Physics:Dimensionless numbers in fluid mechanics

Short description: Dimension

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. 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		ρv2d

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. 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
Archimedes number	Ar	$\mathrm{A}\mathrm{r}=\frac{g{L}^3{\rho }_{\ell }(\rho -{\rho }_{\ell })}{{\mu }^2}$	fluid mechanics (motion of fluids due to density differences)
Atwood number	A	$\mathrm{A}=\frac{{\rho }_1-{\rho }_2}{{\rho }_1+{\rho }_2}$	fluid mechanics (onset of instabilities in fluid mixtures due to density differences)
Bejan number (fluid mechanics)	Be	$\mathrm{B}\mathrm{e}=\frac{\mathrm{\Delta }P{L}^2}{\mu \alpha }$	fluid mechanics (dimensionless pressure drop along a channel)
Bingham number	Bm	$\mathrm{B}\mathrm{m}=\frac{{\tau }_{y}L}{\mu V}$	fluid mechanics, rheology (ratio of yield stress to viscous stress)
Biot number	Bi	$\mathrm{B}\mathrm{i}=\frac{h{L}_C}{k_b}$	heat transfer (surface vs. volume conductivity of solids)
Blake number	Bl or B	$\mathrm{B}=\frac{u\rho }{\mu (1-ϵ)D}$	geology, fluid mechanics, porous media (inertial over viscous forces in fluid flow through porous media)
Bond number	Bo	$\mathrm{B}\mathrm{o}=\frac{\rho a{L}^2}{\gamma }$	geology, fluid mechanics, porous media (buoyant versus capillary forces, similar to the Eötvös number)
Brinkman number	Br	$\mathrm{B}\mathrm{r}=\frac{\mu U^2}{\kappa (T_w-T_0)}$	heat transfer, fluid mechanics (conduction from a wall to a viscous fluid)
Burger number	Bu	$\mathrm{B}\mathrm{u}={\left(\frac{\mathrm{R}\mathrm{o}}{\mathrm{F}\mathrm{r}}\right)}^2$	meteorology, oceanography (density stratification versus Earth's rotation)
Brownell–Katz number	NBK	${\mathrm{N}}_{\mathrm{B}\mathrm{K}}=\frac{u\mu }{k_{\mathrm{r}\mathrm{w}}\sigma }$	fluid mechanics (combination of capillary number and Bond number)
Capillary number	Ca	$\mathrm{C}\mathrm{a}=\frac{\mu V}{\gamma }$	porous media, fluid mechanics (viscous forces versus surface tension)
Cauchy number	Ca	$\mathrm{C}\mathrm{a}=\frac{\rho u^2}{K}$	compressible flows (inertia forces versus compressibility force)
Cavitation number	Ca	$\mathrm{C}\mathrm{a}=\frac{p-p_{\mathrm{v}}}{\frac{1}{2}\rho v^2}$	multiphase flow (hydrodynamic cavitation, pressure over dynamic pressure)
Chandrasekhar number	C	$\mathrm{C}=\frac{B^2{L}^2}{{\mu }_o\mu D_M}$	hydromagnetics (Lorentz force versus viscosity)
Colburn J factors	JM, JH, JD		turbulence; heat, mass, and momentum transfer (dimensionless transfer coefficients)
Damkohler number	Da	$\mathrm{D}\mathrm{a}=k\tau$	chemistry (reaction time scales vs. residence time)
Darcy friction factor	Cf or fD		fluid mechanics (fraction of pressure losses due to friction in a pipe; four times the Fanning friction factor)
Dean number	D	$\mathrm{D}=\frac{\rho Vd}{\mu }{\left(\frac{d}{2R}\right)}^{1/2}$	turbulent flow (vortices in curved ducts)
