho Vd}{\mu }}\left({\frac {d}{2R}}\right)^{1/2}}$	William Reginald Dean	turbulent flow (vortices in curved ducts)
Deborah number	De	${\displaystyle \mathrm {De} ={\frac {t_{\mathrm {c} }}{t_{\mathrm {p} }}}}$	Deborah	rheology (viscoelastic fluids)
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	${\displaystyle {\rm {Du}}={\frac {\kappa ^{\sigma }}{{\mathrm {K} _{m}}a}}.}$	Stanislav and Andrei Dukhin	Fluid heterogeneous systems (surface conductivity to various electrokinetic and electroacoustic effects)
Eckert number	Ec	${\displaystyle \mathrm {Ec} ={\frac {V^{2}}{c_{p}\Delta T}}}$	Ernst R. G. Eckert	convective heat transfer (characterizes dissipation of energy; ratio of kinetic energy to enthalpy)
Ekman number	Ek	${\displaystyle \mathrm {Ek} ={\frac {\nu }{2D^{2}\Omega \sin \varphi }}}$	Vagn Walfrid Ekman	Geophysics (viscosity to Coriolis force ratio)
Eötvös number	Eo	${\displaystyle \mathrm {Eo} ={\frac {\Delta \rho \,g\,L^{2}}{\sigma }}}$	Loránd Eötvös	fluid mechanics (shape of bubbles or drops)
Ericksen number	Er	${\displaystyle \mathrm {Er} ={\frac {\mu vL}{K}}}$	Jerald Ericksen	fluid dynamics (liquid crystal flow behavior; viscous over elastic forces)
Euler number	Eu	${\displaystyle \mathrm {Eu} ={\frac {\Delta {}p}{\rho V^{2}}}}$	Leonhard Euler	hydrodynamics (stream pressure versus inertia forces)
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) [8]
Fanning friction factor	f		John T. Fanning	fluid mechanics (fraction of pressure losses due to friction in a pipe; 1/4th the Darcy friction factor) [9]
Froude number	Fr	${\displaystyle \mathrm {Fr} ={\frac {U}{\sqrt {g\ell }}}}$	William Froude	fluid mechanics (wave and surface behaviour; ratio of a body's inertia to gravitational forces)
Galilei number	Ga	${\displaystyle \mathrm {Ga} ={\frac {g\,L^{3}}{\nu ^{2}}}}$	Galileo Galilei	fluid mechanics (gravitational over viscous forces)
Görtler number	G	${\displaystyle \mathrm {G} ={\frac {U_{e}\theta }{\nu }}\left({\frac {\theta }{R}}\right)^{1/2}}$	Henry Görtler  [ de ]	fluid dynamics (boundary layer flow along a concave wall)
Goucher number  [ fr ]	Go	${\displaystyle \mathrm {Go} =R\left({\frac {\rho g}{2\sigma }}\right)^{1/2}}$	Frederick Shand Goucher (1888–1973)	fluid dynamics (wire coating problems)
Garcia-Atance number	GA	${\displaystyle \mathrm {G_{A}} ={\frac {p-p_{v}}{\rho aL}}}$	Gonzalo Garcia-Atance Fatjo	phase change (ultrasonic cavitation onset, ratio of pressures over pressure due to acceleration)
Graetz number	Gz	${\displaystyle \mathrm {Gz} ={D_{H} \over L}\mathrm {Re} \,\mathrm {Pr} }$	Leo Graetz	heat transfer, fluid mechanics (laminar flow through a conduit; also used in mass transfer)
Grashof number	Gr	${\displaystyle \mathrm {Gr} _{L}={\frac {g\beta (T_{s}-T_{\infty })L^{3}}{\nu ^{2}}}}$	Franz Grashof	heat transfer, natural convection (ratio of the buoyancy to viscous force)
Hartmann number	Ha	${\displaystyle \mathrm {Ha} =BL\left({\frac {\sigma }{\rho \nu }}\right)^{\frac {1}{2}}}$	Julius Hartmann (1881–1951)	magnetohydrodynamics (ratio of Lorentz to viscous forces)
