# Dimensionless Numbers

A. Salih

Dept. of Aerospace Engineering
IIST, Thiruvananthapuram

The nondimensionalization of the governing equations of fluid flow is important for both theoretical and computational reasons. Nondimensional scaling provides a method for developing dimensionless groups that can provide physical insight into the importance of various terms in the system of governing equations. Computationally, dimensionless forms have the added benefit of providing numerical scaling of the system discrete equations, thus providing a physically linked technique for improving the ill-conditioning of the system of equations. Moreover, dimensionless forms also allow us to present the solution in a compact way. Some of the important dimensionless numbers used in fluid mechanics and heat transfer are given below.

• Archimedes Number:
•  Ar  = Re2 Fr
 = gL3ρ(ρs − ρ) μ2
• Atwood Number:
•  A  = (ρ1 − ρ2) (ρ1 + ρ2)
Note: Used in the study of density stratified flows.

• Biot Number:
•  Bi  = hL Ks
 = conductive resistance in solid convective resistance in thermal boundary layer

• Bond Number:
•  Bo  = We Fr
 = ρgL2 σ

• Brinkman Number:
•  Br  = μU2 K(Tw − To)
Note: Brinkman number is related to heat conduction from a wall to a flowing viscous fluid. It is commonly used in polymer processing.

• Capillary Number:
•  Ca  = We Re
 = Uμ σ

• Cauchy Number:
•  Ca  =  M2 = ρU2 K

• Centrifuge Number:
•  Ce  = We Ro2
 = ρΩ2L3 σ

• Dean Number:
•  De  = Re (R ⁄ h)1/2
Note: Dean number deals with the stability of two-dimensional flows in a curved channel with mean radius R and width 2h.

• Deborah Number:
•  De  = τ tp
 = relaxation time characteristic time scale
Note: Commonly used in rheology to characterize how "fluid" a material is. The smaller the De, the more the fluid the material appears.

• Eckert Number:
•  Ec  = U2 cpΔT
Note: Eckert number represents the kinetic energy of the flow relative to the boundary layer enthalpy difference. Ec plays an important role in high speed flows for which viscous dissipation is significant.

• Ekman Number:
•  Ek  = μ ρσL2
 = viscous force Coriolis force
• E�tv�s Number:
•  Eo  = We Fr
 = ΔρgL2 σ

• Euler Number:
•  Eu  = Δp ρU2
 = pressure force inertial force

• Fourier Number:
•  Fo  = αt L2
 = rate of heat conduction rate of thermal energy stored
Note: Fourier number represents the dimensionless time. It may be interpreted as the ratio of current time to time to reach steady-state.

• Froude Number:
•  Fr  = U2 gL
 = inertial force gravitational force

• Galileo Number:
•  Ga  = Re2 Fr
 = ρ2gL3 μ2

• Graetz Number:
•  Gz  = di Pe L
 = Udi ν

• Grashof Number:
•  Gr  = gβ(Thot − Tref)L3 ν2
 = buoyancy force viscous force

• Hagen Number:
•  Hg  = − dp dx
 ρL3 μ2
Note: It is the forced flow equivalent of Grashof number.

• Jakob Number:
•  Ja  = cp(Tw − Tsat) hfg
Note: Jakob number represents the ratio of sensible heat to latent heat absorbed (or released) during the phase change process.

• Knudsen Number:
•  Kn  = λ L
 = length of mean free path characteristic dimension

• Laplace Number:
•  La  = Re2 We
 = ρσL μ2

• Lewis Number:
•  Le  = α DAB
 = thermal diffusivity mass diffusivity

• Mach Number:
•  M  = U a
 = inertial force elastic (compressibility) force

• Marangoni Number:
•  Ma  = − dσ dT
 LΔT μα
Note: Marangoni number is the ratio of thermal surface tension force to the viscous force.

• Morton Number:
•  Mo  = We3 Fr Re4
 = gμ4 Δρσ3

• Nusselt Number:
•  Nu  = hL Kf
Note: Nusselt number represents the dimensionless temperature gradient at the solid surface.

