TABLE 7-5 (from Çengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, McGraw-Hill, 2006) Some common established nondimensional parameters or Πs encountered in fluid mechanics and heat transfer; A is a characteristic area, D is a characteristic diameter, f is a characteristic frequency (Hz), L is a characteristic length, t is a characteristic time, T is a characteristic (absolute) temperature, V is a characteristic velocity, W is a characteristic width,

is a characteristic power, ω is a characteristic radial frequency or angular velocity (radians/s). Other parameters and fluid properties in these Πs include: c = speed of sound, cp, cv = specific heats, dp = particle diameter, DAB = species diffusion coefficient, h = convective heat transfer coefficient, hfg = latent heat of evaporation, k = thermal conductivity and also k = ratio of specific heats, P = pressure, Tsat = saturation temperature, = volume flow rate, α = thermal diffusivity, β = coefficient of thermal expansion, λ = mean free path length, μ = viscosity, ν = kinematic viscosity, ρ = fluid density, ρf = liquid density, ρp = particle density, ρs = solid density, ρv = vapor density, σs = surface tension, and τ = shear stress.

W

V

Name Definition Ratio of significance Archimedes number ( )

3

2Ar ss

gLρρ ρ

μ= − gravitational force

viscous force

aspect ratio AR or L LW D

= length length or width diameter

Biot number Bi hLk

= convective heat transferconduction heat transfer

Bond number ( ) 2

Bo f v

s

g Lρ ρ

σ

−= gravitational forces

surface tension forces

cavitation number ( )

2

2

Ca (sometimes )

2sometimes

vc

v

P PV

P PV

σρ

ρ

−=

⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠

pressure - vapor pressureinertial pressure

Darcy friction factor 2

8 wfV

=τρ

wall friction forcesinertial forces

drag coefficient 21

2

DD

FC

V Aρ= drag force

dynamic force

Eckert number 2

EcP

VC T

= kinetic energyenthalpy

Euler number 2 212

Eu sometimes P PV V

⎛ ⎞Δ Δ= ⎜ ⎟

⎝ ⎠ρ ρ pressure difference

inertial pressure

Fanning friction factor 2

2 wFf V

τρ

= wall friction forcesinertial forces

Fourier number 2Fo (sometimes ) t

Lατ = physical time

thermal diffusion time

Froude number

2

Fr sometimes V VgLgL

⎛ ⎞= ⎜ ⎟

⎝ ⎠ inertial forces

gravitational forces

Grashof number

3 2

2Grg T Lβ ρ

μΔ

= buoyancy forcesviscous forces

Jakob number ( )satJa P

fg

C T Th−

= sensible energylatent energy

Knudsen number Kn

Lλ

= mean free path lengthcharacteristic length

Name Definition Ratio of significance

Lewis number LeP AB AB

kC D D

αρ

= = thermal diffusionspecies diffusion

lift coefficient 212

LL

FC

V Aρ= lift force

dynamic force

Mach number Ma (sometimes M) Vc

= flow speedspeed of sound

Nusselt number Nu Lh

k= convection heat transfer

conduction heat transfer

Peclet number Pe pLVC LVk

ρα

= = bulk heat transferconduction heat transfer

power number P 5 3N WDρ ω

= powerrotational inertia

Prandtl number Pr PC

kμν

α= = viscous diffusion

thermal diffusion

pressure coefficient p 21

2

CP P

Vρ∞−

= static pressure differencedynamic pressure

Rayleigh number

3 2

Ra Pg T L Ck

β ρμ

Δ= buoyancy forces

viscous forces

Reynolds number

Re VL VL= =ρμ ν

inertial forcesviscous forces

Richardson number

5

2Ri L g ρ

ρΔ

=V

buoyancy forcesinertial forces

roughness ratio L

ε roughness heightcharacteristic length

Schmidt number

ScAB ABD D

= =μ ν

ρ viscous diffusion

species diffusion

Sherwood number

ShAB

VLD

= overall mass diffusionspecies diffusion

specific heat ratio

(sometimes ) Pck

cγ =

v

enthalpyinternal energy

Stanton number

Stp

hC Vρ

= heat transferthermal capacity

Stokes number 2

Stk (sometimes St)18

p pd VL

ρμ

= particle relaxation timecharacteristic flow time

Strouhal number St (sometimes S or Sr) fL

V= characteristic flow time

period of oscillation

Weber number 2

Wes

V Lρσ

= inertial forcessurface tension forces