Hagen number	Hg	${\displaystyle \mathrm {Hg} =-{\frac {1}{\rho }}{\frac {\mathrm {d} p}{\mathrm {d} x}}{\frac {L^{3}}{\nu ^{2}}}}$	Gotthilf Hagen	heat transfer (ratio of the buoyancy to viscous force in forced convection)
Iribarren number	Ir	${\displaystyle \mathrm {Ir} ={\frac {\tan \alpha }{\sqrt {H/L_{0}}}}}$	Ramón Iribarren	wave mechanics (breaking surface gravity waves on a slope)
Jakob number	Ja	${\displaystyle \mathrm {Ja} ={\frac {c_{p,f}(T_{w}-T_{sat})}{h_{fg}}}}$	Max Jakob	heat transfer (ratio of sensible heat to latent heat during phase changes)
Jesus number	Je	${\displaystyle \mathrm {Je} ={\frac {\sigma \,L}{M\,g}}}$	Jesus	Surface tension (ratio of surface tension and weight)
Karlovitz number	Ka	${\displaystyle \mathrm {Ka} =kt_{c}}$	Béla Karlovitz	turbulent combustion (characteristic flow time times flame stretch rate)
Kapitza number	Ka	${\displaystyle \mathrm {Ka} ={\frac {\sigma }{\rho (g\sin \beta )^{1/3}\nu ^{4/3}}}}$	Pyotr Kapitsa	fluid mechanics (thin film of liquid flows down inclined surfaces)
Keulegan–Carpenter number	KC	${\displaystyle \mathrm {K_{C}} ={\frac {V\,T}{L}}}$	Garbis H. Keulegan (1890–1989) and Lloyd H. Carpenter	fluid dynamics (ratio of drag force to inertia for a bluff object in oscillatory fluid flow)
Knudsen number	Kn	${\displaystyle \mathrm {Kn} ={\frac {\lambda }{L}}}$	Martin Knudsen	gas dynamics (ratio of the molecular mean free path length to a representative physical length scale)
Kutateladze number	Ku	${\displaystyle \mathrm {Ku} ={\frac {U_{h}\rho _{g}^{1/2}}{\left({\sigma g(\rho _{l}-\rho _{g})}\right)^{1/4}}}}$	Samson Kutateladze	fluid mechanics (counter-current two-phase flow) [10]
Laplace number	La	${\displaystyle \mathrm {La} ={\frac {\sigma \rho L}{\mu ^{2}}}}$	Pierre-Simon Laplace	fluid dynamics (free convection within immiscible fluids; ratio of surface tension to momentum-transport)
Lewis number	Le	${\displaystyle \mathrm {Le} ={\frac {\alpha }{D}}={\frac {\mathrm {Sc} }{\mathrm {Pr} }}}$	Warren K. Lewis	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	${\displaystyle \chi }$	${\displaystyle \chi ={\frac {m_{\ell }}{m_{g}}}{\sqrt {\frac {\rho _{g}}{\rho _{\ell }}}}}$	R. W. Lockhart and Raymond C. Martinelli	two-phase flow (flow of wet gases; liquid fraction) [11]
Mach number	M or Ma	${\displaystyle \mathrm {M} ={\frac {v}{v_{\mathrm {sound} }}}}$	Ernst Mach	gas dynamics (compressible flow; dimensionless velocity)
Marangoni number	Mg	${\displaystyle \mathrm {Mg} =-{\frac {\mathrm {d} \sigma }{\mathrm {d} T}}{\frac {L\Delta T}{\eta \alpha }}}$	Carlo Marangoni	fluid mechanics (Marangoni flow; thermal surface tension forces over viscous forces)
Markstein number	Ma	${\displaystyle \mathrm {Ma} ={\frac {L_{b}}{l_{f}}}}$	George H. Markstein	turbulence, combustion (Markstein length to laminar flame thickness)
Morton number	Mo	${\displaystyle \mathrm {Mo} ={\frac {g\mu _{c}^{4}\,\Delta \rho }{\rho _{c}^{2}\sigma ^{3}}}}$	Rose Morton	fluid dynamics (determination of bubble/drop shape)
Nusselt number	Nu	${\displaystyle \mathrm {Nu} ={\frac {hd}{k}}}$	Wilhelm Nusselt	heat transfer (forced convection; ratio of convective to conductive heat transfer)
