Encyclopedia of dimensionless numbers

About

Mon, 01 Jan 0001 00:00:00 +0000

About#

This project is a small, systematic database of dimensionless numbers and the physical quantities they compare. The goal is not only to make a readable reference, but also to keep every entry in a regular, easily parsable form that can be reused by scripts, calculators, visualizations, teaching material, or other tools.

Each dimensionless number is represented as a ratio of named quantities. Those quantities carry dimensions, SI-unit exponents, symbols, and short explanations, so the database can check dimensional consistency and generate pages without relying on ad hoc prose.

angular velocity

Mon, 01 Jan 0001 00:00:00 +0000

angular velocity#

$$\Omega \sim \text T^{-1}$$

Rate of rotation of a frame, body, or flow. It sets the Coriolis and rotational time scale in geophysical, turbomachinery, and rotating-container flows.

Used in#

Archimedes number

Mon, 01 Jan 0001 00:00:00 +0000

Archimedes number#

Named after: Archimedes (c. 287-c. 212 BCE).

$$\text{Ar} \stackrel{\text{def}}{=} \frac{g L^{3} \rho (\rho_s - \rho_f)}{\mu^{2}} \sim \frac{\text{buoyancy}}{\text{viscous resistance}}$$

Description#

Measures buoyant forcing on a particle or bubble relative to viscous resistance. It indicates the settling or rising regime set by density contrast.

Quantities#

Name	Symbol	SI units	Dimension
gravitational acceleration	$g$	$\mathrm{m}\,\mathrm{s}^{-2}$	$\text L\,\text T^{-2}$
characteristic length	$L$	$\mathrm{m}$	$\text L$
mass density	$\rho$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
solid-fluid density difference	$\rho_s - \rho_f$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
dynamic viscosity	$\mu$	$\mathrm{Pa}\,\mathrm{s}$	$\text L^{-1}\,\text M\,\text T^{-1}$

Regimes#

Particle settling

Bejan number

Mon, 01 Jan 0001 00:00:00 +0000

Bejan number#

Named after: Adrian Bejan (1948-).

$$\text{Be} \stackrel{\text{def}}{=} \frac{(p_1 - p_2) L^{2}}{\mu \alpha} \sim \frac{\text{pressure driving}}{\text{viscous thermal diffusion}}$$

Description#

Measures pressure driving relative to viscous and thermal diffusion. It characterizes forced-convection strength in pressure-driven heat-transfer problems.

Quantities#

Name	Symbol	SI units	Dimension
pressure drop	$p_1 - p_2$	$\mathrm{Pa}$	$\text L^{-1}\,\text M\,\text T^{-2}$
characteristic length	$L$	$\mathrm{m}$	$\text L$
dynamic viscosity	$\mu$	$\mathrm{Pa}\,\mathrm{s}$	$\text L^{-1}\,\text M\,\text T^{-1}$
thermal diffusivity	$\alpha$	$\mathrm{m}^{2}\,\mathrm{s}^{-1}$	$\text L^{2}\,\text T^{-1}$

Regimes#

Forced convection

Biot number

Mon, 01 Jan 0001 00:00:00 +0000

Biot number#

Named after: Jean-Baptiste Biot (1774-1862).

$$\text{Bi} \stackrel{\text{def}}{=} \frac{h L}{k} \sim \frac{\text{internal conduction resistance}}{\text{surface convection resistance}}$$

Description#

Measures internal conduction resistance relative to surface convection resistance. It indicates whether a solid can be treated as nearly uniform in temperature.

Quantities#

Name	Symbol	SI units	Dimension
convective heat transfer coefficient	$h$	$\mathrm{W}\,\mathrm{m}^{-2}\,\mathrm{K}^{-1}$	$\text M\,\Theta^{-1}\,\text T^{-3}$
characteristic length	$L$	$\mathrm{m}$	$\text L$
thermal conductivity	$k$	$\mathrm{W}\,\mathrm{m}^{-1}\,\mathrm{K}^{-1}$	$\text L\,\text M\,\Theta^{-1}\,\text T^{-3}$

Regimes#

Heat conduction in solids

Bond number

Mon, 01 Jan 0001 00:00:00 +0000

Bond number#

Also known as: Eötvös number.

Named after: Wilfrid Noel Bond (1897-1937), Loránd Eötvös (1848-1919).

$$\text{Bo} \stackrel{\text{def}}{=} \frac{\rho g L^{2}}{\sigma} \sim \frac{\text{gravity}}{\text{surface tension}}$$

Description#

Measures gravitational forcing relative to surface tension. It indicates whether interfaces are shaped more by weight or capillarity.

Quantities#

Name	Symbol	SI units	Dimension
mass density	$\rho$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
gravitational acceleration	$g$	$\mathrm{m}\,\mathrm{s}^{-2}$	$\text L\,\text T^{-2}$
characteristic length	$L$	$\mathrm{m}$	$\text L$
surface tension	$\sigma$	$\mathrm{N}\,\mathrm{m}^{-1}$	$\text M\,\text T^{-2}$

Regimes#

Capillary-gravity flow

Brinkman number

Mon, 01 Jan 0001 00:00:00 +0000

Brinkman number#

Named after: Henri Coenraad Brinkman (1908-1961).

$$\text{Br} \stackrel{\text{def}}{=} \frac{\mu U^{2}}{k (T_s - T_\infty)} \sim \frac{\text{viscous heat generation}}{\text{conductive heat removal}}$$

Description#

Measures viscous heat generation relative to conductive heat removal. It indicates whether dissipation heating must be included in an energy balance.

Quantities#

Name	Symbol	SI units	Dimension
dynamic viscosity	$\mu$	$\mathrm{Pa}\,\mathrm{s}$	$\text L^{-1}\,\text M\,\text T^{-1}$
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$
thermal conductivity	$k$	$\mathrm{W}\,\mathrm{m}^{-1}\,\mathrm{K}^{-1}$	$\text L\,\text M\,\Theta^{-1}\,\text T^{-3}$
surface-ambient temperature difference	$T_s - T_\infty$	$\mathrm{K}$	$\Theta$

Regimes#

Viscous heating

bulk modulus

Mon, 01 Jan 0001 00:00:00 +0000

bulk modulus#

$$K \sim \text L^{-1}\,\text M\,\text T^{-2}$$

Elastic resistance to uniform compression, defined as the pressure increase required to produce a relative decrease in volume. It sets the stress scale for compressibility and elastic wave response.

Used in#

Cauchy number

Capillary number

Mon, 01 Jan 0001 00:00:00 +0000

Capillary number#

$$\text{Ca} \stackrel{\text{def}}{=} \frac{\mu U}{\sigma} \sim \frac{\text{viscous stress}}{\text{surface tension}}$$

Description#

Measures viscous stress relative to surface-tension stress. It indicates whether flow can deform interfaces against capillary restoring forces.

Quantities#

Name	Symbol	SI units	Dimension
dynamic viscosity	$\mu$	$\mathrm{Pa}\,\mathrm{s}$	$\text L^{-1}\,\text M\,\text T^{-1}$
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$
surface tension	$\sigma$	$\mathrm{N}\,\mathrm{m}^{-1}$	$\text M\,\text T^{-2}$

Regimes#

Multiphase flow

Range	Regime	Description
0 – 0.001	capillary dominated	Surface tension forces dominate. Droplets and bubbles maintain compact, nearly spherical shapes.
0.001 – 1	transitional	Both surface tension and viscous forces are significant. Interfaces are noticeably deformed.
1 – ∞	viscous dominated	Viscous forces dominate. Interfaces deform freely and droplets may be highly elongated or broken up.

