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A dimensionless number is a quantity which describes a certain physical system and which is a pure number without any physical units In physics and metrology, units are standards for measurement of physical quantities that need clear definitions to be useful. Reproducibility of experimental results is central to the scientific method. To facilitate this we need standards, and to get convenient measures of the standards we need a system of units. Scientific systems of units are a formalization of the concept of weights and measures, initially developed for commercial purposes.
Such a number is typically defined as a product or ratio of quantities which do have units, in such a way that all units cancel.
For example: "one out of every 10 apples I gather is rotten." The rotten-to-gathered ratio is [1 apple] / [10 apples] = 0.1, which is a dimensionless quantity.
Dimensionless numbers are widely applied in the field of mechanical and chemical engineering. According to the Buckingham p-theorem The Buckingham p theorem is a key theorem in dimensional analysis.
The theorem states that the functional dependence between a certain number (e.g.: n) of variables can be reduced by the number (e.g. k) of independent dimensions occurring in those variables to give a set of p = n - k independent, dimensionless numbers. For the purposes of the experimenter, different systems which share the same description by dimensionless numbers are equivalent.
Of dimensional analysis
Dimensional analysis is a mathematical tool often applied in physics, chemistry, and engineering to simplify a problem by reducing the number of variables to the smallest number of "essential" parameters. Systems which share these parameters are called similar and do not have to be studied separately.
The dimension of a physical quantity is the type of unit needed to express it. For instance, the dimension of a speed is distance/time and the dimension of a force is mass×distance/time². In mechanics, every dimension can be expressed in terms of distance (which physicists often call "length"), time, and mass, or alternatively in terms of force, length and mass. Depending on the problem, it may be advantageous to choose one or the other set of, the functional dependence between a certain number (e.g.: n) of variables can be reduced by the number (e.g. k) of independent dimensions occurring in those variables to give a set of p = n - k independent, dimensionless numbers. For the purposes of the experimenter, different systems which share the same description by dimensionless numbers are equivalent.
An example
The power <
Electric power, often known as power or electricity, involves the production and delivery of electrical energy in sufficient quantities to operate domestic appliances, office equipment, industrial machinery and provide sufficient energy for both domestic and commercial lighting, heating, cooking and industrial processes.
History
Although electricity had been known
-consumption of a stirrer with a particular geometry is a function of the <density> Density (ISO 31: volumic mass) is a measure of mass per unit of volume. The higher an object's density, the higher its mass per volume. The average density of an object equals its total mass divided by its total volume. A denser object (such as iron) will have less volume than an equal mass of some less dense substance (such as water).
Again,
D = m ÷ V
where D equals density, m equals total mass, and V equals volume.

Viscosity is a property of a fluid that characterises its perceived "thickness" or resistance to pouring. It describes a fluid's internal resistance to flow and may be thought of as a measure of fluid friction. Thus, methanol is "thin", having a low viscosity, while vegetable oil is "thick" having a high viscosity.
Newton's theory
When a shear stress is applied to a solid
of the fluid to be stirred, the size of the stirrer given by its diameter, and the speed of the stirrer. Therefore, we have n=5 variables representing our example.
Those n=5 variables are built up from k=3 dimensions which are:
Length L [m]
Time T [s]
Mass M [kg]
According to the p-theorem The Buckingham p theorem is a key theorem in dimensional analysis.
The theorem states that the functional dependence between a certain number (e.g.: n) of variables can be reduced by the number (e.g. k) of independent dimensions occurring in those variables to give a set of p = n - k independent, dimensionless numbers. For the purposes of the experimenter, different systems which share the same description by dimensionless numbers are equivalent.
The n=5 variables can be reduced by the k=3 dimensions to form p=n-k=5-3=2 independent dimensionless numbers which are in case of the stirrer
Reynolds number The Reynolds number is the most important dimensionless number in fluid dynamics providing a criterion for dynamic similarity. It is named after Osbourne Reynolds (1842-1912). Typically it is given as follows:
Re = L\\over \\eta}
or
Re = L\\over u} \\; .
With:
vs - mean fluid velocity,
L - characteristic length (equal to diameter 2r if a cross-section is circular),
? - (absolute) dynamic fluid viscosity,
? - kinematic fluid viscosity: ? = ? / ?,
? - fluid density.

