Dimensionless numbers in fluid mechanics explained

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	vd	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	ρvd	Re		Fr
Viscosity	Re^−1	ρν, μ	Oh, Ca, La^−1	Ga^−1
Surface tension		Oh^−1, Ca^−1, La	σ	Bo^−1	We^−1
Gravity	Fr^−1	Ga	Bo	g
Kinetic energy			We		ρvd

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	Field of application
	Ar	Ar = (g L ^{3} \rho _{\ell} (\rho - \rho _{\ell)})/(\mu ^{2})	fluid mechanics (motion of fluids due to density differences)
	A	A = (\rho _{1} - \rho _{2})/(\rho _{1} + \rho _{2})	fluid mechanics (onset of instabilities in fluid mixtures due to density differences)
Bejan number (fluid mechanics)	Be	Be = (\Delta P L ^{2})/(\mu \alpha)	fluid mechanics (dimensionless pressure drop along a channel)^[4]
	Bm	Bm = (\tau _{y} L)/(\mu V)	fluid mechanics, rheology (ratio of yield stress to viscous stress)
	Bi	Bi = (h L _{C})/(k _{b})	heat transfer (surface vs. volume conductivity of solids)
	Bl or B	B = (u \rho)/(\mu ( 1- \epsilon) D)	geology, fluid mechanics, porous media (inertial over viscous forces in fluid flow through porous media)
	Bo	Bo = (\rho a L ^{2})/(\gamma)	geology, fluid mechanics, porous media (buoyant versus capillary forces, similar to the Eötvös number)^[5]
	Br	Br = (\mu U ^{2})/(\kappa (T _{w} - T _{0)})	heat transfer, fluid mechanics (conduction from a wall to a viscous fluid)
Burger number	Bu	Bu = \left(\dfrac{ Ro }\right)^2	meteorology, oceanography (density stratification versus Earth's rotation)
Brownell–Katz number	NBK	N _{BK} = (u \mu)/(k _{rw\sigma})	fluid mechanics (combination of capillary number and Bond number)^[6]
Capillary number	Ca	Ca = (\mu V)/(\gamma)	porous media, fluid mechanics (viscous forces versus surface tension)
Cauchy number	Ca	Ca = (\rho u ^{2})/(K)	compressible flows (inertia forces versus compressibility force)
Cavitation number	Ca	Ca = (p - p _{v})/((1)/(\rho v ^{2} 2))	multiphase flow (hydrodynamic cavitation, pressure over dynamic pressure)
	C	C = (B ^{2} L ^{2})/(\mu _{o} \mu D _{M})	hydromagnetics (Lorentz force versus viscosity)
	JM, JH, JD		turbulence heat, mass, and momentum transfer (dimensionless transfer coefficients)
	Da	Da = k \tau	chemistry (reaction time scales vs. residence time)
	Cf or fD		fluid mechanics (fraction of pressure losses due to friction in a pipe; four times the Fanning friction factor)
	D	D = (\rho V d)/(\mu) \left( (d)/(2 R) \right) ^{1/2}	turbulent flow (vortices in curved ducts)
	De	De = (t _{c})/(t _{p})	rheology (viscoelastic fluids)
	cd	c _{d} = \dfrac{ 2 F _{d} }\,,	aeronautics, fluid dynamics (resistance to fluid motion)
	Ec	Ec = (V ^{2})/(c _{p\Delta} T)	convective heat transfer (characterizes dissipation of energy; ratio of kinetic energy to enthalpy)
	Eo	Eo = (\Delta\rho g L ^{2})/(\sigma)	fluid mechanics (shape of bubbles or drops)
	Er	Er = (\mu v L)/(K)	fluid dynamics (liquid crystal flow behavior; viscous over elastic forces)
	Eu	Eu = (\Delta{)/(p}{\rho) V ^{2}}	hydrodynamics (stream pressure versus inertia forces)
	\Theta _{r}	\Theta _{r} = (c _{p} (T - T _{e)})/(U ^{2 / 2} _{e})	heat transfer, fluid dynamics (change in internal energy versus kinetic energy)^[7]
