Dimensionless Numbers

dimensionless numbers banner 3Dimensionless numbers are mathematical ratios that are used in physics and engineering to describe physical systems or phenomena.  They are typically used to compare physical quantities that have different units, by eliminating the effects of units and focusing on the underlying physical relationships.  These are nondimensional quantity of dimension one and can not be measured on a scale of physical units.  All quantities of dimension one are said to have the SI coherent derived unit "one" with the symbol "1".  It is a pure number, thus always having a dimension of 1.  The number does not change even if the number system you are working in does.  Dimensionless numbers or quantities are used in many disiplines such as chemistry, economics, engineering, mathematics, and physics.

These dimensionless numbers provide important insights into the behavior of physical systems, and they are widely used in engineering and physics to analyze and optimize the performance of various systems.


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Nomenclature & Symbols

Nomenclature & Symbols for Engineering, Mathematics, and Science

Dimensionless numbers Glossary


  • Abbe Number  -  The measure of material's dispersion (variation of refractive index versus wavelength) with high values of V indicating dispersion.  It is used to classify glass and other optically transparent materials.
  • Air–fuel Ratio  -  The mass ratio of air to fuel mixture present in an internal combustion engines.
  • Alfven Number  -  The steady flow past a fixed object of a conduction fluid or characteristic number for the relation between plasma speed and Alfven wave speed.
  • Archimedes Number  -  Analyzes flow as it relates to a system of density differences.  It is used when dealing with gravitational settling of particles in fluid.
  • Atomic Number  -  Equal to the number of protons in an atom's nucleus. The atomic number determines which element an atom is.
  • Atwood Number  -  Describes density difference between two adjacent fluids with a common interface.


  • Bagnold Number  -  The ratio of grain collision stresses to various fluid stresses in a granular flow with interstitial Newtonian fluid.
  • Brinkman Number  -  The heat conduction from a wall to a flowing viscous fluid, comminly used in polymer processes.
  • Brownell-Katz number  -  A combination of Capillary number and Bond number.


  • Cohesion Number  -  Used in partical tchnology by which the cohesivity of different powders can be compared.
  • Colburn J Factors  -  Heat transfer equation to calculate the natural convection moment of heat from vertical surfaces or horizontal cylinders of fluid (gasses or liquids) flowing past these surfaces.
  • Compound Machines Efficiency  -  A simple machine is a mechanical device that changes the direction or magnitude of a force.  A compound machine is machines connected in series.
  • Compression Ratio  -  The ratio of the maximum volume to the minimum volume in a cylinder.
  • Constrictivity  -  A parameter used to describe transportation processes in porous media.
  • Correction Factor  -  A factor multiplied with the result of an equation to correct for a known amount of systemic error.
  • Courant Number  -  The measure of how much information traverses a computational grid cell in a given time interval.


  • Determination Coefficient  -  The porportion of the variation in the dependent variable that is predicated from the independent variables.
  • Drag Coefficient  -  Used to quantify the amount of force or resistance an object experienced as it moves through the fluid stream.
  • Dukhin Number  -  The surface conductivity to various electrokinetic and electroacoustic effects, as well as the electrical conductivity and permittivity of fluid heterogeneous systems.


  • Efficiency  -  Expressed in percentage and always less than 100%.
  • Ekman Number  -  In fluid dynamics it describes the ratio of various forces to Coriolis forces.
  • Elasticity Number  -  The ratio of elastic force to internal force in viscoelastic flow.
  • Emissivity  -  A measure of a material's ability to emit thermal radiation.  It is defined as the ratio of the thermal radiation emitted by a material to the thermal radiation emitted by a blackbody at the same temperature.
  • Energy Efficiency Rating  -  Measure the efficiency with which a product uses energy to function.  It is calculated by dividing a product's BTU output by its wattage.

  • Eotvos Number  -  Measures the importance of gravitational forces compared to surface tension forces for the movement of liquid front.
  • Ericksen Number  -  The ratio of the viscous to elastic forces.

  • Euler Number  -  Used for analyzing flow where the differential pressure between two points is important.


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  • Factor of Safety  -  The ability of a system's structural capacity to be usable beyond it's expected or acrual loads.