Cauchy number

Mon, 01 Jan 0001 00:00:00 +0000

Cauchy number#

Named after: Augustin-Louis Cauchy (1789-1857).

$$\text{Cy} \stackrel{\text{def}}{=} \frac{\rho U^{2}}{K} \sim \frac{\text{inertial stress}}{\text{elastic stress}}$$

Description#

Measures inertial stress relative to elastic or compressibility stress. It indicates when deformation, pressure waves, or material elasticity affect dynamic similarity.

Quantities#

Name	Symbol	SI units	Dimension
mass density	$\rho$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$
bulk modulus	$K$	$\mathrm{Pa}$	$\text L^{-1}\,\text M\,\text T^{-2}$

Regimes#

Dynamic similarity

Cavitation number

Mon, 01 Jan 0001 00:00:00 +0000

Cavitation number#

$$\sigma_c \stackrel{\text{def}}{=} \frac{(p - p_v)}{\rho U^{2}} \sim \frac{\text{pressure margin to vapor pressure}}{\text{dynamic pressure}}$$

Description#

Measures pressure margin above vapor pressure relative to dynamic pressure. It indicates how close a liquid flow is to forming vapor cavities.

Quantities#

Name	Symbol	SI units	Dimension
pressure margin to vapor pressure	$p - p_v$	$\mathrm{Pa}$	$\text L^{-1}\,\text M\,\text T^{-2}$
mass density	$\rho$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$

Regimes#

Cavitating flow

characteristic length

Mon, 01 Jan 0001 00:00:00 +0000

characteristic length#

$$L \sim \text L$$

A geometric scale representative of the problem: pipe diameter, boundary-layer thickness, chord length, or droplet radius. Its choice determines which physical effects are captured by a given dimensionless group.

Used in#

convective heat transfer coefficient

Mon, 01 Jan 0001 00:00:00 +0000

convective heat transfer coefficient#

$$h \sim \text M\,\Theta^{-1}\,\text T^{-3}$$

Rate of convective heat transfer per unit surface area per unit temperature difference between a surface and the adjacent fluid. Encapsulates geometry, flow conditions, and fluid properties into a single engineering quantity.

Used in#

Courant number

Mon, 01 Jan 0001 00:00:00 +0000

Courant number#

Also known as: CFL number.

Named after: Richard Courant (1888-1972).

$$\text{Co} \stackrel{\text{def}}{=} \frac{U t}{L} \sim \frac{\text{distance travelled per step}}{\text{cell length}}$$

Description#

Measures the distance information travels in one time step relative to a cell length. It is a stability and accuracy indicator for time-marching numerical schemes.

Quantities#

Name	Symbol	SI units	Dimension
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$
time	$t$	$\mathrm{s}$	$\text T$
characteristic length	$L$	$\mathrm{m}$	$\text L$

Regimes#

Explicit time integration

diffusivity

Mon, 01 Jan 0001 00:00:00 +0000

diffusivity#

$$D \sim \text L^{2}\,\text T^{-1}$$

Proportionality constant in Fick’s first law, relating the diffusive mass flux of a species to its concentration gradient. Depends strongly on temperature, pressure, and the solute–solvent pair.

Used in#

Drag coefficient

Mon, 01 Jan 0001 00:00:00 +0000

Drag coefficient#

$$C_D \stackrel{\text{def}}{=} \frac{F}{\rho U^{2} A} \sim \frac{\text{drag force}}{\text{dynamic pressure force}}$$

Description#

Measures drag force normalized by dynamic pressure and reference area. It compares the aerodynamic or hydrodynamic resistance of bodies across speeds and scales.

Quantities#

Name	Symbol	SI units	Dimension
force	$F$	$\mathrm{N}$	$\text L\,\text M\,\text T^{-2}$
mass density	$\rho$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$
reference area	$A$	$\mathrm{m}^{2}$	$\text L^{2}$

Regimes#

Body drag

dynamic viscosity

Mon, 01 Jan 0001 00:00:00 +0000

dynamic viscosity#

$$\mu \sim \text L^{-1}\,\text M\,\text T^{-1}$$

Resistance of a fluid to shear deformation. In a Newtonian fluid it relates shear stress linearly to the velocity gradient. It decreases with temperature for liquids and increases for gases.

Used in#

Eckert number

Mon, 01 Jan 0001 00:00:00 +0000

Eckert number#

Named after: Ernst Rudolph Georg Eckert (1904-2004).

$$\text{Ec} \stackrel{\text{def}}{=} \frac{U^{2}}{c_p (T_s - T_\infty)} \sim \frac{\text{kinetic energy}}{\text{sensible enthalpy}}$$

Description#

Measures kinetic energy relative to sensible enthalpy based on a temperature difference. It indicates when viscous or compressible heating affects the thermal field.

Quantities#

Name	Symbol	SI units	Dimension
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$
specific heat capacity	$c_p$	$\mathrm{m}^{2}\,\mathrm{s}^{-2}\,\mathrm{K}^{-1}$	$\text L^{2}\,\Theta^{-1}\,\text T^{-2}$
surface-ambient temperature difference	$T_s - T_\infty$	$\mathrm{K}$	$\Theta$

Regimes#

High speed heat transfer

Ekman number

Mon, 01 Jan 0001 00:00:00 +0000

Ekman number#

Named after: Vagn Walfrid Ekman (1874-1954).

$$\text{Ek} \stackrel{\text{def}}{=} \frac{\nu}{\Omega L^{2}} \sim \frac{\text{viscous diffusion}}{\text{rotation}}$$

Description#

Measures viscous diffusion relative to rotational effects. It indicates the importance of viscous boundary layers in rotating flows.

Quantities#

Name	Symbol	SI units	Dimension
kinematic viscosity	$\nu$	$\mathrm{m}^{2}\,\mathrm{s}^{-1}$	$\text L^{2}\,\text T^{-1}$
angular velocity	$\Omega$	$\mathrm{Hz}$	$\text T^{-1}$
characteristic length	$L$	$\mathrm{m}$	$\text L$

Regimes#

Rotating flow

Range	Regime	Description
0 – 0.01	rotation dominated	Rotation is strong compared with viscous diffusion. Thin Ekman layers control boundary adjustment.
0.01 – 1	mixed viscous-rotating	Viscous diffusion and rotation both influence the flow over the characteristic length.
1 – ∞	viscous dominated	Viscous diffusion acts faster than rotational adjustment. Rotation has a weak dynamical effect.

Euler number

Mon, 01 Jan 0001 00:00:00 +0000

Euler number#

Named after: Leonhard Euler (1707-1783).

$$\text{Eu} \stackrel{\text{def}}{=} \frac{(p_1 - p_2)}{\rho U^{2}} \sim \frac{\text{pressure drop}}{\text{dynamic pressure}}$$

Description#

Measures pressure drop relative to dynamic pressure. It indicates how strongly pressure forces or losses compare with inertial motion in a flow.

Quantities#

Name	Symbol	SI units	Dimension
pressure drop	$p_1 - p_2$	$\mathrm{Pa}$	$\text L^{-1}\,\text M\,\text T^{-2}$
mass density	$\rho$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$

Regimes#

Duct flow

force

Mon, 01 Jan 0001 00:00:00 +0000

force#

$$F \sim \text L\,\text M\,\text T^{-2}$$

Mechanical load exerted on a body or control surface. In aerodynamic and hydrodynamic coefficients it is normalized by dynamic pressure and reference area.

