Dimensionless numbers in fluid mechanics

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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

More information vs., Inertial ...
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
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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

More information vs., Momentum ...
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 Je We−1
Gravity Fr−1 Ga Bo g
Kinetic energy We ρv2d
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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

Summarize
Perspective

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:

More information , ...
Name Standard symbol Definition Named after Field of application
Archimedes numberAr Archimedesfluid mechanics (motion of fluids due to density differences)
Atwood numberA George Atwood[citation needed]fluid mechanics (onset of instabilities in fluid mixtures due to density differences)
Bagnold number Ba Ralph Bagnold Granular flow (grain collision stresses to viscous fluid stresses)
Bejan numberBe Adrian Bejanfluid mechanics (dimensionless pressure drop along a channel)[4]
Bingham numberBm Eugene C. Binghamfluid mechanics, rheology (ratio of yield stress to viscous stress)[5]
Biot numberBi Jean-Baptiste Biotheat transfer (surface vs. volume conductivity of solids)
Blake numberBl or B Frank C. Blake (1892–1926)geology, fluid mechanics, porous media (inertial over viscous forces in fluid flow through porous media)
Bond numberBo Wilfrid Noel Bondgeology, fluid mechanics, porous media (buoyant versus capillary forces, similar to the Eötvös number)[6]
Brinkman numberBr Henri Brinkmanheat transfer, fluid mechanics (conduction from a wall to a viscous fluid)
Burger number Bu Alewyn P. Burger (1927–2003) meteorology, oceanography (density stratification versus Earth's rotation)
Brownell–Katz numberNBK Lloyd E. Brownell and Donald L. Katzfluid mechanics (combination of capillary number and Bond number)[7]
Capillary numberCa porous media, fluid mechanics (viscous forces versus surface tension)
Cauchy numberCa Augustin-Louis Cauchycompressible flows (inertia forces versus compressibility force)
Cavitation numberCa multiphase flow (hydrodynamic cavitation, pressure over dynamic pressure)
Chandrasekhar numberC Subrahmanyan Chandrasekharhydromagnetics (Lorentz force versus viscosity)
Colburn J factorsJM, JH, JD Allan Philip Colburn (1904–1955)turbulence; heat, mass, and momentum transfer (dimensionless transfer coefficients)
Damkohler numberDa Gerhard Damköhlerchemistry (reaction time scales vs. residence time)
Darcy friction factorCf or fD Henry Darcyfluid mechanics (fraction of pressure losses due to friction in a pipe; four times the Fanning friction factor)
Darcy number Da Henry Darcy Fluid dynamics (permeability of the medium versus its cross-sectional area in porous media)
Dean numberD William Reginald Deanturbulent flow (vortices in curved ducts)
Deborah numberDe Deborahrheology (viscoelastic fluids)
Drag coefficientcd aeronautics, fluid dynamics (resistance to fluid motion)
Dukhin number Du Stanislav and Andrei Dukhin Fluid heterogeneous systems (surface conductivity to various electrokinetic and electroacoustic effects)
Eckert numberEc Ernst R. G. Eckertconvective heat transfer (characterizes dissipation of energy; ratio of kinetic energy to enthalpy)
Ekman number Ek Vagn Walfrid Ekman Geophysics (viscosity to Coriolis force ratio)
Eötvös numberEo Loránd Eötvösfluid mechanics (shape of bubbles or drops)
Ericksen numberEr Jerald Ericksenfluid dynamics (liquid crystal flow behavior; viscous over elastic forces)
Euler numberEu Leonhard Eulerhydrodynamics (stream pressure versus inertia forces)
Excess temperature coefficient heat transfer, fluid dynamics (change in internal energy versus kinetic energy)[8]
Fanning friction factorf John T. Fanningfluid mechanics (fraction of pressure losses due to friction in a pipe; 1/4th the Darcy friction factor)[9]
Froude numberFr William Froudefluid mechanics (wave and surface behaviour; ratio of a body's inertia to gravitational forces)
Galilei numberGa Galileo Galileifluid mechanics (gravitational over viscous forces)
Görtler numberG Henry Görtler [de]fluid dynamics (boundary layer flow along a concave wall)
Goucher number [fr] Go Frederick Shand Goucher (1888–1973) fluid dynamics (wire coating problems)
Garcia-Atance numberGA Gonzalo Garcia-Atance Fatjophase change (ultrasonic cavitation onset, ratio of pressures over pressure due to acceleration)
Graetz numberGz Leo Graetzheat transfer, fluid mechanics (laminar flow through a conduit; also used in mass transfer)
Grashof numberGr Franz Grashofheat transfer, natural convection (ratio of the buoyancy to viscous force)
Hartmann numberHa Julius Hartmann (1881–1951)magnetohydrodynamics (ratio of Lorentz to viscous forces)
Hagen numberHg Gotthilf Hagenheat transfer (ratio of the buoyancy to viscous force in forced convection)