Ohnesorge number	Oh	${\displaystyle \mathrm {Oh} ={\frac {\mu }{\sqrt {\rho \sigma L}}}={\frac {\sqrt {\mathrm {We} }}{\mathrm {Re} }}}$	Wolfgang von Ohnesorge	fluid dynamics (atomization of liquids, Marangoni flow)
Péclet number	Pe	${\displaystyle \mathrm {Pe} ={\frac {Lu}{D}}}$ or ${\displaystyle \mathrm {Pe} ={\frac {Lu}{\alpha }}}$	Jean Claude Eugène Péclet	fluid mechanics (ratio of advective transport rate over molecular diffusive transport rate), heat transfer (ratio of advective transport rate over thermal diffusive transport rate)
Prandtl number	Pr	${\displaystyle \mathrm {Pr} ={\frac {\nu }{\alpha }}={\frac {c_{p}\mu }{k}}}$	Ludwig Prandtl	heat transfer (ratio of viscous diffusion rate over thermal diffusion rate)
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)
Rayleigh number	Ra	${\displaystyle \mathrm {Ra} _{x}={\frac {g\beta }{\nu \alpha }}(T_{s}-T_{\infty })x^{3}}$	John William Strutt, 3rd Baron Rayleigh	heat transfer (buoyancy versus viscous forces in free convection)
Reynolds number	Re	${\displaystyle \mathrm {Re} ={\frac {UL\rho }{\mu }}={\frac {UL}{\nu }}}$	Osborne Reynolds	fluid mechanics (ratio of fluid inertial and viscous forces) [5]
Richardson number	Ri	${\displaystyle \mathrm {Ri} ={\frac {gh}{U^{2}}}={\frac {1}{\mathrm {Fr} ^{2}}}}$	Lewis Fry Richardson	fluid dynamics (effect of buoyancy on flow stability; ratio of potential over kinetic energy) [12]
Roshko number	Ro	${\displaystyle \mathrm {Ro} ={fL^{2} \over \nu }=\mathrm {St} \,\mathrm {Re} }$	Anatol Roshko	fluid dynamics (oscillating flow, vortex shedding)
Rossby number	Ro	${\displaystyle {\text{Ro}}={\frac {U}{Lf}},}$	Carl-Gustaf Rossby	fluid flow (geophysics, ratio of inertial force to Coriolis force)
Rouse number	P	${\displaystyle \mathrm {P} ={\frac {w_{s}}{\kappa u_{*}}}}$	Hunter Rouse	Fluid dynamics (concentration profile of suspended sediment)
Schmidt number	Sc	${\displaystyle \mathrm {Sc} ={\frac {\nu }{D}}}$	Ernst Heinrich Wilhelm Schmidt (1892–1975)	mass transfer (viscous over molecular diffusion rate) [13]
Scruton number	Sc	${\displaystyle \mathrm {Sc} ={\frac {2\delta _{s}m_{e}}{\rho b_{\text{ref}}^{2}}}}$	Christopher 'Kit' Scruton	Fluid dynamics (vortex resonance)
Shape factor	H	${\displaystyle H={\frac {\delta ^{*}}{\theta }}}$		boundary layer flow (ratio of displacement thickness to momentum thickness)
Sherwood number	Sh	${\displaystyle \mathrm {Sh} ={\frac {KL}{D}}}$	Thomas Kilgore Sherwood	mass transfer (forced convection; ratio of convective to diffusive mass transport)
Shields parameter	θ	${\displaystyle \theta ={\frac {\tau }{(\rho _{s}-\rho )gD}}}$	Albert F. Shields	Fluid dynamics (motion of sediment)
Sommerfeld number	S	${\displaystyle \mathrm {S} =\left({\frac {r}{c}}\right)^{2}{\frac {\mu N}{P}}}$	Arnold Sommerfeld	hydrodynamic lubrication (boundary lubrication) [14]
Stanton number	St	${\displaystyle \mathrm {St} ={\frac {h}{c_{p}\rho V}}={\frac {\mathrm {Nu} }{\mathrm {Re} \,\mathrm {Pr} }}}$	Thomas Ernest Stanton	heat transfer and fluid dynamics (forced convection)