(This is the most important dimensionless number; it describes the fluid flow regime)
Power number The power number Np (also known as Newton number) is a dimensionless number relating the resistance force to the inertia force. In engineering, this number, along with the Reynolds number, is one of the most widely employed dimensionless numbers.
The power-number has different specifications according to the field of application. E.g., for stirrers the power number is defined as:
(Describes the stirrer and also involves the density of the fluid)
Listing of dimensionless numbers
There are literally thousands (to be precise: infinite) dimensionless numbers including those being used most often: (in alphabetical order, indicating their field of use)
Abbe number In physics and optics, the Abbe number, also known as the V-number or constringence of a transparent material is a measure of the material's dispersion (variation of refractive index with wavelength). Named for Ernst Abbe (1840-1905), German physicist.
The Abbe number V of a material is defined as:
V = \\frac
where nD, nF and nC are the refractive indices of the material at the wavelengths of the Fraunhofer D-, F- and C- spectral lines (589.2 nm, 486.1 nm and 656.3 nm respectively). Low dispersion materials have high values of V.
Dispersion in optical materials
Archimedes number An Archimedes number, to determine the motion of fluids due to density differences, is a number in the form Ar = \\frac
Where
g - gravity acceleration (9.81 m/s2)
?l - density of the fluid
? - density of the body
µ - fluid absolute viscosity

Motion of fluids due to density differences
Biot number The Biot number (Bi) is a dimensionless number used in unsteady-state and heat transfer calculations. It relates the heat transfer resistance inside and at the surface of a body.
It is defined as follows:
Bi = \\frac
Where:
h - overall heat transfer coefficient
L - charcteristic length
?b - Thermal conductivity of the body
Values of the Biot number larger than 1 imply that the heat conduction inside the body is slower than at its surface, and temperature gradients are non-negligible inside it.
Surface vs volume conductivity of solids
Bodenstein number: residence-time distribution
Capillary number: fluid flow influenced by surface tension
Damköhler numbers: reaction time scales vs transport phenomena
Deborah number The Deborah number is a dimensionless number which characterizes how "fluid" a material is. Even solids "flow" if they are observed long enough; the origin of the name is the line "The mountains flowed before the Lord" in a song by prophetess Deborah recorded in the Bible.
Formally, the Deborah number is defined as the ratio of the polymer characteristic relaxation time (lambda) and the
Rheology is the study of the deformation and flow of matter. The term rheology was coined by Eugene Bingham, a professor at Lehigh University, in 1920, from a suggestion by Markus Reiner, inspired by Heraclitus' famous expression panta rhei, "everything flows".
In practice, rheology is principally concerned with extending the relatively straightforward disciplines of elasticity and Newtonian fluid mechanics to more complicated and realistic materials.
of viscoelastic fluids
Drag coefficient The drag coefficient is a number that describes a characteristic amount of aerodynamic drag caused by fluid flow, used in the drag equation. Different objects with the same drag coefficient will behave in similar ways, after scaling for differences in size.
A cylinder is given a default drag coefficient of one. That means that two cylinders of the same size will have the same drag, one twice as large will have twice the drag. Less streamlined shapes will have higher values, while smoother shapes will have lower values.
Flow resistance
Ekman number The Ekman number, named for V. Walfrid Ekman, is a dimensionless number used in describing geophysical phenomena in the oceans and atmosphere. It characterises the ratio of viscous forces in a fluid to the fictitious forces arising from planetary rotation.
It is defined as:
Ek=\\frac
- where D is a characteristic (usually vertical) length scale of a phenomenon; ?, the kinematic eddy viscosity; O, the angular velocity of planetary rotation; and f, the latitude. The term 2 O sin f is the Coriolis acceleration.
Frictional
Viscosity is a property of a fluid that characterises its perceived "thickness" or resistance to pouring. It describes a fluid's internal resistance to flow and may be thought of as a measure of fluid friction. Thus, methanol is "thin", having a low viscosity, while vegetable oil is "thick" having a high viscosity.
Newton's theory
When a shear stress is applied to a solid
forces in Geophysics, the study of the earth by quantitative physical methods, especially by seismic reflection and refraction, gravity, magnetic, electrical, electromagnetic, and radioactivity methods.
It includes the branches of:
Seismology (earthquakes and elastic waves)
Gravity and geodesy (the earth's gravitational field and the size and form of the earth)
Atmospheric electricity and terrestrial magnetism (including ionosphere, Van Allen belts, telluric currents, etc.)
Geothermometry (heating of the earth, heat flow, volcanology, and hot springs)
Hydrology (ground and surface water, sometimes including glaciology)
Physical oceanography
Meteorology
Tectonophysics (geological processes in the earth)
Exploration and engineering geophysics


Euler number The Euler numbers are a sequence En of integers defined by the following Taylor series expansion:
\\frac = \\sum_ ^ \\frac \\cdot t^n
(Note that e, the base of the natural logarithm, is also occasionally called Euler's number, as is the Euler characteristic.)
The odd-indexed Euler numbers are all zero. The even-indexed ones (sequence A000364 in OEIS) have alternating signs. Some values are:
Hydrodynamics (pressure forces vs. inertia forces)
Friction factor The Darcy friction factor is a dimensionless number used in internal flow calculations. It expresses the linear relationship between mean flow velocity and pressure gradient.
It is defined as:
f = \\frac ) D_h}
where:
\\frac is the pressure drop per unit length
D_h is the hydraulic diameter
? is the fluid density

