Name
Standard symbol
Definition
Field of application
Archimedes number Ar
A
r
=
g
L
3
ρ
ℓ
(
ρ
−
ρ
ℓ
)
μ
2
{\displaystyle \mathrm {Ar} ={\frac {gL^{3}\rho _{\ell }(\rho -\rho _{\ell })}{\mu ^{2}}}}
fluid mechanics (motion of fluids due to density differences)
Asakuma number As
A
s
=
W
α
ρ
d
p
H
{\displaystyle \mathrm {As} ={\frac {W}{\alpha \rho d_{p}H}}}
heat transfer (ratio of heat generation of microwave dielectric heating to thermal diffusion[ 6]
Atwood number A
A
=
ρ
1
−
ρ
2
ρ
1
+
ρ
2
{\displaystyle \mathrm {A} ={\frac {\rho _{1}-\rho _{2}}{\rho _{1}+\rho _{2}}}}
fluid mechanics (onset of instabilities in fluid mixtures due to density differences)
Bagnold number Ba
B
a
=
ρ
d
2
λ
1
/
2
γ
˙
μ
{\displaystyle \mathrm {Ba} ={\frac {\rho d^{2}\lambda ^{1/2}{\dot {\gamma }}}{\mu }}}
fluid mechanics , geology (ratio of grain collision stresses to viscous fluid stresses in flow of a granular material such as grain and sand )[ 7]
Bejan number (fluid mechanics ) Be
B
e
=
Δ
P
L
2
μ
α
{\displaystyle \mathrm {Be} ={\frac {\Delta PL^{2}}{\mu \alpha }}}
fluid mechanics (dimensionless pressure drop along a channel )[ 8]
Bejan number (thermodynamics ) Be
B
e
=
S
˙
g
e
n
,
Δ
T
′
S
˙
g
e
n
,
Δ
T
′
+
S
˙
g
e
n
,
Δ
p
′
{\displaystyle \mathrm {Be} ={\frac {{\dot {S}}'_{\mathrm {gen} ,\,\Delta T}}{{\dot {S}}'_{\mathrm {gen} ,\,\Delta T}+{\dot {S}}'_{\mathrm {gen} ,\,\Delta p}}}}
thermodynamics (ratio of heat transfer irreversibility to total irreversibility due to heat transfer and fluid friction)[ 9]
Bingham number Bm
B
m
=
τ
y
L
μ
V
{\displaystyle \mathrm {Bm} ={\frac {\tau _{y}L}{\mu V}}}
fluid mechanics , rheology (ratio of yield stress to viscous stress)[ 1]
Biot number Bi
B
i
=
h
L
C
k
b
{\displaystyle \mathrm {Bi} ={\frac {hL_{C}}{k_{b}}}}
heat transfer (surface vs. volume conductivity of solids)
Blake number Bl or B
B
=
u
ρ
μ
(
1
−
ϵ
)
D
{\displaystyle \mathrm {B} ={\frac {u\rho }{\mu (1-\epsilon )D}}}
geology , fluid mechanics , porous media (inertial over viscous forces in fluid flow through porous media)
Bond number Bo
B
o
=
ρ
a
L
2
γ
{\displaystyle \mathrm {Bo} ={\frac {\rho aL^{2}}{\gamma }}}
geology , fluid mechanics , porous media (buoyant versus capillary forces, similar to the Eötvös number ) [ 10]
Brinkman number Br
B
r
=
μ
U
2
κ
(
T
w
−
T
0
)
{\displaystyle \mathrm {Br} ={\frac {\mu U^{2}}{\kappa (T_{w}-T_{0})}}}
heat transfer , fluid mechanics (conduction from a wall to a viscous fluid )
Brownell–Katz number NBK
N
B
K
=
u
μ
k
r
w
σ
{\displaystyle \mathrm {N} _{\mathrm {BK} }={\frac {u\mu }{k_{\mathrm {rw} }\sigma }}}
fluid mechanics (combination of capillary number and Bond number ) [ 11]
Capillary number Ca
C
a
=
μ
V
γ
{\displaystyle \mathrm {Ca} ={\frac {\mu V}{\gamma }}}
porous media , fluid mechanics (viscous forces versus surface tension )