Used in#

Fourier number

Mon, 01 Jan 0001 00:00:00 +0000

Fourier number#

Named after: Jean-Baptiste Joseph Fourier (1768-1830).

$$\text{Fo} \stackrel{\text{def}}{=} \frac{\alpha t}{L^{2}} \sim \frac{\text{thermal diffusion spread}}{\text{length squared}}$$

Description#

Measures elapsed time relative to the thermal diffusion time across a length scale. It indicates how far a transient temperature disturbance has penetrated.

Quantities#

Name	Symbol	SI units	Dimension
thermal diffusivity	$\alpha$	$\mathrm{m}^{2}\,\mathrm{s}^{-1}$	$\text L^{2}\,\text T^{-1}$
time	$t$	$\mathrm{s}$	$\text T$
characteristic length	$L$	$\mathrm{m}$	$\text L$

Regimes#

Transient heat conduction

frequency

Mon, 01 Jan 0001 00:00:00 +0000

frequency#

$$f \sim \text T^{-1}$$

Number of complete oscillation cycles per unit time. In fluid mechanics it characterises periodic phenomena such as vortex shedding, acoustic resonance, or externally imposed oscillations.

Used in#

Froude number

Mon, 01 Jan 0001 00:00:00 +0000

Froude number#

Named after: William Froude (1810-1879).

$$\text{Fr} \stackrel{\text{def}}{=} \frac{U}{\sqrt{g} \sqrt{L}} \sim \frac{\text{flow velocity}}{\text{gravity wave speed}}$$

Description#

Measures flow speed relative to gravity wave speed. It indicates whether gravity can communicate disturbances upstream in free-surface and stratified flows.

Quantities#

Name	Symbol	SI units	Dimension
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$
gravitational acceleration	$g$	$\mathrm{m}\,\mathrm{s}^{-2}$	$\text L\,\text T^{-2}$
characteristic length	$L$	$\mathrm{m}$	$\text L$

Regimes#

Open channel flow

Galileo number

Mon, 01 Jan 0001 00:00:00 +0000

Galileo number#

Named after: Galileo Galilei (1564-1642).

$$\text{Ga} \stackrel{\text{def}}{=} \frac{g L^{3}}{\nu^{2}} \sim \frac{\text{gravity driven motion}}{\text{viscous diffusion}}$$

Description#

Measures gravity-driven motion relative to viscous diffusion. It is useful for settling, rising bubbles, and natural motion set by body forces.

Quantities#

Name	Symbol	SI units	Dimension
gravitational acceleration	$g$	$\mathrm{m}\,\mathrm{s}^{-2}$	$\text L\,\text T^{-2}$
characteristic length	$L$	$\mathrm{m}$	$\text L$
kinematic viscosity	$\nu$	$\mathrm{m}^{2}\,\mathrm{s}^{-1}$	$\text L^{2}\,\text T^{-1}$

Regimes#

Settling and natural motion

Grashof number

Mon, 01 Jan 0001 00:00:00 +0000

Grashof number#

Named after: Franz Grashof (1826-1893).

$$\text{Gr} \stackrel{\text{def}}{=} \frac{g \beta (T_s - T_\infty) L^{3}}{\nu^{2}} \sim \frac{\text{buoyancy}}{\text{viscous damping}}$$

Description#

Measures buoyancy forcing relative to viscous damping. It is the natural-convection analogue of a Reynolds-number scale based on temperature-induced density differences.

Quantities#

Name	Symbol	SI units	Dimension
gravitational acceleration	$g$	$\mathrm{m}\,\mathrm{s}^{-2}$	$\text L\,\text T^{-2}$
thermal expansion coefficient	$\beta$	$\mathrm{K}^{-1}$	$\Theta^{-1}$
surface-ambient temperature difference	$T_s - T_\infty$	$\mathrm{K}$	$\Theta$
characteristic length	$L$	$\mathrm{m}$	$\text L$
kinematic viscosity	$\nu$	$\mathrm{m}^{2}\,\mathrm{s}^{-1}$	$\text L^{2}\,\text T^{-1}$

Regimes#

Natural convection

gravitational acceleration

Mon, 01 Jan 0001 00:00:00 +0000

gravitational acceleration#

$$g \sim \text L\,\text T^{-2}$$

Acceleration imparted by gravity at the Earth’s surface (~9.81 m/s²). Sets the restoring force for surface waves and drives buoyancy-induced flows wherever density gradients are present.

Used in#

Helmholtz number

Mon, 01 Jan 0001 00:00:00 +0000

Helmholtz number#

Named after: Hermann von Helmholtz (1821-1894).

$$\text{He} \stackrel{\text{def}}{=} \frac{f L}{c} \sim \frac{\text{acoustic length scale}}{\text{sound speed}}$$

Description#

Measures acoustic length scale relative to wavelength through frequency and sound speed. It indicates whether a body or domain is acoustically compact.

Quantities#

Name	Symbol	SI units	Dimension
frequency	$f$	$\mathrm{Hz}$	$\text T^{-1}$
characteristic length	$L$	$\mathrm{m}$	$\text L$
speed of sound	$c$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$

Regimes#

Acoustics

kinematic viscosity

Mon, 01 Jan 0001 00:00:00 +0000

kinematic viscosity#

$$\nu \sim \text L^{2}\,\text T^{-1}$$

Momentum diffusivity of a fluid, equal to dynamic viscosity divided by density. It measures how quickly viscous momentum disturbances spread through the fluid.

Used in#

Knudsen number

Mon, 01 Jan 0001 00:00:00 +0000

Knudsen number#

Named after: Martin Knudsen (1871-1949).

$$\text{Kn} \stackrel{\text{def}}{=} \frac{\lambda}{L} \sim \frac{\text{molecular length}}{\text{continuum length}}$$

Description#

Measures molecular mean free path relative to a macroscopic length. It indicates whether continuum fluid assumptions are valid.

Quantities#

Name	Symbol	SI units	Dimension
mean free path	$\lambda$	$\mathrm{m}$	$\text L$
characteristic length	$L$	$\mathrm{m}$	$\text L$

Regimes#

Rarefied gas flow

Range	Regime	Description
0 – 0.01	continuum	Molecular mean free path is much smaller than the geometry. Navier-Stokes continuum models are usually appropriate.
0.01 – 0.1	slip	Continuum behavior mostly holds, but velocity slip and temperature jump at boundaries can be important.
0.1 – 10	transition	Continuum assumptions break down and kinetic effects must be modeled explicitly.
10 – ∞	free molecular	Molecular collisions with boundaries dominate over intermolecular collisions. Free molecular or kinetic descriptions are required.

Laplace number

Mon, 01 Jan 0001 00:00:00 +0000

Laplace number#

Also known as: Suratman number.

Named after: Pierre-Simon Laplace (1749-1827).

$$\text{La} \stackrel{\text{def}}{=} \frac{\rho \sigma L}{\mu^{2}} \sim \frac{\text{inertia and surface tension}}{\text{viscous damping}}$$

Description#

Measures inertial and capillary effects relative to viscous damping. It is the inverse-square counterpart of the Ohnesorge number for free-surface motion.

