Dimensionless numbers in fluid mechanics are a set of
dimensionless quantities
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
that have an important role in analyzing the behavior of
fluids. Common examples include the
Reynolds or the
Mach numbers, which describe as ratios the relative magnitude of fluid and physical system characteristics, such as
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
,
viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
,
speed of sound,
flow speed, etc.
Diffusive numbers in transport phenomena
As a general example of how dimensionless numbers arise in fluid mechanics, the classical numbers in
transport phenomena
In engineering, physics, and chemistry, the study of transport phenomena concerns the exchange of mass, energy, charge, momentum and angular momentum between observed and studied systems. While it draws from fields as diverse as continuum mecha ...
of
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
,
momentum, and
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
are principally analyzed by the ratio of effective
diffusivities in each transport mechanism. The six
dimensionless numbers
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
give the relative strengths of the different phenomena of
inertia
Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
,
viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
,
conductive heat transport, and diffusive
mass transport
Public transport (also known as public transportation, public transit, mass transit, or simply transit) is a system of transport for passengers by group travel systems available for use by the general public unlike private transport, typica ...
. (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
Droplet formation mostly depends on momentum, viscosity and surface tension.
In
inkjet printing
Inkjet printing is a type of computer printing that recreates a digital image by propelling droplets of ink onto paper and plastic substrates. Inkjet printers were the most commonly used type of printer in 2008, and range from small inexpen ...
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
Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a unit ...
. See other article for extensive
list of dimensionless quantities This is a list of well-known dimensionless quantities illustrating their variety of forms and applications. The tables also include pure numbers, dimensionless ratios, or dimensionless physical constants; these topics are discussed in the article.
...
. Certain
dimensionless quantities
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
of some importance to
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
are given below:
, ,
gas dynamics
Compressible flow (or gas dynamics) is the branch of fluid mechanics that deals with flows having significant changes in fluid density. While all flows are compressible, flows are usually treated as being incompressible when the Mach number (the ...
(
compressible flow
Compressible flow (or gas dynamics) is the branch of fluid mechanics that deals with flows having significant changes in fluid density. While all flows are compressible, flows are usually treated as being incompressible when the Mach number (the r ...
; dimensionless
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
)
, -
,
Manning roughness coefficient , , ''n'' , , , ,
open channel flow
In fluid mechanics and hydraulics, open-channel flow is a type of liquid flow within a conduit with a free surface, known as a channel. The other type of flow within a conduit is pipe flow. These two types of flow are similar in many ways but di ...
(flow driven by
gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
)
, -
,
Marangoni number , , Mg , ,
, ,
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
(
Marangoni flow; thermal
surface tension forces over
viscous
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
forces)
, -
,
Markstein number In combustion engineering and explosion studies, the Markstein number characterizes the effect of local heat release of a propagating flame on variations in the surface topology along the flame and the associated local flame front curvature. The di ...
, , Ma , ,
, ,
turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
,
combustion
Combustion, or burning, is a high-temperature exothermic redox chemical reaction between a fuel (the reductant) and an oxidant, usually atmospheric oxygen, that produces oxidized, often gaseous products, in a mixture termed as smoke. Combus ...
(Markstein length to laminar flame thickness)
, -
,
Morton number :
In fluid dynamics, the Morton number (Mo) is a dimensionless number used together with the Eötvös number or Bond number to characterize the shape of bubbles or drops moving in a surrounding fluid or continuous phase, ''c''.
It is named after ...
, , Mo , ,
, ,
fluid dynamics (determination of
bubble
Bubble, Bubbles or The Bubble may refer to:
Common uses
* Bubble (physics), a globule of one substance in another, usually gas in a liquid
** Soap bubble
* Economic bubble, a situation where asset prices are much higher than underlying funda ...
/
drop shape)
, -
,
Nusselt number
In thermal fluid dynamics, the Nusselt number (, after Wilhelm Nusselt) is the ratio of convective to conductive heat transfer at a boundary in a fluid. Convection includes both advection (fluid motion) and diffusion (conduction). The conductiv ...
, , Nu , ,
, ,
heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
(forced
convection
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the conve ...