Chandrasekhar number Q
Q
=
B
0
2
d
2
μ
0
ρ
ν
λ
{\displaystyle \mathrm {Q} ={\frac {{B_{0}}^{2}d^{2}}{\mu _{0}\rho \nu \lambda }}}
magnetohydrodynamics (ratio of the Lorentz force to the viscosity in magnetic convection )
Colburn J factors J M , J H , J D turbulence ; heat , mass , and momentum transfer (dimensionless transfer coefficients)
Darcy friction factor C f or f D fluid mechanics (fraction of pressure losses due to friction in a pipe ; four times the Fanning friction factor )
Dean number D
D
=
ρ
V
d
μ
(
d
2
R
)
1
/
2
{\displaystyle \mathrm {D} ={\frac {\rho Vd}{\mu }}\left({\frac {d}{2R}}\right)^{1/2}}
turbulent flow (vortices in curved ducts)
Deborah number De
D
e
=
t
c
t
p
{\displaystyle \mathrm {De} ={\frac {t_{\mathrm {c} }}{t_{\mathrm {p} }}}}
rheology (viscoelastic fluids)
Drag coefficient c d
c
d
=
2
F
d
ρ
v
2
A
,
{\displaystyle c_{\mathrm {d} }={\dfrac {2F_{\mathrm {d} }}{\rho v^{2}A}}\,,}
aeronautics , fluid dynamics (resistance to fluid motion)
Eckert number Ec
E
c
=
V
2
c
p
Δ
T
{\displaystyle \mathrm {Ec} ={\frac {V^{2}}{c_{p}\Delta T}}}
convective heat transfer (characterizes dissipation of energy ; ratio of kinetic energy to enthalpy )
Ekman number Ek
E
k
=
ν
2
D
2
Ω
sin
φ
{\displaystyle \mathrm {Ek} ={\frac {\nu }{2D^{2}\Omega \sin \varphi }}}
geophysics (viscous versus Coriolis forces )
Eötvös number Eo
E
o
=
Δ
ρ
g
L
2
σ
{\displaystyle \mathrm {Eo} ={\frac {\Delta \rho \,g\,L^{2}}{\sigma }}}
fluid mechanics (shape of bubbles or drops )
Ericksen number Er
E
r
=
μ
v
L
K
{\displaystyle \mathrm {Er} ={\frac {\mu vL}{K}}}
fluid dynamics (liquid crystal flow behavior; viscous over elastic forces)
Euler number Eu
E
u
=
Δ
p
ρ
V
2
{\displaystyle \mathrm {Eu} ={\frac {\Delta {}p}{\rho V^{2}}}}
hydrodynamics (stream pressure versus inertia forces)
Excess temperature coefficient
Θ
r
{\displaystyle \Theta _{r}}
Θ
r
=
c
p
(
T
−
T
e
)
U
e
2
/
2
{\displaystyle \Theta _{r}={\frac {c_{p}(T-T_{e})}{U_{e}^{2}/2}}}
heat transfer , fluid dynamics (change in internal energy versus kinetic energy )[ 12]
Fanning friction factor f fluid mechanics (fraction of pressure losses due to friction in a pipe ; 1/4th the Darcy friction factor )[ 13]
Fourier number Fo
F
o
=
α
t
L
2
{\displaystyle \mathrm {Fo} ={\frac {\alpha t}{L^{2}}}}
heat transfer , mass transfer (ratio of diffusive rate versus storage rate)
Froude number Fr
F
r
=
v
g
ℓ
{\displaystyle \mathrm {Fr} ={\frac {v}{\sqrt {g\ell }}}}
fluid mechanics (wave and surface behaviour; ratio of a body's inertia to gravitational forces )
Galilei number Ga
G
a
=
g
L
3
ν
2
{\displaystyle \mathrm {Ga} ={\frac {g\,L^{3}}{\nu ^{2}}}}
fluid mechanics (gravitational over viscous forces)
Görtler number G
G
=
U
e
θ
ν
(
θ
R
)
1
/
2
{\displaystyle \mathrm {G} ={\frac {U_{e}\theta }{\nu }}\left({\frac {\theta }{R}}\right)^{1/2}}