Quantities#

Name	Symbol	SI units	Dimension
mass density	$\rho$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
surface tension	$\sigma$	$\mathrm{N}\,\mathrm{m}^{-1}$	$\text M\,\text T^{-2}$
characteristic length	$L$	$\mathrm{m}$	$\text L$
dynamic viscosity	$\mu$	$\mathrm{Pa}\,\mathrm{s}$	$\text L^{-1}\,\text M\,\text T^{-1}$

Regimes#

Free surface flow

Lewis number

Mon, 01 Jan 0001 00:00:00 +0000

Lewis number#

Named after: Warren K. Lewis (1882-1975).

$$\text{Le} \stackrel{\text{def}}{=} \frac{\alpha}{D} \sim \frac{\text{thermal diffusion}}{\text{mass diffusion}}$$

Description#

Measures thermal diffusivity relative to mass diffusivity. It indicates whether heat or species concentration spreads faster.

Quantities#

Name	Symbol	SI units	Dimension
thermal diffusivity	$\alpha$	$\mathrm{m}^{2}\,\mathrm{s}^{-1}$	$\text L^{2}\,\text T^{-1}$
diffusivity	$D$	$\mathrm{m}^{2}\,\mathrm{s}^{-1}$	$\text L^{2}\,\text T^{-1}$

Regimes#

Coupled heat and mass transfer

Range	Regime	Description
0 – 1	mass diffuses faster	Species concentration disturbances spread faster than temperature disturbances. Mass transfer boundary layers tend to be thicker than thermal boundary layers.
1 – ∞	heat diffuses faster	Temperature disturbances spread faster than species concentration disturbances. Thermal boundary layers tend to be thicker than concentration boundary layers.

Lift coefficient

Mon, 01 Jan 0001 00:00:00 +0000

Lift coefficient#

$$C_L \stackrel{\text{def}}{=} \frac{F}{\rho U^{2} A} \sim \frac{\text{lift force}}{\text{dynamic pressure force}}$$

Description#

Measures lift force normalized by dynamic pressure and reference area. It compares lifting performance across geometries, speeds, and fluid densities.

Quantities#

Name	Symbol	SI units	Dimension
force	$F$	$\mathrm{N}$	$\text L\,\text M\,\text T^{-2}$
mass density	$\rho$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$
reference area	$A$	$\mathrm{m}^{2}$	$\text L^{2}$

Regimes#

Lifting surfaces

local-reference pressure difference

Mon, 01 Jan 0001 00:00:00 +0000

local-reference pressure difference#

$$p - p_\infty \sim \text L^{-1}\,\text M\,\text T^{-2}$$

Difference between local static pressure and a reference or free-stream static pressure. It is the pressure contrast used to express local pressure loading on a body or surface.

Used in#

Pressure coefficient

Mach number

Mon, 01 Jan 0001 00:00:00 +0000

Mach number#

Named after: Ernst Mach (1838-1916).

$$\text{Ma} \stackrel{\text{def}}{=} \frac{U}{c} \sim \frac{\text{flow speed}}{\text{sound speed}}$$

Description#

Measures flow speed relative to the speed of sound. It indicates whether compressibility, acoustic waves, and shocks are important.

Quantities#

Name	Symbol	SI units	Dimension
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$
speed of sound	$c$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$

Regimes#

Compressible flow

Range	Regime	Description
0 – 0.3	incompressible	Density variations are negligible. Flow can be treated as incompressible.
0.3 – 0.8	weakly compressible	Compressibility effects are small but may be noticeable in some cases.
0.8 – 1.2	transonic	Flow speed approaches the speed of sound. Shock waves and strong compressibility effects occur.
1.2 – 5	supersonic	Flow speed exceeds the speed of sound. Strong shock waves and compressibility effects dominate.
5 – ∞	hypersonic	Flow speed is much greater than the speed of sound. Extreme shock waves, high temperatures, and ionization may occur.

mass density

Mon, 01 Jan 0001 00:00:00 +0000

mass density#

$$\rho \sim \text L^{-3}\,\text M$$

Mass per unit volume of a substance. A fundamental thermodynamic property that depends on temperature and pressure; its spatial variation underpins buoyancy-driven flows and stratification.

Used in#

Mass Fourier number

Mon, 01 Jan 0001 00:00:00 +0000

Mass Fourier number#

Named after: Jean-Baptiste Joseph Fourier (1768-1830).

$$\text{Fo}_m \stackrel{\text{def}}{=} \frac{D t}{L^{2}} \sim \frac{\text{mass diffusion spread}}{\text{length squared}}$$

Description#

Measures elapsed time relative to the mass diffusion time across a length scale. It indicates how far a transient concentration disturbance has penetrated.

Quantities#

Name	Symbol	SI units	Dimension
diffusivity	$D$	$\mathrm{m}^{2}\,\mathrm{s}^{-1}$	$\text L^{2}\,\text T^{-1}$
time	$t$	$\mathrm{s}$	$\text T$
characteristic length	$L$	$\mathrm{m}$	$\text L$

Regimes#

Transient diffusion

Mass Péclet number

Mon, 01 Jan 0001 00:00:00 +0000

Mass Péclet number#

Also known as: Bodenstein number.

Named after: Jean Claude Eugène Péclet (1793-1857).

$$\text{Pe}_m \stackrel{\text{def}}{=} \frac{U L}{D} \sim \frac{\text{advection}}{\text{mass diffusion}}$$

Description#

Measures advective species transport relative to molecular diffusion. It indicates whether concentration fields are carried mainly by flow or smoothed by diffusion.

Quantities#

Name	Symbol	SI units	Dimension
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$
characteristic length	$L$	$\mathrm{m}$	$\text L$
diffusivity	$D$	$\mathrm{m}^{2}\,\mathrm{s}^{-1}$	$\text L^{2}\,\text T^{-1}$

Regimes#

Mass transfer

mass transfer coefficient

Mon, 01 Jan 0001 00:00:00 +0000

mass transfer coefficient#

$$k_m \sim \text L\,\text T^{-1}$$

Convective mass-transfer analogue of the heat-transfer coefficient. Relates the convective species flux to the concentration difference between an interface and the bulk fluid; depends on geometry, flow conditions, and diffusivity.

Used in#

Sherwood number

mean free path

Mon, 01 Jan 0001 00:00:00 +0000

mean free path#

$$\lambda \sim \text L$$

Average distance a molecule travels between collisions. It provides the microscopic length scale used to decide whether a gas can be treated as a continuum.

Used in#

Knudsen number

Nusselt number

Mon, 01 Jan 0001 00:00:00 +0000

Nusselt number#

Named after: Wilhelm Nusselt (1882-1957).

$$\text{Nu} \stackrel{\text{def}}{=} \frac{h L}{k} \sim \frac{\text{convective heat transfer}}{\text{conductive heat transfer}}$$

Description#

Measures convective heat transfer relative to pure conduction across the same length scale. It indicates how much motion enhances heat transfer above the conductive reference.

