Dimensionless Numbers in Fluid Mechanics | Definition & List

By | January 25, 2019

In this article we will understand concept of ‘Dimensionless Numbers’. After understanding it we will discuss some important dimensionless (Non-Dimensional) numbers related to fluid mechanics field.

  1. What is a dimensionless number?

As the name indicates dimensionless numbers are not associated with any dimensions, like m, kg, sec etc.

Non-dimensional numbers are the ratios of two numbers which have same dimensions. Hence dimensions get cancelled.

For example:

If we take ratio of pressure to stress then the number obtained will be dimensionless. Because both pressure and stress have same dimensions i.e. N/m2.

Let us now understand some very important dimensionless numbers related to fluid mechanics.

  1. Five important dimensionless numbers in fluid mechanics
  • Mach’s number (M)
  • Weber’s number (We)
  • Euler’s number (Eu)
  • Froude’s number (Fe)
  • Reynold’s number (Re)

2.1. What is Mach’s number (M)?

Mach’s number is defined as square root of ratio of inertia force to elastic force of moving fluid.

M = (Inertia force/Elastic force)1/2

After putting values and solving above equation we will get.

M = V/C

Where

V = Velocity of fluid

C = Velocity of sound in that fluid

Alternatively, many people define Mach’s number as a ratio of velocity of object to velocity of sound.

2.2. What is Weber’s Number (We)?

Weber’s number is defined as square root of ratio of inertia force to surface tension force of moving fluid.

We = (Inertia force/Surface tension force)1/2

After putting values and solving above equation we will get.

We = V/(σ/ρ*L)1/2

Where

V = Velocity of fluid

σ = Coefficient of surface tension

ρ = Density of fluid

2.3. What is Euler’s Number (Eu)?

Euler’s number is defined as square root of ratio of inertia force to pressure force of moving fluid.

Eu = (Inertia force/Pressure force)1/2

After putting values and solving above equation we will get.

Eu = V/(P/ρ)1/2

Where

V = Velocity of fluid

ρ = Density of fluid

P = Pressure of flowing fluid

2.4. What is Froude’s Number (Fe)?

Froude’s number is defined as square root of ratio of inertia force to gravity force of moving fluid.

Fe = (Inertia force/gravity force)1/2

After putting values and solving above equation we will get.

Fe = V/(L*g)1/2

Where

V = Velocity of fluid

g = Gravitational acceleration

2.5. What is Reynold’s Number (Re)?

Reynold’s number is defined as ratio of inertia force to viscous force of moving fluid.

Re = Inertia force/viscous force

After putting values and solving above equation we will get.

Re = V*L/ʋ

Where

V = Velocity of fluid

ʋ = Kinematic viscosity

  1. List of other important dimensionless numbers in fluid mechanics

  • Zel’dovich number
  • Womersley number
  • Weissenberg number
  • Weaver flame speed number
  • Wallis parameter
  • Ursell number
  • Taylor number
  • Stuart number
  • Strouhal number
  • Stokes number
  • Stanton number
  • Sommerfeld number
  • Sherwood number
  • Shape factor
  • Schmidt number
  • Roshko number
  • Richardson number
  • Rayleigh number
  • Pressure coefficient
  • Prandtl number
  • Péclet number
  • Ohnesorge number
  • Nusselt number
  • Morton number
  • Markstein number
  • Marangoni number
  • Manning roughness coefficient
  • Lockhart–Martinelli parameter
  • Lift coefficient
  • Lewis number
  • Laplace number
  • Kutateladze number
  • Knudsen number
  • Keulegan–Carpenter number
  • Karlovitz number
  • Iribarren number
  • Hagen number
  • Hartmann number
  • Grashof number
  • Graetz number
  • Görtler number
  • Galilei number
  • Fanning friction factor
  • Excess temperature coefficient
  • Ericksen number
  • Eötvös number
  • Eckert number
  • Drag coefficient
  • Deborah number
  • Dean number
  • Darcy friction factor
  • Damkohler number
  • Colburn J factors
  • Chandrasekhar number
  • Capillary number
  • Brownell–Katz number
  • Brinkman number
  • Bond number
  • Blake number
  • Biot number
  • Bingham number
  • Bejan number
  • Atwood number
  • Archimedes number

Featured image attributions: By US Navy – http://www.hnn.navy.mil/Archives/030314/images_031403/originals/Los%20Angeles.JPG (from page http://www.hnn.navy.mil/Archives/030314/losangeles_031403.htm), Public Domain, https://commons.wikimedia.org/w/index.php?curid=647962

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