	f		fluid mechanics (fraction of pressure losses due to friction in a pipe; 1/4th the Darcy friction factor)^[8]
	Fr	Fr = (U)/(\sqrt{g\ell) }	fluid mechanics (wave and surface behaviour; ratio of a body's inertia to gravitational forces)
	Ga	Ga = (g L ^{3})/(\nu ^{2})	fluid mechanics (gravitational over viscous forces)
	G	G = (U _{e} \theta)/(\nu) \left( (\theta)/(R) \right) ^{1/2}	fluid dynamics (boundary layer flow along a concave wall)
	GA	G _{A} = (p - p _{v})/(\rho a L)	phase change (ultrasonic cavitation onset, ratio of pressures over pressure due to acceleration)
	Gz	Gz = {D _{H} \over L} Re Pr	heat transfer, fluid mechanics (laminar flow through a conduit; also used in mass transfer)
	Gr	Gr _{L} = (g \beta (T _{s} - T _{infty} ) L ^{3})/(\nu ^{2})	heat transfer, natural convection (ratio of the buoyancy to viscous force)
	Ha	Ha = BL \left( (\sigma)/(\rho\nu) \right) ^{(1)/(2)}	magnetohydrodynamics (ratio of Lorentz to viscous forces)
	Hg	Hg =- (1)/(\rho) (d p)/(d x) (L ^{3})/(\nu ^{2})	heat transfer (ratio of the buoyancy to viscous force in forced convection)
	Ir	Ir = (\tan \alpha)/(\sqrt{H/L _{0}) }	wave mechanics (breaking surface gravity waves on a slope)
	Ja	Ja = (c _{p,f} (T _{w} - T _{sat} ))/(h _{fg})	heat transfer (ratio of sensible heat to latent heat during phase changes)
	Ka	Ka = k t _{c}	turbulent combustion (characteristic flow time times flame stretch rate)
	Ka	Ka = (\sigma)/(\rho(g\sin\beta) ^{1/3} \nu ^{4/3})	fluid mechanics (thin film of liquid flows down inclined surfaces)
	KC	K _{C} = (VT)/(L)	fluid dynamics (ratio of drag force to inertia for a bluff object in oscillatory fluid flow)
	Kn	Kn = (λ)/(L)	gas dynamics (ratio of the molecular mean free path length to a representative physical length scale)
	Ku	Ku = (U _{h} \rho ^{1 / 2} _{g})/(\left({\sigma g (\rho _{l} - \rho _{g)}) \right) ^{1/4} }	fluid mechanics (counter-current two-phase flow)^[9]
	La	La = (\sigma \rho L)/(\mu ^{2})	fluid dynamics (free convection within immiscible fluids; ratio of surface tension to momentum-transport)
	Le	Le = (\alpha)/(D) = (Sc)/(Pr)	heat and mass transfer (ratio of thermal to mass diffusivity)
	CL	C _{L} = (L)/(qS)	aerodynamics (lift available from an airfoil at a given angle of attack)
	\chi	\chi = (m _{\ell})/(\sqrt{ m _{g}) (\rho _{g})/(\rho _{\ell}) }	two-phase flow (flow of wet gases; liquid fraction)^[10]
	M or Ma	M = ({v)/() }	gas dynamics (compressible flow; dimensionless velocity)
	Mg	Mg = - { (d \sigma)/(d T) }\frac	fluid mechanics (Marangoni flow; thermal surface tension forces over viscous forces)
	Ma	Ma = (L _{b})/(l _{f})	turbulence, combustion (Markstein length to laminar flame thickness)
	Mo	Mo = (g \mu ^{4} _{c} \Delta \rho)/(\rho ^{2} _{c} \sigma ^{3})	fluid dynamics (determination of bubble/drop shape)
	Nu	Nu = (hd)/(k)	heat transfer (forced convection; ratio of convective to conductive heat transfer)
	Oh	Oh = (\mu)/(\sqrt{\rho \sigma L) } = \frac	fluid dynamics (atomization of liquids, Marangoni flow)
	Pe	Pe = (L u)/(D) or Pe = (L u)/(\alpha)	fluid mechanics (ratio of advective transport rate over molecular diffusive transport rate), heat transfer (ratio of advective transport rate over thermal diffusive transport rate)