Quantities#

Name	Symbol	SI units	Dimension
convective heat transfer coefficient	$h$	$\mathrm{W}\,\mathrm{m}^{-2}\,\mathrm{K}^{-1}$	$\text M\,\Theta^{-1}\,\text T^{-3}$
characteristic length	$L$	$\mathrm{m}$	$\text L$
thermal conductivity	$k$	$\mathrm{W}\,\mathrm{m}^{-1}\,\mathrm{K}^{-1}$	$\text L\,\text M\,\Theta^{-1}\,\text T^{-3}$

Regimes#

Forced convection

Ohnesorge number

Mon, 01 Jan 0001 00:00:00 +0000

Ohnesorge number#

Named after: Wolfgang von Ohnesorge (1901-1976).

$$\text{Oh} \stackrel{\text{def}}{=} \frac{\mu}{\sqrt{\rho} \sqrt{\sigma} \sqrt{L}} \sim \frac{\text{viscous damping}}{\text{inertia and surface tension}}$$

Description#

Measures viscous damping relative to inertial and capillary effects. It indicates how strongly viscosity suppresses droplet oscillation, pinch off, and breakup.

Quantities#

Name	Symbol	SI units	Dimension
dynamic viscosity	$\mu$	$\mathrm{Pa}\,\mathrm{s}$	$\text L^{-1}\,\text M\,\text T^{-1}$
mass density	$\rho$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
surface tension	$\sigma$	$\mathrm{N}\,\mathrm{m}^{-1}$	$\text M\,\text T^{-2}$
characteristic length	$L$	$\mathrm{m}$	$\text L$

Regimes#

Droplet dynamics

Péclet number

Mon, 01 Jan 0001 00:00:00 +0000

Péclet number#

Named after: Jean Claude Eugène Péclet (1793-1857).

$$\text{Pe} \stackrel{\text{def}}{=} \frac{U L}{\alpha} \sim \frac{\text{advection}}{\text{thermal diffusion}}$$

Description#

Measures advective heat transport relative to thermal diffusion. It indicates whether temperature is carried mainly by flow or smoothed by diffusion.

Quantities#

Name	Symbol	SI units	Dimension
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$
characteristic length	$L$	$\mathrm{m}$	$\text L$
thermal diffusivity	$\alpha$	$\mathrm{m}^{2}\,\mathrm{s}^{-1}$	$\text L^{2}\,\text T^{-1}$

Regimes#

Heat transfer

Prandtl number

Mon, 01 Jan 0001 00:00:00 +0000

Prandtl number#

Named after: Ludwig Prandtl (1875-1953).

$$\text{Pr} \stackrel{\text{def}}{=} \frac{\nu}{\alpha} \sim \frac{\text{momentum diffusion}}{\text{heat diffusion}}$$

Description#

Measures momentum diffusivity relative to thermal diffusivity. It indicates the relative thickness of velocity and thermal boundary layers.

Quantities#

Name	Symbol	SI units	Dimension
kinematic viscosity	$\nu$	$\mathrm{m}^{2}\,\mathrm{s}^{-1}$	$\text L^{2}\,\text T^{-1}$
thermal diffusivity	$\alpha$	$\mathrm{m}^{2}\,\mathrm{s}^{-1}$	$\text L^{2}\,\text T^{-1}$

Regimes#

Heat transfer

Range	Regime	Description
0 – 0.1	liquid metal	Thermal diffusivity dominates over momentum diffusion. Thermal boundary layers are much thicker than velocity boundary layers, typical of liquid metals.
0.1 – 10	gas	Momentum and thermal diffusivities are comparable. Velocity and thermal boundary layers have similar thicknesses, typical of many gases.
10 – ∞	oil or viscous liquid	Momentum diffusion dominates over thermal diffusion. Thermal boundary layers are thinner than velocity boundary layers, typical of oils and viscous liquids.

Pressure coefficient

Mon, 01 Jan 0001 00:00:00 +0000

Pressure coefficient#

$$C_p \stackrel{\text{def}}{=} \frac{(p - p_\infty)}{\rho U^{2}} \sim \frac{\text{local pressure difference}}{\text{dynamic pressure}}$$

Description#

Measures static pressure variation normalized by dynamic pressure. It maps local pressure loading on bodies and surfaces in a flow.

Quantities#

Name	Symbol	SI units	Dimension
local-reference pressure difference	$p - p_\infty$	$\mathrm{Pa}$	$\text L^{-1}\,\text M\,\text T^{-2}$
mass density	$\rho$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$

Regimes#

External flow

pressure drop

Mon, 01 Jan 0001 00:00:00 +0000

pressure drop#

$$p_1 - p_2 \sim \text L^{-1}\,\text M\,\text T^{-2}$$

Static pressure decrease between an upstream station and a downstream station. It acts as the driving force for conduit flow and measures losses due to friction, fittings, or geometry changes.

Used in#

pressure margin to vapor pressure

Mon, 01 Jan 0001 00:00:00 +0000

pressure margin to vapor pressure#

$$p - p_v \sim \text L^{-1}\,\text M\,\text T^{-2}$$

Difference between the local or reference static pressure and the liquid vapor pressure. It measures how much pressure head remains before vapor cavities can form.

Used in#

Cavitation number

Rayleigh number

Mon, 01 Jan 0001 00:00:00 +0000

Rayleigh number#

Named after: John William Strutt, 3rd Baron Rayleigh (1842-1919).

$$\text{Ra} \stackrel{\text{def}}{=} \frac{g \beta (T_s - T_\infty) L^{3}}{\nu \alpha} \sim \frac{\text{buoyancy}}{\text{momentum and heat diffusion}}$$

Description#

Measures buoyancy forcing relative to combined momentum and thermal diffusion. It indicates the onset and strength of natural convection.

Quantities#

Name	Symbol	SI units	Dimension
gravitational acceleration	$g$	$\mathrm{m}\,\mathrm{s}^{-2}$	$\text L\,\text T^{-2}$
thermal expansion coefficient	$\beta$	$\mathrm{K}^{-1}$	$\Theta^{-1}$
surface-ambient temperature difference	$T_s - T_\infty$	$\mathrm{K}$	$\Theta$
characteristic length	$L$	$\mathrm{m}$	$\text L$
kinematic viscosity	$\nu$	$\mathrm{m}^{2}\,\mathrm{s}^{-1}$	$\text L^{2}\,\text T^{-1}$
thermal diffusivity	$\alpha$	$\mathrm{m}^{2}\,\mathrm{s}^{-1}$	$\text L^{2}\,\text T^{-1}$

Regimes#

Buoyancy driven heat transfer

reference area

Mon, 01 Jan 0001 00:00:00 +0000

reference area#

$$A \sim \text L^{2}$$

Characteristic surface or projected area used to normalize forces, fluxes, or loads. The appropriate reference area is set by the geometry and convention for the coefficient being formed.

Used in#

Reynolds number

Mon, 01 Jan 0001 00:00:00 +0000

Reynolds number#

Named after: Osborne Reynolds (1842-1912).

$$\text{Re} \stackrel{\text{def}}{=} \frac{\rho U L}{\mu} \sim \frac{\text{inertia}}{\text{viscosity}}$$

Description#

Measures inertial transport relative to viscous resistance. It is the primary indicator of laminar, transitional, and turbulent flow behavior.

Quantities#

Name	Symbol	SI units	Dimension
mass density	$\rho$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$
characteristic length	$L$	$\mathrm{m}$	$\text L$
dynamic viscosity	$\mu$	$\mathrm{Pa}\,\mathrm{s}$	$\text L^{-1}\,\text M\,\text T^{-1}$

Regimes#

Pipe flow

Richardson number

Mon, 01 Jan 0001 00:00:00 +0000

Richardson number#

Named after: Lewis Fry Richardson (1881-1953).

$$\text{Ri} \stackrel{\text{def}}{=} \frac{g \beta (T_s - T_\infty) L}{U^{2}} \sim \frac{\text{buoyancy}}{\text{inertia}}$$

Description#

Measures buoyancy relative to inertial forcing. It indicates whether a flow behaves more like forced, mixed, or natural convection.