• Ohnesorge Number:
•  Oh  = We1/2 Re
 = μ (ρσL)1/2

• Peclet Number:
•  Pe  = UL α
 = inertia (convection) Diffusion

• Prandtl Number:
•  Pr  = ν α
 = momentum diffusivity thermal diffusivity

• Rayleigh Number:
•  Ra  = Gr Pr = gβ(Thot − Tref)L3 να
 = buoyancy viscous × rate of heat diffusion

• Reynolds Number:
•  Re  = ρUL μ
 = inertial force viscous force

• Richardson Number:
•  Ri  = Gr Re2
 = gβ(Thot − Tref)L U2
 = buoyancy force inertial force

• Rossby Number:
•  Ro  = U ΩL
 = inertial force Coriolis force

• Rotating Froude Number:
•  FrR  = Fr Ro2
 = Ω2L g

• Schmidt Number:
•  Sc  =   Le Pr   = ν DAB
 = momentum diffusivity mass diffusivity

• Sherwood Number:
•  Sh  = hmL DAB
Note: Sherwood number represents the dimensionless concentration gradient at the solid surface.

• Stanton Number:
•  St  = Nu Re Pr
 = h ρUcp
Note: Stanton number is the modified Nusselt number. It is used in analogy between heat transfer and viscous transport in boundary layers.

• Stefan Number:
•  St  = cpdT Lm
 = specific heat latent heat
Note: Stefan number is useful in the study of heat transfer during phase change.

• Stokes Number:
•  Stk  = τUo dc
 = stopping distance of a particle characteristic dimension of the obstacle
Note: Commonly used in particles suspended in fluid.
For Stk << 1, the particle negotiates the obstacle.
For Stk >> 1, the particle travels in straightline and eventually collides with obstacle.

• Strouhal Number (for oscillatory flow):
•  St  = L Utref
 = inertia (local) inertia (convection)
Note: If tref is taken as the reciprocal of the circular frequency ω of the system, then
 St  = ωL U
• Taylor Number:
•  Ta  = ρ2Ωi2L4 μ2
where L = [ri(rori)3]1/4

• Weber Number:
•  We  = ρU2L σ
 = inertial force surface tension force

• Womersley Number:
•  α  =   (π Re St)1/2 =   L (ρω)1/2 μ1/2
Note: Womersley number is used in biofluid mechanics. It is a dimensionless expression of the pulsatile flow frequency in relation to the viscous effects.

Nomenclature:
 a → speed of sound cp → specific heat at constant pressure DAB → mass diffusivity coefficient dT → temperature difference between phases dc → characteristic dimension of the obstacle di → hydraulic diameter of the duct g → gravitational acceleration h → heat transfer coefficient h → width of the channel hfg → latent heat of condensation hm → mass transfer coefficient K → bulk modulus of elasticity K → thermal conductivity of fluid Kf → thermal conductivity of fluid Ks → thermal conductivity of solid L → characteristic length scale Lm → latent heat of melting R → radius of the channel ri → radius of the inner cylinder ro → radius of the outer cylinder Thot → temperature of the hot wall Tref → reference temperature To → bulk fluid temperature Tsat → saturation temperature Tw → wall temperature T∞ → quiescent temperature of the fluid t → time tref → reference time tp → characteristic time scale U → characteristic velocity scale Uo → fluid velocity far away from the object dp/dx → pressure gradient dσ/dT → rate of change of surface tension with temperature α → thermal diffusivity of fluid β → volumetric thermal expansion coefficient Δp → characteristic pressure difference of flow ΔT → characteristic temperature difference Δρ → difference in density of the two phases λ → length of mean free path μ → viscosity of fluid ν → kinematic viscosity of fluid ρ → density of fluid ρ1 → density of heavier fluid ρ2 → density of lighter fluid ρl → density of solid σ → surface tension τ → relaxation time Ω → angular velocity ω → circular frequency ωi → angular velocity of inner cylinder