Iribarren numberIr Ramón Iribarrenwave mechanics (breaking surface gravity waves on a slope)
Jakob numberJa Max Jakobheat transfer (ratio of sensible heat to latent heat during phase changes)
Jesus number Je Jesus Surface tension (ratio of surface tension and weight)
Karlovitz numberKa Béla Karlovitzturbulent combustion (characteristic flow time times flame stretch rate)
Kapitza numberKa Pyotr Kapitsafluid mechanics (thin film of liquid flows down inclined surfaces)
Keulegan–Carpenter numberKC Garbis H. Keulegan (1890–1989) and Lloyd H. Carpenterfluid dynamics (ratio of drag force to inertia for a bluff object in oscillatory fluid flow)
Knudsen numberKn Martin Knudsengas dynamics (ratio of the molecular mean free path length to a representative physical length scale)
Kutateladze numberKu Samson Kutateladzefluid mechanics (counter-current two-phase flow)[10]
Laplace numberLa Pierre-Simon Laplacefluid dynamics (free convection within immiscible fluids; ratio of surface tension to momentum-transport)
Lewis numberLe Warren K. Lewisheat and mass transfer (ratio of thermal to mass diffusivity)
Lift coefficientCL aerodynamics (lift available from an airfoil at a given angle of attack)
Lockhart–Martinelli parameter R. W. Lockhart and Raymond C. Martinellitwo-phase flow (flow of wet gases; liquid fraction)[11]
Mach numberM or Ma Ernst Machgas dynamics (compressible flow; dimensionless velocity)
Marangoni numberMg Carlo Marangonifluid mechanics (Marangoni flow; thermal surface tension forces over viscous forces)
Markstein numberMa George H. Marksteinturbulence, combustion (Markstein length to laminar flame thickness)
Morton numberMo Rose Mortonfluid dynamics (determination of bubble/drop shape)
Nusselt numberNu Wilhelm Nusseltheat transfer (forced convection; ratio of convective to conductive heat transfer)
Ohnesorge numberOh Wolfgang von Ohnesorgefluid dynamics (atomization of liquids, Marangoni flow)
Péclet numberPe or Jean Claude Eugène Pécletfluid mechanics (ratio of advective transport rate over molecular diffusive transport rate), heat transfer (ratio of advective transport rate over thermal diffusive transport rate)
Prandtl numberPr Ludwig Prandtlheat transfer (ratio of viscous diffusion rate over thermal diffusion rate)
Pressure coefficientCP aerodynamics, hydrodynamics (pressure experienced at a point on an airfoil; dimensionless pressure variable)
Rayleigh numberRa John William Strutt, 3rd Baron Rayleighheat transfer (buoyancy versus viscous forces in free convection)
Reynolds numberRe Osborne Reynoldsfluid mechanics (ratio of fluid inertial and viscous forces)[5]
Richardson numberRi Lewis Fry Richardsonfluid dynamics (effect of buoyancy on flow stability; ratio of potential over kinetic energy)[12]
Roshko numberRo Anatol Roshkofluid dynamics (oscillating flow, vortex shedding)
Rossby numberRo Carl-Gustaf Rossbyfluid flow (geophysics, ratio of inertial force to Coriolis force)
Rouse number P Hunter Rouse Fluid dynamics (concentration profile of suspended sediment)
Schmidt numberSc Ernst Heinrich Wilhelm Schmidt (1892–1975)mass transfer (viscous over molecular diffusion rate)[13]
Scruton number Sc Christopher 'Kit' Scruton Fluid dynamics (vortex resonance)
Shape factorH boundary layer flow (ratio of displacement thickness to momentum thickness)
Sherwood numberSh Thomas Kilgore Sherwoodmass transfer (forced convection; ratio of convective to diffusive mass transport)
Shields parameter θ Albert F. Shields Fluid dynamics (motion of sediment)
Sommerfeld numberS Arnold Sommerfeldhydrodynamic lubrication (boundary lubrication)[14]
Stanton numberSt Thomas Ernest Stantonheat transfer and fluid dynamics (forced convection)
Stokes numberStk or Sk Sir George Stokes, 1st Baronetparticles suspensions (ratio of characteristic time of particle to time of flow)
Strouhal numberSt Vincenc StrouhalVortex shedding (ratio of characteristic oscillatory velocity to ambient flow velocity)
Stuart numberN John Trevor Stuartmagnetohydrodynamics (ratio of electromagnetic to inertial forces)
Taylor numberTa G. I. Taylorfluid dynamics (rotating fluid flows; inertial forces due to rotation of a fluid versus viscous forces)
Thoma number σ Dieter Thoma (1881–1942) multiphase flow (hydrodynamic cavitation, pressure over dynamic pressure)
Ursell numberU Fritz Ursellwave mechanics (nonlinearity of surface gravity waves on a shallow fluid layer)
Wallis parameterj Graham B. Wallismultiphase flows (nondimensional superficial velocity)[15]
Weber numberWe Moritz Webermultiphase flow (strongly curved surfaces; ratio of inertia to surface tension)
Weissenberg numberWi Karl Weissenbergviscoelastic flows (shear rate times the relaxation time)[16]
Womersley number John R. Womersleybiofluid mechanics (continuous and pulsating flows; ratio of pulsatile flow frequency to viscous effects)[17]
Zeldovich number Yakov Zeldovichfluid dynamics, Combustion (Measure of activation energy)
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References

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