Quantities#

Name	Symbol	SI units	Dimension
gravitational acceleration	$g$	$\mathrm{m}\,\mathrm{s}^{-2}$	$\text L\,\text T^{-2}$
thermal expansion coefficient	$\beta$	$\mathrm{K}^{-1}$	$\Theta^{-1}$
surface-ambient temperature difference	$T_s - T_\infty$	$\mathrm{K}$	$\Theta$
characteristic length	$L$	$\mathrm{m}$	$\text L$
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$

Regimes#

Mixed convection

Roshko number

Mon, 01 Jan 0001 00:00:00 +0000

Roshko number#

Named after: Anatol Roshko (1923-2017).

$$\text{Ro}_s \stackrel{\text{def}}{=} \frac{f L^{2}}{\nu} \sim \frac{\text{vortex shedding}}{\text{viscous diffusion}}$$

Description#

Measures vortex-shedding frequency relative to viscous diffusion. It combines Strouhal and Reynolds effects for wake oscillations.

Quantities#

Name	Symbol	SI units	Dimension
frequency	$f$	$\mathrm{Hz}$	$\text T^{-1}$
characteristic length	$L$	$\mathrm{m}$	$\text L$
kinematic viscosity	$\nu$	$\mathrm{m}^{2}\,\mathrm{s}^{-1}$	$\text L^{2}\,\text T^{-1}$

Regimes#

Wake oscillations

Range	Regime	Description
0 – 1	viscous wake	Viscous diffusion is fast relative to the shedding time scale, suppressing organized vortex formation.
1 – 10	periodic shedding	Unsteady shedding and viscous diffusion are comparable, supporting coherent periodic wakes.
10 – ∞	high frequency shedding	Shedding is rapid compared with viscous diffusion and wake dynamics are dominated by inertia.

Rossby number

Mon, 01 Jan 0001 00:00:00 +0000

Rossby number#

Named after: Carl-Gustaf Rossby (1898-1957).

$$\text{Ro} \stackrel{\text{def}}{=} \frac{U}{\Omega L} \sim \frac{\text{inertia}}{\text{rotation}}$$

Description#

Measures inertial motion relative to rotational or Coriolis effects. It indicates whether rotation controls the flow dynamics.

Quantities#

Name	Symbol	SI units	Dimension
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$
angular velocity	$\Omega$	$\mathrm{Hz}$	$\text T^{-1}$
characteristic length	$L$	$\mathrm{m}$	$\text L$

Regimes#

Rotating flow

Range	Regime	Description
0 – 0.1	rotation dominated	Coriolis effects dominate inertia. Flow tends toward geostrophic or columnar balance.
0.1 – 1	mixed	Rotation and inertia both influence trajectories, waves, and instabilities.
1 – ∞	inertia dominated	Inertial motion dominates over rotation. Coriolis deflection is a secondary correction.

Schmidt number

Mon, 01 Jan 0001 00:00:00 +0000

Schmidt number#

Named after: Ernst Heinrich Wilhelm Schmidt (1892-1975).

$$\text{Sc} \stackrel{\text{def}}{=} \frac{\nu}{D} \sim \frac{\text{momentum diffusion}}{\text{mass diffusion}}$$

Description#

Measures momentum diffusivity relative to mass diffusivity. It indicates the relative thickness of velocity and concentration boundary layers.

Quantities#

Name	Symbol	SI units	Dimension
kinematic viscosity	$\nu$	$\mathrm{m}^{2}\,\mathrm{s}^{-1}$	$\text L^{2}\,\text T^{-1}$
diffusivity	$D$	$\mathrm{m}^{2}\,\mathrm{s}^{-1}$	$\text L^{2}\,\text T^{-1}$

Regimes#

Molecular diffusion

Range	Regime	Description
0 – 0.2	liquid metal	Mass diffusivity dominates over momentum diffusivity. Occurs in liquid metals and dilute gases.
0.2 – 4	gas	Momentum and mass diffusivities are comparable. Common in many gases.
4 – ∞	liquid	Typical for liquids, e.g. 600 to 700 for water.

shear stress

Mon, 01 Jan 0001 00:00:00 +0000

shear stress#

$$\tau \sim \text L^{-1}\,\text M\,\text T^{-2}$$

Tangential stress transmitted across a surface by viscous or turbulent momentum transport. It is commonly normalized by dynamic pressure in wall-friction and sediment-transport parameters.

Used in#

Sherwood number

Mon, 01 Jan 0001 00:00:00 +0000

Sherwood number#

Named after: Thomas Kilgore Sherwood (1903-1976).

$$\text{Sh} \stackrel{\text{def}}{=} \frac{k_m L}{D} \sim \frac{\text{convective mass transfer}}{\text{molecular diffusion}}$$

Description#

Measures convective mass transfer relative to molecular diffusion. It indicates how much flow enhances species transport above the diffusive reference.

Quantities#

Name	Symbol	SI units	Dimension
mass transfer coefficient	$k_m$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$
characteristic length	$L$	$\mathrm{m}$	$\text L$
diffusivity	$D$	$\mathrm{m}^{2}\,\mathrm{s}^{-1}$	$\text L^{2}\,\text T^{-1}$

Regimes#

Mass transfer

Shields parameter

Mon, 01 Jan 0001 00:00:00 +0000

Shields parameter#

Also known as: Shields number.

Named after: Albert Frank Shields (1908-1974).

$$\theta \stackrel{\text{def}}{=} \frac{\tau}{(\rho_s - \rho_f) g L} \sim \frac{\text{bed shear stress}}{\text{submerged particle weight}}$$

Description#

Measures bed shear stress relative to submerged particle weight. It indicates whether grains on a bed remain stable or begin to move.

Quantities#

Name	Symbol	SI units	Dimension
shear stress	$\tau$	$\mathrm{Pa}$	$\text L^{-1}\,\text M\,\text T^{-2}$
solid-fluid density difference	$\rho_s - \rho_f$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
gravitational acceleration	$g$	$\mathrm{m}\,\mathrm{s}^{-2}$	$\text L\,\text T^{-2}$
characteristic length	$L$	$\mathrm{m}$	$\text L$

Regimes#

Sediment transport

Skin-friction coefficient

Mon, 01 Jan 0001 00:00:00 +0000

Skin-friction coefficient#

$$C_f \stackrel{\text{def}}{=} \frac{\tau}{\rho U^{2}} \sim \frac{\text{wall shear stress}}{\text{dynamic pressure}}$$

Description#

Measures wall shear stress normalized by dynamic pressure. It indicates the frictional part of momentum loss along a surface.

Quantities#

Name	Symbol	SI units	Dimension
shear stress	$\tau$	$\mathrm{Pa}$	$\text L^{-1}\,\text M\,\text T^{-2}$
mass density	$\rho$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$

Regimes#

Wall bounded flow

Range	Regime	Description
0 – 0.001	low friction	Wall shear is small compared with the dynamic pressure scale, as in high Reynolds number streamlined flows.
0.001 – 0.01	moderate friction	Wall shear is a measurable part of the momentum balance.
0.01 – ∞	high friction	Wall shear is large relative to dynamic pressure, often indicating strong viscous effects or roughness.

solid-fluid density difference

Mon, 01 Jan 0001 00:00:00 +0000

solid-fluid density difference#

$$\rho_s - \rho_f \sim \text L^{-3}\,\text M$$

Density of a solid particle or sediment grain minus the surrounding fluid density. It sets the submerged-weight contrast for settling, suspension, and sediment transport.