	Pr	Pr = (\nu)/(\alpha) = (c _{p} \mu)/(k)	heat transfer (ratio of viscous diffusion rate over thermal diffusion rate)
	CP	C _{p} = {p - p _{infty} \over (1)/(2) \rho _{infty} V ^{2 }} _{inf ty}	aerodynamics, hydrodynamics (pressure experienced at a point on an airfoil; dimensionless pressure variable)
	Ra	Ra _{x} = (g \beta)/(\nu \alpha) (T _{s} - T _{infin)} x ^{3}	heat transfer (buoyancy versus viscous forces in free convection)
	Re	Re = (U L\rho)/(= \mu) (U L)/(\nu)	fluid mechanics (ratio of fluid inertial and viscous forces)^[11]
	Ri	Ri = (gh)/(U ^{2}) = (1)/(Fr ^{2})	fluid dynamics (effect of buoyancy on flow stability; ratio of potential over kinetic energy)^[12]
	Ro	Ro = {f L ^{2} \over \nu} = St Re	fluid dynamics (oscillating flow, vortex shedding)
	Ro	Ro = (U)/(Lf) ,	fluid flow (geophysics, ratio of inertial force to Coriolis force)
	Sc	Sc = (\nu)/(D)	mass transfer (viscous over molecular diffusion rate)^[13]
	H	H = (\delta ^{*})/(\theta)	boundary layer flow (ratio of displacement thickness to momentum thickness)
	Sh	Sh = (K L)/(D)	mass transfer (forced convection; ratio of convective to diffusive mass transport)
	S	S = \left( (r)/(c) \right) ^{2} (\mu N)/(P)	hydrodynamic lubrication (boundary lubrication)^[14]
	St	St = (h)/(c _{p} \rho V) = (Nu)/(Re Pr)	heat transfer and fluid dynamics (forced convection)
	Stk or Sk	Stk = (\tau U _{o})/(d _{c})	particles suspensions (ratio of characteristic time of particle to time of flow)
	St	St = (f L)/(U)	Vortex shedding (ratio of characteristic oscillatory velocity to ambient flow velocity)
	N	N = (B ^{2} L _{c} \sigma)/(\rho U) = (Ha ^{2})/(Re)	magnetohydrodynamics (ratio of electromagnetic to inertial forces)
	Ta	Ta = (4 \Omega ^{2} R ^{4})/(\nu ^{2})	fluid dynamics (rotating fluid flows; inertial forces due to rotation of a fluid versus viscous forces)
	U	U = (H λ ^{2})/(h ^{3})	wave mechanics (nonlinearity of surface gravity waves on a shallow fluid layer)
	j	j ^{*} = R \left( (\omega \rho)/(\mu) \right) ^{(1)/(2)}	multiphase flows (nondimensional superficial velocity)^[15]
	We	We = (\rho v ^{2} l)/(\sigma)	multiphase flow (strongly curved surfaces; ratio of inertia to surface tension)
	Wi	Wi = • \gamma λ	viscoelastic flows (shear rate times the relaxation time)^[16]
	\alpha	\alpha = R \left( (\omega \rho)/(\mu) \right) ^{(1)/(2)}	biofluid mechanics (continuous and pulsating flows; ratio of pulsatile flow frequency to viscous effects)^[17]
	\beta	\beta = (E)/(RT _{f}) (T _{f-T} _{o})/(T _{f})	fluid dynamics, Combustion (Measure of activation energy)

References

• Book: Springer Handbook of Experimental Fluid Mechanics. Tropea . C. . Yarin . A.L. . Foss . J.F. . Springer-Verlag. 2007.

Notes and References

1. Web site: ISO 80000-1:2009. International Organization for Standardization. 2019-09-15.
2. Book: Dijksman . J. Frits . Pierik . Anke . Hutchings . Ian M. . Martin . Graham D. . Inkjet Technology for Digital Fabrication . John Wiley & Sons . 9780470681985 . 45–86 . Dynamics of Piezoelectric Print-Heads . 2012 . 10.1002/9781118452943.ch3.