Stokes number	Stk or Sk	${\displaystyle \mathrm {Stk} ={\frac {\tau U_{o}}{d_{c}}}}$	Sir George Stokes, 1st Baronet	particles suspensions (ratio of characteristic time of particle to time of flow)
Strouhal number	St	${\displaystyle \mathrm {St} ={\frac {fL}{U}}}$	Vincenc Strouhal	Vortex shedding (ratio of characteristic oscillatory velocity to ambient flow velocity)
Stuart number	N	${\displaystyle \mathrm {N} ={\frac {B^{2}L_{c}\sigma }{\rho U}}={\frac {\mathrm {Ha} ^{2}}{\mathrm {Re} }}}$	John Trevor Stuart	magnetohydrodynamics (ratio of electromagnetic to inertial forces)
Taylor number	Ta	${\displaystyle \mathrm {Ta} ={\frac {4\Omega ^{2}R^{4}}{\nu ^{2}}}}$	G. I. Taylor	fluid dynamics (rotating fluid flows; inertial forces due to rotation of a fluid versus viscous forces)
Thoma number	σ	${\displaystyle \mathrm {\sigma } ={\frac {\mathrm {NPSH} }{h_{\mathrm {pump} }}}}$	Dieter Thoma (1881–1942)	multiphase flow (hydrodynamic cavitation, pressure over dynamic pressure)
Ursell number	U	${\displaystyle \mathrm {U} ={\frac {H\,\lambda ^{2}}{h^{3}}}}$	Fritz Ursell	wave mechanics (nonlinearity of surface gravity waves on a shallow fluid layer)
Wallis parameter	j∗	${\displaystyle j^{*}=R\left({\frac {\omega \rho }{\mu }}\right)^{\frac {1}{2}}}$	Graham B. Wallis	multiphase flows (nondimensional superficial velocity) [15]
Weber number	We	${\displaystyle \mathrm {We} ={\frac {\rho v^{2}l}{\sigma }}}$	Moritz Weber	multiphase flow (strongly curved surfaces; ratio of inertia to surface tension)
Weissenberg number	Wi	${\displaystyle \mathrm {Wi} ={\dot {\gamma }}\lambda }$	Karl Weissenberg	viscoelastic flows (shear rate times the relaxation time) [16]
Womersley number	${\displaystyle \alpha }$	${\displaystyle \alpha =R\left({\frac {\omega \rho }{\mu }}\right)^{\frac {1}{2}}}$	John R. Womersley	biofluid mechanics (continuous and pulsating flows; ratio of pulsatile flow frequency to viscous effects) [17]
Zeldovich number	${\displaystyle \beta }$	${\displaystyle \beta ={\frac {E}{RT_{f}}}{\frac {T_{f}-T_{o}}{T_{f}}}}$	Yakov Zeldovich	fluid dynamics, Combustion (Measure of activation energy)

References

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12. Richardson number Archived 2015-03-02 at the Wayback Machine
13. Schmidt number Archived 2010-01-24 at the Wayback Machine
14. Ekerfors, Lars O. (1985). Boundary lubrication in screw-nut transmissions (PDF) (PhD). Luleå University of Technology. ISSN   0348-8373.
15. 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 (1): 75–84. Bibcode:1999NuEnD.188...75P. doi:10.1016/S0029-5493(99)00005-9.
16. Smith, Douglas E.; Babcock, Hazen P.; Chu, Steven (12 March 1999). "Single-Polymer Dynamics in Steady Shear Flow" (PDF). Science. 283 (5408). American Association for the Advancement of Science: 1724–1727. Bibcode:1999Sci...283.1724S. doi:10.1126/science.283.5408.1724. PMID   10073935. Archived from the original (PDF) on 1 November 2006.
17. Bookbinder; Engler; Hong; Miller (May 2001). "Comparison of Flow Measure Techniques during Continuous and Pulsatile Flow". 2001 BE Undergraduate Projects. Department of Bioengineering, University of Pennsylvania.

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