Fluid Flow
Froude number The Froude number is the reciprocal of the square root of the Richardson number.
It is usually used in the context of the Boussinesq approximation and is defined as
}
where u is a representative speed, g' the reduced gravity (see Boussinesq approximation), and h a representative vertical lengthscale. Strictly, this is known as the densimetric Froude number.
Wave and surface behaviour
Grashof number The Grashof number is a dimensionless number which approximates the ratio of buoyancy force to the viscous force acting on a fluid.
Gr = ( g ß (Ts - Tinf) L3 ) / ?2
g gravity ß volumetric thermal expansion coefficient Ts source temperature Tinf quiescent temperature L characteristic length ? kinematic viscosity
Free convection
The Knudsen number is the ratio of the molecular mean free path length to a representative physical length scale.
Continuum approximation in fluids
The Laplace number (La) is a dimensionless number used in the characterisation of free surface fluid dynamics. It is related to the ratio of the surface tension to the momentum-transport inside a fluid.
It is defined as follows:
La = \\frac
where:
s = surface tension
? = density
L = characteristic length
µ = absolute viscosity

Free convection with inmiscible fluids
Lift consists of the sum of all the aerodynamic forces normal to the direction of the external airflow.
Lift is created by forcing air downward. The pushing (accelerating) of the air downward creates an equal and opposing force upward on wing (see Newton's third law.) The displacement of air downward during the creation of lift is known as downwash. The diversion of the airflow downwards can be seen to create a higher pressure below the wing and a lower one above it. An aerofoil is so shaped to accomplish this as efficiently as possible. One puzzle is why the airflow "sticks" to the wing as it changes direction - this is known as the Coanda Effect, but the reason for it is not fully understood.
An airfoil (or aerofoil in British English) is a specially shaped cross-section of a wing or blade, used to provide lift or downforce, depending on its application. Airfoils have a characteristic shape which is that of a curved streamline, with a rounded leading edge and a sharp trailing edge.
For an understanding of the various ways of explaining lift, see lift. This force can be harnessed to lift an aircraft, or, in an inverted position, to hold a car or other vehicle to the ground. Airfoils are also found in propellors, fans, and turbines.

Angle of attack is a term used in aerodynamics to describe the angle between the wing's chord and the direction of the relative wind, effectively the direction in which the aircraft is currently moving. The amount of lift generated by a wing is directly related to the angle of attack, with greater angles generating more lift. This remains true up to the stall point, where lift starts to decrease again because of airflow separation. Planes flying at high angles of attack can suddenly enter a stall if, for example, a strong wind gust changes the direction of the relative wind, an effect that is seen primarily in low-speed aircraft.
If an object travels through a medium, then its Mach number is the ratio of the object's speed to the speed of sound in that medium. It is a dimensionless number, typically used to describe the speed of aircraft. Mach 1 is equal to the speed of sound, Mach 2 is twice that speed, etc. Since the speed of sound increases as the temperature increases, the actual speed of an object traveling at Mach 1 will depend on its altitude and the atmospheric conditions. High speed flight can be classified in six categories:
Subsonic M < 1 Sonic M = 1 Supersonic M > 1 Transsonic 0.8 < M < 1.3 Hypersonic 5 < M < 10 Hypervelocity M > 10 and above

The Nusselt number is a dimensionless number equal to the dimensionless temperature gradient at the surface in a convection situation. It therefore measures the heat transfer occurring at that surface.
Nu_L = \\frac
where
L = characteristic length
kf = thermal conductivity of the fluid
h = convection heat transfer coefficient


Ohnesorge number: Atomization of liquids
In physics, the Péclet number is a dimensionless number relating the forced convection of a system to its heat conduction. It is equivalent to the product of the Reynolds number with the Prandtl number.
There are various definitions of the Péclet number. The most typical are as follows:
Pe = l * v / a
Pe = l * v * ? * cp / ?
Pe = l2 * ? * cp / ? / t
..... Click the link for more information. : Forced convection
Power number: Power consumption by agitators
Prandtl number Forced and Free convection
Rayleigh number : Buoyancy and viscous forces in free convection
Reynolds number : Characterizing the flow behaviour
Richardson number : whether buoyancy is important
Rockwell scale : Mechanical
Rossby number : Inertial forces in
Sherwood number : Mass transfer with forced convection
Stokes number: Dynamics of particles
Strouhal number : Oscillatory flows
Weber number: Characterization of mulitphase flow with strongly bended surfaces
Weissenberg number : Viscoelastic flows
Dimensionless physical constants
The system of natural units chooses its base units in such a way as to make several physical constants such as the speed of light into simple dimensionless constants by definition. However, other dimensionless physical constants cannot be eliminated, and have to be discovered experimentally. These are often called fundamental physical constants These include:
the fine structure constant
the electromagnetic coupling constant
the strong coupling constant
the gravitational fine structure constant