; ratio of
convective
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the convec ...
to
conductive
In physics and electrical engineering, a conductor is an object or type of material that allows the flow of charge (electric current) in one or more directions. Materials made of metal are common electrical conductors. Electric current is gene ...
heat transfer)
, -
,
Ohnesorge number , , Oh , ,
, ,
fluid dynamics (atomization of liquids,
Marangoni flow)
, -
,
Péclet number
In continuum mechanics, the Péclet number (, after Jean Claude Eugène Péclet) is a class of dimensionless numbers relevant in the study of transport phenomena in a continuum. It is defined to be the ratio of the rate of advection of a physical ...
, , Pe , ,
or
, ,
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
(ratio of advective transport rate over molecular diffusive transport rate),
heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
(ratio of advective transport rate over thermal diffusive transport rate)
, -
,
Prandtl number , , Pr , ,
, ,
heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
(ratio of
viscous diffusion rate over
thermal diffusion rate)
, -
,
Pressure coefficient The pressure coefficient is a dimensionless number which describes the relative pressures throughout a flow field in fluid dynamics. The pressure coefficient is used in aerodynamics and hydrodynamics. Every point in a fluid flow field has its own ...
, , ''C
P'' , ,
, ,
aerodynamics
Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dy ...
,
hydrodynamics (
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
experienced at a point on an
airfoil; dimensionless pressure variable)
, -
,
Rayleigh number
In fluid mechanics, the Rayleigh number (, after Lord Rayleigh) for a fluid is a dimensionless number associated with buoyancy-driven flow, also known as free (or natural) convection. It characterises the fluid's flow regime: a value in a certai ...
, , Ra , ,
, ,
heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
(
buoyancy
Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the ...
versus
viscous forces
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inter ...
in
free convection
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the convec ...
)
, -
,
Reynolds number , , Re , ,
, ,
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
(ratio of fluid
inertia
Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
l and
viscous
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
forces)
, -
,
Richardson number The Richardson number (Ri) is named after Lewis Fry Richardson (1881–1953). It is the dimensionless number that expresses the ratio of the buoyancy term to the flow shear term:
:
\mathrm = \frac = \frac \frac
where g is gravity, \rho is de ...
, , Ri , ,
, ,
fluid dynamics (effect of
buoyancy
Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the ...
on flow stability; ratio of
potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
over
kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...
)
, -
,
Roshko number , , Ro , ,
, ,
fluid dynamics (oscillating flow,
vortex
In fluid dynamics, a vortex ( : vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in ...
shedding)
, -
,
Schmidt number
Schmidt number (Sc) is a dimensionless number defined as the ratio of momentum diffusivity ( kinematic viscosity) and mass diffusivity, and it is used to characterize fluid flows in which there are simultaneous momentum and mass diffusion convec ...
, , Sc , ,
, ,
mass transfer
Mass transfer is the net movement of mass from one location (usually meaning stream, phase, fraction or component) to another. Mass transfer occurs in many processes, such as absorption, evaporation, drying, precipitation, membrane filtration ...
(
viscous
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
over molecular
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemica ...
rate)
, -
,
Shape factor , , ''H'' , ,
, ,
boundary layer flow
In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary cond ...
(ratio of displacement thickness to momentum thickness)
, -
,
Sherwood number
The Sherwood number (Sh) (also called the mass transfer Nusselt number) is a dimensionless number used in mass-transfer operation. It represents the ratio of the convective mass transfer to the rate of diffusive mass transport, and is named in h ...
, , Sh , ,
, ,
mass transfer
Mass transfer is the net movement of mass from one location (usually meaning stream, phase, fraction or component) to another. Mass transfer occurs in many processes, such as absorption, evaporation, drying, precipitation, membrane filtration ...
(
forced convection
Forced convection is a mechanism, or type of transport, in which fluid motion is generated by an external source (like a pump, fan, suction device, etc.). Alongside natural convection, thermal radiation, and thermal conduction it is one of the m ...
; ratio of
convective
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the convec ...
to
diffusive mass transport)
, -
,
Sommerfeld number , , S , ,
, ,
hydrodynamic lubrication (boundary
lubrication)
, -
,
Stanton number , , St , ,
, ,
heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
and
fluid dynamics (forced
convection
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the conve ...