Used in#

specific heat capacity

Mon, 01 Jan 0001 00:00:00 +0000

specific heat capacity#

$$c_p \sim \text L^{2}\,\Theta^{-1}\,\text T^{-2}$$

Energy required to raise the temperature of unit mass by one kelvin at constant pressure. It sets the sensible heat storage scale in convective heat transfer and compressible-flow heating estimates.

Used in#

speed of sound

Mon, 01 Jan 0001 00:00:00 +0000

speed of sound#

$$c \sim \text L\,\text T^{-1}$$

Phase speed of small-amplitude compressional waves in a medium. Determined by the ratio of the isentropic bulk modulus to density; sets the compressibility scale and serves as the reference speed in the Mach number.

Used in#

Stanton number

Mon, 01 Jan 0001 00:00:00 +0000

Stanton number#

Named after: Thomas Ernest Stanton (1865-1931).

$$\text{St}_h \stackrel{\text{def}}{=} \frac{h}{\rho c_p U} \sim \frac{\text{surface heat transfer}}{\text{convective heat capacity flow}}$$

Description#

Measures surface heat-transfer rate relative to the thermal capacity advected by a flow. It indicates what fraction of the moving fluid thermal capacity is exchanged.

Quantities#

Name	Symbol	SI units	Dimension
convective heat transfer coefficient	$h$	$\mathrm{W}\,\mathrm{m}^{-2}\,\mathrm{K}^{-1}$	$\text M\,\Theta^{-1}\,\text T^{-3}$
mass density	$\rho$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
specific heat capacity	$c_p$	$\mathrm{m}^{2}\,\mathrm{s}^{-2}\,\mathrm{K}^{-1}$	$\text L^{2}\,\Theta^{-1}\,\text T^{-2}$
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$

Regimes#

Convective heat transfer

Strouhal number

Mon, 01 Jan 0001 00:00:00 +0000

Strouhal number#

Named after: Vincenc Strouhal (1850-1922).

$$\text{St} \stackrel{\text{def}}{=} \frac{f L}{U} \sim \frac{\text{oscillation speed}}{\text{flow speed}}$$

Description#

Measures oscillation time scale relative to advective flow time scale. It connects periodic motion such as vortex shedding to the speed and size of the flow.

Quantities#

Name	Symbol	SI units	Dimension
frequency	$f$	$\mathrm{Hz}$	$\text T^{-1}$
characteristic length	$L$	$\mathrm{m}$	$\text L$
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$

Regimes#

Fluid oscillations

surface tension

Mon, 01 Jan 0001 00:00:00 +0000

surface tension#

$$\sigma \sim \text M\,\text T^{-2}$$

Energy per unit interfacial area (equivalently, force per unit contact-line length) arising from the imbalance of intermolecular cohesive forces at a phase boundary. Drives capillary rise, droplet formation, and interface stabilisation.

Used in#

surface-ambient temperature difference

Mon, 01 Jan 0001 00:00:00 +0000

surface-ambient temperature difference#

$$T_s - T_\infty \sim \Theta$$

Temperature difference between a surface and the surrounding or free-stream fluid. It is the driving temperature contrast for many convection, buoyancy, and wall-heating estimates.

Used in#

Taylor number

Mon, 01 Jan 0001 00:00:00 +0000

Taylor number#

Named after: Geoffrey Ingram Taylor (1886-1975).

$$\text{Ta} \stackrel{\text{def}}{=} \frac{\Omega^{2} L^{4}}{\nu^{2}} \sim \frac{\text{rotation}}{\text{viscous diffusion}}$$

Description#

Measures rotational forcing relative to viscous diffusion. It indicates when rotating shear flows become susceptible to Taylor-type instabilities.

Quantities#

Name	Symbol	SI units	Dimension
angular velocity	$\Omega$	$\mathrm{Hz}$	$\text T^{-1}$
characteristic length	$L$	$\mathrm{m}$	$\text L$
kinematic viscosity	$\nu$	$\mathrm{m}^{2}\,\mathrm{s}^{-1}$	$\text L^{2}\,\text T^{-1}$

Regimes#

Rotating instability

Range	Regime	Description
0 – 1	viscous dominated	Viscous diffusion damps rotational shear before strong inertial structures develop.
1 – 1700	rotation important	Rotation competes with viscous diffusion and can alter stability and momentum transport.
1700 – ∞	instability prone	Rotational forcing is strong enough that Taylor-like vortices or related instabilities may occur, depending on geometry.

thermal conductivity

Mon, 01 Jan 0001 00:00:00 +0000

thermal conductivity#

$$k \sim \text L\,\text M\,\Theta^{-1}\,\text T^{-3}$$

Intrinsic ability of a material to transfer heat by conduction. Relates the conductive heat flux to the local temperature gradient via Fourier’s law; generally weakly temperature-dependent for solids, but strongly so for gases and liquids.

Used in#

thermal diffusivity

Mon, 01 Jan 0001 00:00:00 +0000

thermal diffusivity#

$$\alpha \sim \text L^{2}\,\text T^{-1}$$

Ratio of thermal conductivity to volumetric heat capacity; governs the rate at which temperature disturbances propagate through a material. A high value indicates rapid thermal equilibration relative to heat storage.

Used in#

thermal expansion coefficient

Mon, 01 Jan 0001 00:00:00 +0000

thermal expansion coefficient#

$$\beta \sim \Theta^{-1}$$

Fractional density or volume change per unit temperature change. In buoyancy-driven flow it converts a temperature difference into a density contrast under the Boussinesq approximation.

Used in#

time

Mon, 01 Jan 0001 00:00:00 +0000

time#

$$t \sim \text T$$

Elapsed time, used as the independent variable in transient analyses. Sets the temporal scale against which diffusion, convection, or oscillation rates are compared in unsteady dimensionless groups.

Used in#

velocity

Mon, 01 Jan 0001 00:00:00 +0000

velocity#

$$U \sim \text L\,\text T^{-1}$$

Rate of displacement of a fluid element or body. In dimensionless analysis it typically represents a characteristic (bulk, free-stream, or mean) speed that sets the advective time scale.

Used in#

Weber number

Mon, 01 Jan 0001 00:00:00 +0000

Weber number#

Named after: Moritz Weber (1871-1951).

$$\text{We} \stackrel{\text{def}}{=} \frac{\rho U^{2}}{\sigma/L} \sim \frac{\text{inertia}}{\text{surface tension}}$$

Description#

Measures inertial stress relative to surface-tension stress. It indicates whether interfaces, jets, and droplets remain cohesive or break up.

Quantities#

Name	Symbol	SI units	Dimension
mass density	$\rho$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$
surface tension	$\sigma$	$\mathrm{N}\,\mathrm{m}^{-1}$	$\text M\,\text T^{-2}$
characteristic length	$L$	$\mathrm{m}$	$\text L$

Regimes#

Liquid jet breakup

Womersley number

Mon, 01 Jan 0001 00:00:00 +0000

Womersley number#

Named after: John Ronald Womersley (1907-1958).

$$\alpha \stackrel{\text{def}}{=} \frac{L \sqrt{f} \sqrt{\rho}}{\sqrt{\mu}} \sim \frac{\text{characteristic length}}{\text{viscous penetration depth}}$$

Description#

Measures oscillatory inertia relative to viscous diffusion in unsteady internal flow. It indicates whether velocity profiles can adjust within each cycle.