  • Fan Efficiency  -  The ratio of input and output power.
  • Fresnel Number  -  Occuring in optics, in paticular in scalar diffraction theory.


  • Gas Compressibility Factor  -  A factor independent of the quantity of gas and determined by the character of the gas, the temperature, and pressure.
  • Glide Ratio  -  Heavier-than-air flight without the use of thrust.
  • Golden Ratio (Divine Porportion, Golden Mean, Golden Section)  -  A mathematical ratio commonly found in nature and design.
  • Gortler Number  -  The secondary flows that appear in the boundary layer flow along a concave wall.
  • Graetz Number  -  Characterizes laminar flow with heat transfer in a conduit.  This number is used to determine the thermal development on the entrance to ducts.
  • Grashof Number  -  The ratio of buoyant to viscous forces.

  • Gravimetric Factor  -  The number by which the weight of a gravimetric determination has to be multiplied in order to obtain the mass of the analyte it contains.


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  • Hagen Number  -  Used in forced flow calculations.  The equation is similar to the Grashof Number but for forced flow rather than natural convection.

  • Hartmann Number  -  The ratio of electromagnetic force to the viscous.

  • Hatta Number  -  Compares the rate of reaction in a liquid film to the rate of diffusion through the film.
  • Hazen-Williams Coefficient  -  Used in the Hazen-Williams Equation.  The lower the coefficient, the smoother the pipe is.  The higher the coefficient, the less fluid flow is restricted.
  • Head Friction Loss in Fittings and Valves  -  The pressure drop caused by fittings and valves.
  • Hydraulic Efficiency  -  Is the ratio of hydrodynamic energy in the form of fluid to the amount of mechanical energy delivered to the rotor.
  • Hydraulic Gradient  -  The change in height (pressure) to length between any two points.


  • Iribarren Number  -  Used to model several effects of surface gravity waves on beaches and coastal surfaces.



  • Karlovitz Number  -  The ratio of chemical time scale to Kolmogorov time scale.
  • Keulegan–Carpenter Number  -  Describes the relative importance of the drag forces over inertia forces for bluff objects in an oscillatory fluid flow.
  • Knudsen Number  -  Used in fluid dynamics to determine what type of analysis should be used to model an object moving through a fluid.  It is defined as the ratio of the molecular mean free path length to a representative physical length scale.
  • Kutateladze Number  -  Is the reciprocal of Jakok Number.  Note that there is a second Kutateladze Number that is related to electric arcs.


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  • Laplace Number  -  Used in the charcaterization of free surface fluid dynamics.  It is a ratio of surface tension to the momentum transport inside a fluid.

  • Lewis Number  -  The ratio of thermal diffusivity to mass diffusivity.  It is used to characterize fluid flows where there is simultaneous heat and mass transfer.

  • Lift Coefficient  -  Represents the lift generated by a lifting body to the fluid density around the body, the density of the fluid around the body, velocity of the fulid and an associated reference area.
  • Lockhart-Martinelli Parameter  -  Used in two-phase flow calculations.  It expresses the liquid fraction of a flowing fluid.
  • Love Number  -  Parameters that measure the rigidity of a planetary body and the susceptibility of its shape to change in responce to a tidal potential.
  • Lundquist Number  -  A ratio that compares the timescale of an Alfven wave crossing to the timescale of resistive diffusion.


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  • Mach Number  -  The ratio of the velocity of flow to the velocity of sound.  The speed of sound in this equation is dependent on the density of the medium that the sound is traveling through.

  • Machinability Index  -  Used to compare the machinability of different materials in the various cutting process.
  • Marangoni Number  -  Compares the rate of transport due to Marangoni flows, with a rate of transport of diffusion.
  • Margin of Safety  -  The factor of safety minus 1.
  • 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.
  • Mass Transfer Coefficient  -  A porportional constant to the difference in the concentrations and the rate of mass transfer.
  • Moisture Content  -  The quantity of water contained in a material.
  • Mole Fraction  -  The number of molecules (moles) of a solution in a mixture divided by the total number of moles present in the solution.



  • Octane Number  -  The antiknock properties of a liquid motor fuel with a higher number indicating a smaller likelihood of knocking.
  • Ohnesorge Number  -  Relates the viscous force to inertial forces and surface tension forces.