Deborah number	De	$\mathrm{D}\mathrm{e}=\frac{t_{\mathrm{c}}}{t_{\mathrm{p}}}$	rheology (viscoelastic fluids)
Drag coefficient	cd	$c_{\mathrm{d}}=\frac{2F_{\mathrm{d}}}{\rho v^{2}A},$	aeronautics, fluid dynamics (resistance to fluid motion)
Eckert number	Ec	$\mathrm{E}\mathrm{c}=\frac{V^2}{c_p\mathrm{\Delta }T}$	convective heat transfer (characterizes dissipation of energy; ratio of kinetic energy to enthalpy)
Eötvös number	Eo	$\mathrm{E}\mathrm{o}=\frac{\mathrm{\Delta }\rho g{L}^2}{\sigma }$	fluid mechanics (shape of bubbles or drops)
Ericksen number	Er	$\mathrm{E}\mathrm{r}=\frac{\mu vL}{K}$	fluid dynamics (liquid crystal flow behavior; viscous over elastic forces)
Euler number	Eu	$\mathrm{E}\mathrm{u}=\frac{\mathrm{\Delta }p}{\rho V^2}$	hydrodynamics (stream pressure versus inertia forces)
Excess temperature coefficient	${\mathrm{\Theta }}_r$	${\mathrm{\Theta }}_r=\frac{c_p(T-T_e)}{U_e^2/2}$	heat transfer, fluid dynamics (change in internal energy versus kinetic energy)
Fanning friction factor	f		fluid mechanics (fraction of pressure losses due to friction in a pipe; 1/4th the Darcy friction factor)
Froude number	Fr	$\mathrm{F}\mathrm{r}=\frac{U}{\sqrt{g\ell }}$	fluid mechanics (wave and surface behaviour; ratio of a body's inertia to gravitational forces)
Galilei number	Ga	$\mathrm{G}\mathrm{a}=\frac{g{L}^3}{{\nu }^2}$	fluid mechanics (gravitational over viscous forces)
Görtler number	G	$\mathrm{G}=\frac{U_e\theta }{\nu }{\left(\frac{\theta }{R}\right)}^{1/2}$	fluid dynamics (boundary layer flow along a concave wall)
Garcia-Atance number	GA	${\mathrm{G}}_{\mathrm{A}}=\frac{p-p_v}{\rho aL}$	phase change (ultrasonic cavitation onset, ratio of pressures over pressure due to acceleration)
Graetz number	Gz	$\mathrm{G}\mathrm{z}=\frac{D_H}{L}\mathrm{R}\mathrm{e}\mathrm{P}\mathrm{r}$	heat transfer, fluid mechanics (laminar flow through a conduit; also used in mass transfer)
Grashof number	Gr	${\mathrm{G}\mathrm{r}}_L=\frac{g\beta (T_s-T_{\mathrm{\infty }})L^3}{{\nu }^2}$	heat transfer, natural convection (ratio of the buoyancy to viscous force)
Hartmann number	Ha	$\mathrm{H}\mathrm{a}=BL{\left(\frac{\sigma }{\rho \nu }\right)}^{\frac{1}{2}}$	magnetohydrodynamics (ratio of Lorentz to viscous forces)
Hagen number	Hg	$\mathrm{H}\mathrm{g}=-\frac{1}{\rho }\frac{\mathrm{d}p}{\mathrm{d}x}\frac{L^3}{{\nu }^2}$	heat transfer (ratio of the buoyancy to viscous force in forced convection)
Iribarren number	Ir	$\mathrm{I}\mathrm{r}=\frac{\tan \alpha }{\sqrt{H/L_0}}$	wave mechanics (breaking surface gravity waves on a slope)
Jakob number	Ja	$\mathrm{J}\mathrm{a}=\frac{c_{p,f}(T_w-T_{sat})}{h_{fg}}$	heat transfer (ratio of sensible heat to latent heat during phase changes)
Karlovitz number	Ka	$\mathrm{K}\mathrm{a}=k{t}_c$	turbulent combustion (characteristic flow time times flame stretch rate)
Kapitza number	Ka	$\mathrm{K}\mathrm{a}=\frac{\sigma }{\rho (g\sin \beta {)}^{1/3}{\nu }^{4/3}}$	fluid mechanics (thin film of liquid flows down inclined surfaces)
