Dimensionless numbers in fluid mechanics

Dimensionless numbers have an important role in analyzing the behavior of fluids and their flow as well as in other transport phenomena. 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 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

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

Droplet formation mostly depends on momentum, viscosity and surface tension. 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. These dimensionless ratios can be obtained by relating each term in a consistent form, such as energy per volume or pressure. For example, the ratios of characteristic pressures for inertial, viscous, gravity, and surface tension effects gives dimensionless ratios of each pair. More fundamentally, these and other dimensionless numbers are derived from dimensional analysis and/or non-dimensionalization of the Navier-Stokes equations. 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		Archimedes	fluid mechanics
Atwood number	A		George Atwood	fluid mechanics
Bagnold number	Ba		Ralph Bagnold	Granular flow
Bejan number	Be		Adrian Bejan	fluid mechanics
Bingham number	Bm		Eugene C. Bingham	fluid mechanics, rheology
Biot number	Bi		Jean-Baptiste Biot	heat transfer
Blake number	Bl or B		Frank C. Blake	geology, fluid mechanics, porous media
Bond number	Bo		Wilfrid Noel Bond	geology, fluid mechanics, porous media
Brinkman number	Br		Henri Brinkman	heat transfer, fluid mechanics
Burger number	Bu		Alewyn P. Burger	meteorology, oceanography
Brownell–Katz number	N_BK		Lloyd E. Brownell and Donald L. Katz	fluid mechanics
Capillary number	Ca			porous media, fluid mechanics
Cauchy number	Ca		Augustin-Louis Cauchy	compressible flows
Cavitation number	Ca			multiphase flow
Chandrasekhar number	C		Subrahmanyan Chandrasekhar	hydromagnetics
Colburn J factors	J_M, J_H, J_D		Allan Philip Colburn	turbulence; heat, mass, and momentum transfer
Damkohler number	Da		Gerhard Damköhler	chemistry
Darcy friction factor	C_f or f_D		Henry Darcy	fluid mechanics
Darcy number	Da		Henry Darcy	Fluid dynamics
Dean number	D		William Reginald Dean	turbulent flow
Deborah number	De		Deborah	rheology
Drag coefficient	c_d			aeronautics, fluid dynamics
Dukhin number	Du		Stanislav and Andrei Dukhin	Fluid heterogeneous systems
Eckert number	Ec		Ernst R. G. Eckert	convective heat transfer
Ekman number	Ek		Vagn Walfrid Ekman	Geophysics
Eötvös number	Eo		Loránd Eötvös	fluid mechanics
Ericksen number	Er		Jerald Ericksen	fluid dynamics
Euler number	Eu		Leonhard Euler	hydrodynamics
Excess temperature coefficient				heat transfer, fluid dynamics
Fanning friction factor	f		John T. Fanning	fluid mechanics
Froude number	Fr		William Froude	fluid mechanics
Galilei number	Ga		Galileo Galilei	fluid mechanics
Görtler number	G			fluid dynamics
	Go		Frederick Shand Goucher	fluid dynamics
Garcia-Atance number	G_A		Gonzalo Garcia-Atance Fatjo	phase change
Graetz number	Gz		Leo Graetz	heat transfer, fluid mechanics
Grashof number	Gr		Franz Grashof	heat transfer, natural convection
Hartmann number	Ha		Julius Hartmann	magnetohydrodynamics
Hagen number	Hg		Gotthilf Hagen	heat transfer
Iribarren number	Ir		Ramón Iribarren	wave mechanics
Jakob number	Ja		Max Jakob	heat transfer
Jesus number	Je		Jesus	Surface tension
Karlovitz number	Ka		Béla Karlovitz	turbulent combustion
Kapitza number	Ka		Pyotr Kapitsa	fluid mechanics
Keulegan–Carpenter number	K_C		Garbis H. Keulegan and Lloyd H. Carpenter	fluid dynamics
Knudsen number	Kn		Martin Knudsen	gas dynamics
Kutateladze number	Ku		Samson Kutateladze	fluid mechanics
Laplace number	La		Pierre-Simon Laplace	fluid dynamics
Lewis number	Le		Warren K. Lewis	heat and mass transfer
Lift coefficient	C_L			aerodynamics
Lockhart–Martinelli parameter			R. W. Lockhart and Raymond C. Martinelli	two-phase flow
Mach number	M or Ma		Ernst Mach	gas dynamics
Marangoni number	Mg		Carlo Marangoni	fluid mechanics
Markstein number	Ma		George H. Markstein	turbulence, combustion
Morton number	Mo		Rose Morton	fluid dynamics
Nusselt number	Nu		Wilhelm Nusselt	heat transfer
Ohnesorge number	Oh		Wolfgang von Ohnesorge	fluid dynamics
Péclet number	Pe	or	Jean Claude Eugène Péclet	fluid mechanics, heat transfer
Prandtl number	Pr		Ludwig Prandtl	heat transfer
Pressure coefficient	C_P			aerodynamics, hydrodynamics
Rayleigh number	Ra		John William Strutt, 3rd Baron Rayleigh	heat transfer
Reynolds number	Re		Osborne Reynolds	fluid mechanics
Richardson number	Ri		Lewis Fry Richardson	fluid dynamics
Roshko number	Ro		Anatol Roshko	fluid dynamics
Rossby number	Ro		Carl-Gustaf Rossby	fluid flow
Rouse number	P		Hunter Rouse	Fluid dynamics
Schmidt number	Sc		Ernst Heinrich Wilhelm Schmidt	mass transfer
Scruton number	Sc		Christopher 'Kit' Scruton	Fluid dynamics
Shape factor	H			boundary layer flow
Sherwood number	Sh		Thomas Kilgore Sherwood	mass transfer
Shields parameter	θ		Albert F. Shields	Fluid dynamics
Sommerfeld number	S		Arnold Sommerfeld	hydrodynamic lubrication
Stanton number	St		Thomas Ernest Stanton	heat transfer and fluid dynamics
Stokes number	Stk or S_k		Sir George Stokes, 1st Baronet	particles suspensions
Strouhal number	St		Vincenc Strouhal	Vortex shedding
Stuart number	N		John Trevor Stuart	magnetohydrodynamics
Taylor number	Ta		G. I. Taylor	fluid dynamics
Thoma number	σ		Dieter Thoma	multiphase flow
Ursell number	U		Fritz Ursell	wave mechanics
Wallis parameter	j		Graham B. Wallis	multiphase flows
Weber number	We		Moritz Weber	multiphase flow
Weissenberg number	Wi		Karl Weissenberg	viscoelastic flows
Womersley number			John R. Womersley	biofluid mechanics
Zeldovich number			Yakov Zeldovich	fluid dynamics, Combustion