Nomenclature
 Archimedes Number:
Ar =
Re2
gL3ρ(ρs − ρ)
=
Fr
μ2
 Atwood Number:
(ρ1 − ρ2)
(ρ1 + ρ2)
Note: Used in study of density stratified flows.
A =
 Biot Number:
hL
conductive resistance in solid
Bi =
=
Ks
convective resistance in thermal boundary layer
 Bond Number:
We
ρgL2
=
Fr
σ
Bo =
 Brinkman Number:
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.
Br =
 Capillary Number:
Ca =
We
Uμ
=
Re
σ
 Cauchy Number:
Ca = M2 =
ρU2
K
 Centrifuge Number:
Ce =
We
ρΩ2L3
=
Ro2
σ
 Dean Number:
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.
De =
 Deborah Number:
τ
relaxation time
=
tp
characteristic time scale
Note: Commonly used in rheology to characterize how "fluid" a material
De =
is. The smaller the De, the more the fluid the material appears.
 Eckert Number:
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.
Ec =
 Ekman Number:
Ek =
μ
viscous force
2 =
ρσL
Coriolis force
 Eötvös Number:
Eo =
We
ΔρgL2
=
Fr
σ
 Euler Number:
Eu =
Δp
pressure force
2 =
ρU
inertial force
 Fourier Number:
αt
rate of heat conduction
=
L2
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.
Fo =
 Froude Number:
Fr =
U2
inertial force
=
gL
gravitational force
 Galileo Number:
Ga =
Re2
ρ2gL3
=
Fr
μ2
Gz =
di Pe
Udi
=
L
ν
 Graetz Number:
 Grashof Number:
Gr =
gβ(Thot − Tref)L3
buoyancy force
=
2
ν
viscous force
 Hagen Number:
dp ρL3
dx μ2
Note: It is the forced flow equivalent of Grashof number.
Hg = −
 Jakob Number:
cp(Tw − Tsat)
hfg
Note: Jakob number represents the ratio of sensible heat to latent heat
absorbed (or released) during the phase change process.
Ja =
 Knudsen Number:
Kn =
λ
length of mean free path
=
L
characteristic dimension
 Laplace Number:
La =
Re2
ρσL
= 2
We
μ
 Lewis Number:
Le =
α
thermal diffusivity
=
DAB
mass diffusivity
 Mach Number:
M =
U
inertial force
=
a
elastic (compressibility) force
 Marangoni Number:
dσ LΔT
dT μα
Note: Marangoni number is the ratio of thermal surface tension force to
the viscous force.
Ma = −
 Morton Number:
Mo =
We3
gμ4
=
Fr Re4
Δρσ3
 Nusselt Number:
hL
Kf
Note: Nusselt number represents the dimensionless temperature gradient at
the solid surface.
Nu =
 Ohnesorge Number:
Oh =
We1/2
μ
=
Re
(ρσL)1/2
 Peclet Number:
Pe =
UL
inertia (convection)
=
α
Diffusion
 Prandtl Number:
Pr =
ν
momentum diffusivity
=
α
thermal diffusivity
 Rayleigh Number:
gβ(Thot − Tref)L3
buoyancy
Ra = Gr Pr =
=
να
viscous × rate of heat diffusion
 Reynolds Number:
Re =
ρUL
inertial force
=
viscous force
μ
 Richardson Number:
Gr
gβ(Thot − Tref)L
buoyancy force
Ri =
=
2 =
2
Re
U
inertial force
 Rossby Number:
Ro =
U
inertial force
=
ΩL
Coriolis force
 Rotating Froude Number:
FrR =
Fr
Ω2L
=
Ro2
g
 Schmidt Number:
Sc = Le Pr =
ν
momentum diffusivity
=
DAB
mass diffusivity
 Sherwood Number:
hmL
DAB
Note: Sherwood number represents the dimensionless concentration
gradient at the solid surface.
Sh =
 Stanton Number:
Nu
h
=
Re Pr
ρUcp
Note: Stanton number is the modified Nusselt number. It is used in
analogy between heat transfer and viscous transport in boundary layers.
St =
 Stefan Number:
cpdT
specific heat
=
Lm
latent heat
Note: Stefan number is useful in the study of heat transfer during phase
change.
St =
 Stokes Number:
τUo
stopping distance of a particle
Stk =
=
dc
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):
L
inertia (local)
St =
=
Utref
inertia (convection)
Note: If tref is taken as the reciprocal of the circular frequency ω of the
system, then
ωL
St =
U
 Taylor Number:
ρ2Ωi2L4
Ta =
μ2
3 1/4
where L = [ri(ro − ri) ]
 Weber Number:
We =
ρU2L
inertial force
=
σ
surface tension force
 Womersley Number:
(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.
α = (π Re St)1/2 = L
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