Keulegan–Carpenter number	KC	${\mathrm{K}}_{\mathrm{C}}=\frac{VT}{L}$	fluid dynamics (ratio of drag force to inertia for a bluff object in oscillatory fluid flow)
Knudsen number	Kn	$\mathrm{K}\mathrm{n}=\frac{\lambda }{L}$	gas dynamics (ratio of the molecular mean free path length to a representative physical length scale)
Kutateladze number	Ku	$\mathrm{K}\mathrm{u}=\frac{U_h{\rho }_g^{1/2}}{{\left(\sigma g({\rho }_l-{\rho }_g)\right)}^{1/4}}$	fluid mechanics (counter-current two-phase flow)
Laplace number	La	$\mathrm{L}\mathrm{a}=\frac{\sigma \rho L}{{\mu }^2}$	fluid dynamics (free convection within immiscible fluids; ratio of surface tension to momentum-transport)
Lewis number	Le	$\mathrm{L}\mathrm{e}=\frac{\alpha }{D}=\frac{\mathrm{S}\mathrm{c}}{\mathrm{P}\mathrm{r}}$	heat and mass transfer (ratio of thermal to mass diffusivity)
Lift coefficient	CL	$C_{\mathrm{L}}=\frac{L}{qS}$	aerodynamics (lift available from an airfoil at a given angle of attack)
Lockhart–Martinelli parameter	$\chi$	$\chi =\frac{m_{\ell }}{m_g}\sqrt{\frac{{\rho }_g}{{\rho }_{\ell }}}$	two-phase flow (flow of wet gases; liquid fraction)
Mach number	M or Ma	$\mathrm{M}=\frac{v}{v_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}}$	gas dynamics (compressible flow; dimensionless velocity)
Marangoni number	Mg	$\mathrm{M}\mathrm{g}=-\frac{\mathrm{d}\sigma }{\mathrm{d}T}\frac{L\mathrm{\Delta }T}{\eta \alpha }$	fluid mechanics (Marangoni flow; thermal surface tension forces over viscous forces)
Markstein number	Ma	$\mathrm{M}\mathrm{a}=\frac{L_b}{l_f}$	turbulence, combustion (Markstein length to laminar flame thickness)
Morton number	Mo	$\mathrm{M}\mathrm{o}=\frac{g{\mu }_c^4\mathrm{\Delta }\rho }{{\rho }_c^2{\sigma }^3}$	fluid dynamics (determination of bubble/drop shape)
Nusselt number	Nu	$\mathrm{N}\mathrm{u}=\frac{hd}{k}$	heat transfer (forced convection; ratio of convective to conductive heat transfer)
Ohnesorge number	Oh	$\mathrm{O}\mathrm{h}=\frac{\mu }{\sqrt{\rho \sigma L}}=\frac{\sqrt{\mathrm{W}\mathrm{e}}}{\mathrm{R}\mathrm{e}}$	fluid dynamics (atomization of liquids, Marangoni flow)
Péclet number	Pe	$\mathrm{P}\mathrm{e}=\frac{Lu}{D}$ or $\mathrm{P}\mathrm{e}=\frac{Lu}{\alpha }$	fluid mechanics (ratio of advective transport rate over molecular diffusive transport rate), heat transfer (ratio of advective transport rate over thermal diffusive transport rate)
Prandtl number	Pr	$\mathrm{P}\mathrm{r}=\frac{\nu }{\alpha }=\frac{c_p\mu }{k}$	heat transfer (ratio of viscous diffusion rate over thermal diffusion rate)
Pressure coefficient	CP	$C_p=\frac{p-p_{\mathrm{\infty }}}{\frac{1}{2}{\rho }_{\mathrm{\infty }}{V}_{\mathrm{\infty }}^2}$	aerodynamics, hydrodynamics (pressure experienced at a point on an airfoil; dimensionless pressure variable)
Rayleigh number	Ra	${\mathrm{R}\mathrm{a}}_x=\frac{g\beta }{\nu \alpha }(T_s-T_{\mathrm{\infty }})x^3$	heat transfer (buoyancy versus viscous forces in free convection)