  • Overall Efficiency  -  Measures the ratio of the output of actual power to the input of actual power.  It determines the amount of energy lost overall.


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  • Péclet Number  -  Defined as a ratio of heat transport by convection to heat transport by conduction.

  • Pi  -  Archimedes' constant number that never ends and never repeats. The ratio of the circumference of a circle to the diameter.
  • Plastic Index  -  The range of water content over which the soil remains in the plastic state.
  • Plastic Limit  -  The water content at which the soil changes from semi-solid state to solid state.
  • Poisson's Ratio  -  The elastic ratio between lateral strain and longitudinal strain.
  • Porosity  -  The ratio between the pore volume of a substance and its total volume.  Porosity of a measure of a rocks capacity to store fluids.
  • Prandtl Number  -  In fluid dynamics is used to calculate force by the ratio of momentum diffusivity (kinematic viscosity) and thermal diffusivities.
  • Pump Efficiency  -  The ratio of total horsepower output to shaft horsepower input of the pump or power output to power input.


  • Quality Factor  -  The underdamped condition of an oscillator or resonator.



  • Safety Factor  -  See Factor of Safety
  • Schmidt Number  -  Used in analyzing fluid flows where there is an interface between two different fluids. 

  • Shape Factor  -  Used in structural it is the strength of the shape no matter the scale for a given load, bending, torsion, or twisting.

  • Shear Strain  -  Is opposing forces acting parrallel to the cross-section of a body.
  • Sherwood Number  -  The ratio between convective mass transfer and diffusive mass transfer.

  • Shrinkage Limit  -  The maximum water content at which further reduction in the water content will not cause decrease in volume of soil.
  • Soil Erodibility Factor  -  A quantitative description of soil detatchment by runoff and raindrop impact of a particular soil.
  • Sommerfield Number  -  Used extensively in hydrodynamic lubrication analysis.

  • Specific Gravity  -  The density or ratio of any substance to another substance.
  • Stokes Number  -  The behavior of particles suspended in a fluid flow.

  • Stuart Number  -  The ratio of electromagnetic to inertial forces.

  • Strain  -  A linear strain or longitudinal strain, is the deformation, stretched or compressed, of a material compared to its original length.
  • Strouhal Number  -  Used to describe oscillating flow.  It is used in fluid dynamics and is the ratio of inertial forces due to the unsteadiness of the flow or acceleration of inertial forces due to the changes in velocity between points in a flow field.
  • Support Practice Factor  -  Control particles that reduce the erosion potential of runoff by their influence on drainage paterns.


  • Tayor Number  -  The quantity that characterizes the importance of centrifugal forces due to the rotation of a fluid about an axis, relative to viscous forces.

  • Thermal Efficiency  -  The fraction of heat that is converted to work or desired output divided by required input.
  • Tortuosity  -  The dispersion of fluid flow in porous media.
  • Turbulent Prandtl Number  -  The ratio between the momentum eddy diffusivity and the heat transfer eddy diffusivity.


  • Uniformity Coefficient  -  Classifies a soil as well graded or poorly graded.
  • Ursell Number  -  Indicates the nonlinearity of long surface gravity waves on a fluid layer.


  • Void Ratio  -  The ratio of the volume of voids to the volume of solids.
  • Volumetric Efficiency  -  The more air inside the combustion chamber, the more fuel that can be burned and the higher the output engine torque and power.
  • Von Karman Constant  -  The logarithmic law describing the distribution of the longitudinal velocity in the wall-normal direction of a turbulent fluid flow near a boundary with a no-slip condition.


  • Water Content  -  The ratio of weight of soil to the weight of water or the quantity of water contained in a material.
  • Wahl Correction Factor  -  A method to find out the effect of direct sheat and change in coil curvature in a spring.
  • Weber Number  -  Used in fluid mechanics is often useful in analysing fluid flows where there is an interface between two different fluids, especially for multiphase flows with strongly curved surfaces.
  • Weissenberg Number  -  Compares the elastic forces to the viscous forces.

  • Womersley Number  -  The pulsatile flow frequency in relation to viscous effects.





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Abbe Number
Aeration Number
Archimedes Number
Arrhenius Number
Atomic Number

Tags: Glossaries