Quantities#

Name	Symbol	SI units	Dimension
characteristic length	$L$	$\mathrm{m}$	$\text L$
frequency	$f$	$\mathrm{Hz}$	$\text T^{-1}$
mass density	$\rho$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
dynamic viscosity	$\mu$	$\mathrm{Pa}\,\mathrm{s}$	$\text L^{-1}\,\text M\,\text T^{-1}$

Regimes#

Oscillatory pipe flow

Name	Symbol	SI units	Dimension
gravitational acceleration	\(g\)	\(\mathrm{m}\,\mathrm{s}^{-2}\)	\(\text L\,\text T^{-2}\)
characteristic length	\(L\)	\(\mathrm{m}\)	\(\text L\)
mass density	\(\rho\)	\(\mathrm{kg}\,\mathrm{m}^{-3}\)	\(\text L^{-3}\,\text M\)
solid-fluid density difference	\(\rho_s - \rho_f\)	\(\mathrm{kg}\,\mathrm{m}^{-3}\)	\(\text L^{-3}\,\text M\)
dynamic viscosity	\(\mu\)	\(\mathrm{Pa}\,\mathrm{s}\)	\(\text L^{-1}\,\text M\,\text T^{-1}\)

Name	Symbol	SI units	Dimension
pressure drop	\(p_1 - p_2\)	\(\mathrm{Pa}\)	\(\text L^{-1}\,\text M\,\text T^{-2}\)
characteristic length	\(L\)	\(\mathrm{m}\)	\(\text L\)
dynamic viscosity	\(\mu\)	\(\mathrm{Pa}\,\mathrm{s}\)	\(\text L^{-1}\,\text M\,\text T^{-1}\)
thermal diffusivity	\(\alpha\)	\(\mathrm{m}^{2}\,\mathrm{s}^{-1}\)	\(\text L^{2}\,\text T^{-1}\)

Name	Symbol	SI units	Dimension
convective heat transfer coefficient	\(h\)	\(\mathrm{W}\,\mathrm{m}^{-2}\,\mathrm{K}^{-1}\)	\(\text M\,\Theta^{-1}\,\text T^{-3}\)
characteristic length	\(L\)	\(\mathrm{m}\)	\(\text L\)
thermal conductivity	\(k\)	\(\mathrm{W}\,\mathrm{m}^{-1}\,\mathrm{K}^{-1}\)	\(\text L\,\text M\,\Theta^{-1}\,\text T^{-3}\)

Name	Symbol	SI units	Dimension
mass density	\(\rho\)	\(\mathrm{kg}\,\mathrm{m}^{-3}\)	\(\text L^{-3}\,\text M\)
gravitational acceleration	\(g\)	\(\mathrm{m}\,\mathrm{s}^{-2}\)	\(\text L\,\text T^{-2}\)
characteristic length	\(L\)	\(\mathrm{m}\)	\(\text L\)
surface tension	\(\sigma\)	\(\mathrm{N}\,\mathrm{m}^{-1}\)	\(\text M\,\text T^{-2}\)

Name	Symbol	SI units	Dimension
dynamic viscosity	\(\mu\)	\(\mathrm{Pa}\,\mathrm{s}\)	\(\text L^{-1}\,\text M\,\text T^{-1}\)
velocity	\(U\)	\(\mathrm{m}\,\mathrm{s}^{-1}\)	\(\text L\,\text T^{-1}\)
thermal conductivity	\(k\)	\(\mathrm{W}\,\mathrm{m}^{-1}\,\mathrm{K}^{-1}\)	\(\text L\,\text M\,\Theta^{-1}\,\text T^{-3}\)
surface-ambient temperature difference	\(T_s - T_\infty\)	\(\mathrm{K}\)	\(\Theta\)

Name	Symbol	SI units	Dimension
pressure margin to vapor pressure	\(p - p_v\)	\(\mathrm{Pa}\)	\(\text L^{-1}\,\text M\,\text T^{-2}\)
mass density	\(\rho\)	\(\mathrm{kg}\,\mathrm{m}^{-3}\)	\(\text L^{-3}\,\text M\)
velocity	\(U\)	\(\mathrm{m}\,\mathrm{s}^{-1}\)	\(\text L\,\text T^{-1}\)

Name	Symbol	SI units	Dimension
force	\(F\)	\(\mathrm{N}\)	\(\text L\,\text M\,\text T^{-2}\)
mass density	\(\rho\)	\(\mathrm{kg}\,\mathrm{m}^{-3}\)	\(\text L^{-3}\,\text M\)
velocity	\(U\)	\(\mathrm{m}\,\mathrm{s}^{-1}\)	\(\text L\,\text T^{-1}\)
reference area	\(A\)	\(\mathrm{m}^{2}\)	\(\text L^{2}\)

Name	Symbol	SI units	Dimension
kinematic viscosity	\(\nu\)	\(\mathrm{m}^{2}\,\mathrm{s}^{-1}\)	\(\text L^{2}\,\text T^{-1}\)
angular velocity	\(\Omega\)	\(\mathrm{Hz}\)	\(\text T^{-1}\)
characteristic length	\(L\)	\(\mathrm{m}\)	\(\text L\)

Name	Symbol	SI units	Dimension
mean free path	\(\lambda\)	\(\mathrm{m}\)	\(\text L\)
characteristic length	\(L\)	\(\mathrm{m}\)	\(\text L\)

Name	Symbol	SI units	Dimension
diffusivity	\(D\)	\(\mathrm{m}^{2}\,\mathrm{s}^{-1}\)	\(\text L^{2}\,\text T^{-1}\)
time	\(t\)	\(\mathrm{s}\)	\(\text T\)
characteristic length	\(L\)	\(\mathrm{m}\)	\(\text L\)

Name	Symbol	SI units	Dimension
local-reference pressure difference	\(p - p_\infty\)	\(\mathrm{Pa}\)	\(\text L^{-1}\,\text M\,\text T^{-2}\)
mass density	\(\rho\)	\(\mathrm{kg}\,\mathrm{m}^{-3}\)	\(\text L^{-3}\,\text M\)
velocity	\(U\)	\(\mathrm{m}\,\mathrm{s}^{-1}\)	\(\text L\,\text T^{-1}\)

Name	Symbol	SI units	Dimension
mass transfer coefficient	\(k_m\)	\(\mathrm{m}\,\mathrm{s}^{-1}\)	\(\text L\,\text T^{-1}\)
characteristic length	\(L\)	\(\mathrm{m}\)	\(\text L\)
diffusivity	\(D\)	\(\mathrm{m}^{2}\,\mathrm{s}^{-1}\)	\(\text L^{2}\,\text T^{-1}\)

Name	Symbol	SI units	Dimension
shear stress	\(\tau\)	\(\mathrm{Pa}\)	\(\text L^{-1}\,\text M\,\text T^{-2}\)
solid-fluid density difference	\(\rho_s - \rho_f\)	\(\mathrm{kg}\,\mathrm{m}^{-3}\)	\(\text L^{-3}\,\text M\)
gravitational acceleration	\(g\)	\(\mathrm{m}\,\mathrm{s}^{-2}\)	\(\text L\,\text T^{-2}\)
characteristic length	\(L\)	\(\mathrm{m}\)	\(\text L\)