Reynolds number	Re	$\mathrm{R}\mathrm{e}=\frac{UL\rho }{\mu }=\frac{UL}{\nu }$	fluid mechanics (ratio of fluid inertial and viscous forces)
Richardson number	Ri	$\mathrm{R}\mathrm{i}=\frac{gh}{U^2}=\frac{1}{{\mathrm{F}\mathrm{r}}^2}$	fluid dynamics (effect of buoyancy on flow stability; ratio of potential over kinetic energy)
Roshko number	Ro	$\mathrm{R}\mathrm{o}=\frac{f{L}^2}{\nu }=\mathrm{S}\mathrm{t}\mathrm{R}\mathrm{e}$	fluid dynamics (oscillating flow, vortex shedding)
Rossby number	Ro	$\text{Ro}=\frac{U}{Lf},$	fluid flow (geophysics, ratio of inertial force to Coriolis force)
Schmidt number	Sc	$\mathrm{S}\mathrm{c}=\frac{\nu }{D}$	mass transfer (viscous over molecular diffusion rate)
Shape factor	H	$H=\frac{{\delta }^{*}}{\theta }$	boundary layer flow (ratio of displacement thickness to momentum thickness)
Sherwood number	Sh	$\mathrm{S}\mathrm{h}=\frac{KL}{D}$	mass transfer (forced convection; ratio of convective to diffusive mass transport)
Sommerfeld number	S	$\mathrm{S}={\left(\frac{r}{c}\right)}^2\frac{\mu N}{P}$	hydrodynamic lubrication (boundary lubrication)
Stanton number	St	$\mathrm{S}\mathrm{t}=\frac{h}{c_p\rho V}=\frac{\mathrm{N}\mathrm{u}}{\mathrm{R}\mathrm{e}\mathrm{P}\mathrm{r}}$	heat transfer and fluid dynamics (forced convection)
Stokes number	Stk or Sk	$\mathrm{S}\mathrm{t}\mathrm{k}=\frac{\tau U_o}{d_c}$	particles suspensions (ratio of characteristic time of particle to time of flow)
Strouhal number	St	$\mathrm{S}\mathrm{t}=\frac{fL}{U}$	Vortex shedding (ratio of characteristic oscillatory velocity to ambient flow velocity)
Stuart number	N	$\mathrm{N}=\frac{B^2{L}_c\sigma }{\rho U}=\frac{{\mathrm{H}\mathrm{a}}^2}{\mathrm{R}\mathrm{e}}$	magnetohydrodynamics (ratio of electromagnetic to inertial forces)
Taylor number	Ta	$\mathrm{T}\mathrm{a}=\frac{4{\mathrm{\Omega }}^2{R}^4}{{\nu }^2}$	fluid dynamics (rotating fluid flows; inertial forces due to rotation of a fluid versus viscous forces)
Ursell number	U	$\mathrm{U}=\frac{H{\lambda }^2}{h^3}$	wave mechanics (nonlinearity of surface gravity waves on a shallow fluid layer)
Wallis parameter	j∗	$j^{*}=R{\left(\frac{\omega \rho }{\mu }\right)}^{\frac{1}{2}}$	multiphase flows (nondimensional superficial velocity)
Weber number	We	$\mathrm{W}\mathrm{e}=\frac{\rho v^{2}l}{\sigma }$	multiphase flow (strongly curved surfaces; ratio of inertia to surface tension)
Weissenberg number	Wi	$\mathrm{W}\mathrm{i}=\dot{\gamma }\lambda$	viscoelastic flows (shear rate times the relaxation time)
Womersley number	$\alpha$	$\alpha =R{\left(\frac{\omega \rho }{\mu }\right)}^{\frac{1}{2}}$	biofluid mechanics (continuous and pulsating flows; ratio of pulsatile flow frequency to viscous effects)
Zel'dovich number	$\beta$	$\beta =\frac{E}{R{T}_f}\frac{T_f-T_o}{T_f}$	fluid dynamics, Combustion (Measure of activation energy)

References

• Tropea, C.; Yarin, A.L.; Foss, J.F. (2007). Springer Handbook of Experimental Fluid Mechanics. Springer-Verlag.