)
, -
,
Stokes number , , Stk or S
k , ,
, ,
particles suspensions (ratio of characteristic
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, ...
of particle to time of flow)
, -
,
Strouhal number , , St , ,
, ,
Vortex shedding
In fluid dynamics, vortex shedding is an oscillating flow that takes place when a fluid such as air or water flows past a bluff (as opposed to streamlined) body at certain velocities, depending on the size and shape of the body. In this flow, v ...
(ratio of characteristic oscillatory velocity to ambient flow velocity)
, -
,
Stuart number , , N , ,
, ,
magnetohydrodynamics
Magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydromagnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magnetofluids include plasmas, liquid metals, ...
(ratio of
electromagnetic
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
to inertial forces)
, -
,
Taylor number
In fluid dynamics, the Taylor number (Ta) is a dimensionless quantity that characterizes the importance of centrifugal "forces" or so-called inertial forces due to rotation of a fluid about an axis, relative to viscous forces.
In 1923 Geoffre ...
, , Ta , ,
, ,
fluid dynamics (rotating fluid flows; inertial forces due to
rotation of a
fluid versus
viscous forces
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inter ...
)
, -
,
Ursell number , , U , ,
, ,
wave
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
mechanics (nonlinearity of
surface gravity waves on a shallow fluid layer)
, -
,
Wallis parameter , , ''j'' , ,
, ,
multiphase flow
In fluid mechanics, multiphase flow is the simultaneous flow of materials with two or more thermodynamic phases. Virtually all processing technologies from cavitating pumps and turbines to paper-making and the construction of plastics involve so ...
s (nondimensional
superficial velocity Superficial velocity (or superficial flow velocity), in engineering of multiphase flows and flows in porous media, is a hypothetical (artificial) flow velocity calculated as if the given phase or fluid were the only one flowing or present in a give ...
)
, -
,
Weaver flame speed number , , Wea , ,
, ,
combustion
Combustion, or burning, is a high-temperature exothermic redox chemical reaction between a fuel (the reductant) and an oxidant, usually atmospheric oxygen, that produces oxidized, often gaseous products, in a mixture termed as smoke. Combus ...
(
laminar
Laminar means "flat". Laminar may refer to:
Terms in science and engineering:
* Laminar electronics or organic electronics, a branch of material sciences dealing with electrically conductive polymers and small molecules
* Laminar armour or "band ...
burning
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
relative to
hydrogen
Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic ...
gas)
, -
,
Weber number
The Weber number (We) is a dimensionless number in fluid mechanics that is often useful in analysing fluid flows where there is an interface between two different fluids, especially for multiphase flows with strongly curved surfaces. It is named ...
, , We , ,
, ,
multiphase flow
In fluid mechanics, multiphase flow is the simultaneous flow of materials with two or more thermodynamic phases. Virtually all processing technologies from cavitating pumps and turbines to paper-making and the construction of plastics involve so ...
(strongly curved surfaces; ratio of
inertia
Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
to
surface tension)
, -
,
Weissenberg number , , Wi , ,
, ,
viscoelastic
In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly ...
flows (
shear rate
In physics, shear rate is the rate at which a progressive shearing deformation is applied to some material.
Simple shear
The shear rate for a fluid flowing between two parallel plates, one moving at a constant speed and the other one stationary ...
times the relaxation time)
, -
,
Womersley number , ,
, ,
, ,
biofluid mechanics (continuous and pulsating flows; ratio of
pulsatile flow frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
to
viscous effects)
, -
,
Zel'dovich number , ,
, ,
, ,
fluid dynamics,
Combustion
Combustion, or burning, is a high-temperature exothermic redox chemical reaction between a fuel (the reductant) and an oxidant, usually atmospheric oxygen, that produces oxidized, often gaseous products, in a mixture termed as smoke. Combus ...
(Measure of
activation energy
In chemistry and physics, activation energy is the minimum amount of energy that must be provided for compounds to result in a chemical reaction. The activation energy (''E''a) of a reaction is measured in joules per mole (J/mol), kilojoules p ...
)
References
*
{{Dimensionless numbers in fluid mechanics
Dimensionless numbers of fluid mechanics
Fluid dynamics