3. Derby. Brian. Brian Derby. Inkjet Printing of Functional and Structural Materials: Fluid Property Requirements, Feature Stability, and Resolution. Annual Review of Materials Research. 40. 1. 2010. 395–414. 1531-7331. 10.1146/annurev-matsci-070909-104502. 2010AnRMS..40..395D . 138001742 .
4. The formation of wall jet near a high temperature wall under microgravity environment . Subrata . Bhattacharje . William L. . Grosshandler . 1988 . National Heat Transfer Conference . Harold R. . Jacobs . 1 . American Society of Mechanical Engineers . Houston, TX . 711–716 . 1988nht.....1..711B.
5. Mahajan . Milind P. . Tsige . Mesfin . Zhang . Shiyong . Alexander . J. Iwan D. . Taylor . P. L. . Rosenblatt . Charles . Collapse Dynamics of Liquid Bridges Investigated by Time-Varying Magnetic Levitation . Physical Review Letters . 10 January 2000 . 84 . 2 . 338–341 . 10.1103/PhysRevLett.84.338 . 11015905 . 2000PhRvL..84..338M . https://web.archive.org/web/20120305114521/http://ising.phys.cwru.edu/plt/PapersInPdf/181BridgeCollapse.pdf . 5 March 2012.
6. Web site: Home . OnePetro . 2015-05-04 . 2015-05-08.
7. Book: Schetz, Joseph A.. Boundary Layer Analysis. limited. 1993. Prentice-Hall, Inc.. Englewood Cliffs, NJ. 0-13-086885-X. 132–134.
8. Web site: Fanning friction factor . 2015-06-25 . https://web.archive.org/web/20131220032423/http://www.engineering.uiowa.edu/~cee081/Exams/Final/Final.htm . 2013-12-20 . dead .
9. Tan . R. B. H. . Sundar . R. . 10.1016/S0009-2509(01)00247-0 . On the froth–spray transition at multiple orifices . Chemical Engineering Science . 56 . 21–22 . 6337 . 2001 . 2001ChEnS..56.6337T .
10. Stewart . David . The Evaluation of Wet Gas Metering Technologies for Offshore Applications, Part 1 – Differential Pressure Meters . Flow Measurement Guidance Note . February 2003 . 40 . https://web.archive.org/web/20061117065355/http://www.flowprogramme.co.uk:80/publications/guidancenotes/GN40.pdf . 17 November 2006 . National Engineering Laboratory . Glasgow, UK.
11. Web site: Table of Dimensionless Numbers . 2009-11-05.
12. http://apollo.lsc.vsc.edu/classes/met455/notes/section4/2.html Richardson number
13. http://www.ent.ohiou.edu/~hbwang/fluidynamics.htm Schmidt number
14. Ekerfors . Lars O. . 1985 . Boundary lubrication in screw-nut transmissions . PhD . Luleå University of Technology . 0348-8373.
15. Petritsch . G. . Mewes . D. . 10.1016/S0029-5493(99)00005-9 . Experimental investigations of the flow patterns in the hot leg of a pressurized water reactor . Nuclear Engineering and Design . 188 . 75–84 . 1999 .
16. Smith . Douglas E. . Babcock . Hazen P. . Chu . Steven . Single-Polymer Dynamics in Steady Shear Flow . Science . 12 March 1999 . 283 . 5408 . 1724–1727 . 10.1126/science.283.5408.1724 . American Association for the Advancement of Science . 10073935 . 1999Sci...283.1724S . https://web.archive.org/web/20061101152745/http://physics.ucsd.edu/~des/Shear1999.pdf . 1 November 2006.
17. Web site: Bookbinder . Engler . Hong . Miller . Comparison of Flow Measure Techniques during Continuous and Pulsatile Flow . 2001 BE Undergraduate Projects . Department of Bioengineering, University of Pennsylvania . May 2001.