fluid dynamics (boundary layer flow along a concave wall)
Graetz number Gz
G
z
=
D
H
L
R
e
P
r
{\displaystyle \mathrm {Gz} ={D_{H} \over L}\mathrm {Re} \,\mathrm {Pr} }
heat transfer , fluid mechanics (laminar flow through a conduit; also used in mass transfer )
Grashof number Gr
G
r
L
=
g
β
(
T
s
−
T
∞
)
L
3
ν
2
{\displaystyle \mathrm {Gr} _{L}={\frac {g\beta (T_{s}-T_{\infty })L^{3}}{\nu ^{2}}}}
heat transfer , natural convection (ratio of the buoyancy to viscous force)
Hagen number Hg
H
g
=
−
1
ρ
d
p
d
x
L
3
ν
2
{\displaystyle \mathrm {Hg} =-{\frac {1}{\rho }}{\frac {\mathrm {d} p}{\mathrm {d} x}}{\frac {L^{3}}{\nu ^{2}}}}
heat transfer (ratio of the buoyancy to viscous force in forced convection )
Hydraulic gradient i
i
=
d
h
d
l
=
h
2
−
h
1
l
e
n
g
t
h
{\displaystyle i={\frac {\mathrm {d} h}{\mathrm {d} l}}={\frac {h_{2}-h_{1}}{\mathrm {length} }}}
fluid mechanics , groundwater flow (pressure head over distance)
Karlovitz number Ka
K
a
=
t
F
t
η
{\displaystyle \mathrm {Ka} ={\frac {t_{F}}{t_{\eta }}}}
turbulent combustion (characteristic chemical time scale to Kolmogorov time scale)
Keulegan–Carpenter number KC
K
C
=
V
T
L
{\displaystyle \mathrm {K_{C}} ={\frac {V\,T}{L}}}
fluid dynamics (ratio of drag force to inertia for a bluff object in oscillatory fluid flow)
Knudsen number Kn
K
n
=
λ
L
{\displaystyle \mathrm {Kn} ={\frac {\lambda }{L}}}
gas dynamics (ratio of the molecular mean free path length to a representative physical length scale)
Kutateladze number Ku
K
u
=
U
h
ρ
g
1
/
2
(
σ
g
(
ρ
l
−
ρ
g
)
)
1
/
4
{\displaystyle \mathrm {Ku} ={\frac {U_{h}\rho _{g}^{1/2}}{\left({\sigma g(\rho _{l}-\rho _{g})}\right)^{1/4}}}}
fluid mechanics (counter-current two-phase flow )[ 14]
Laplace number La
L
a
=
σ
ρ
L
μ
2
{\displaystyle \mathrm {La} ={\frac {\sigma \rho L}{\mu ^{2}}}}
fluid dynamics (free convection within immiscible fluids; ratio of surface tension to momentum -transport)
Lewis number Le
L
e
=
α
D
=
S
c
P
r
{\displaystyle \mathrm {Le} ={\frac {\alpha }{D}}={\frac {\mathrm {Sc} }{\mathrm {Pr} }}}
heat and mass transfer (ratio of thermal to mass diffusivity )
Lift coefficient C L
C
L
=
L
q
S
{\displaystyle C_{\mathrm {L} }={\frac {L}{q\,S}}}
aerodynamics (lift available from an airfoil at a given angle of attack )
Lockhart–Martinelli parameter
χ
{\displaystyle \chi }
χ
=
m
ℓ
m
g
ρ
g
ρ
ℓ
{\displaystyle \chi ={\frac {m_{\ell }}{m_{g}}}{\sqrt {\frac {\rho _{g}}{\rho _{\ell }}}}}
two-phase flow (flow of wet gases ; liquid fraction)[ 15]
Mach number M or Ma
M
=
v
v
s
o
u
n
d
{\displaystyle \mathrm {M} ={\frac {v}{v_{\mathrm {sound} }}}}
gas dynamics (compressible flow ; dimensionless velocity )
Magnetic Reynolds number Rm
R
m
=
U
L
η
{\displaystyle \mathrm {R} _{\mathrm {m} }={\frac {UL}{\eta }}}
magnetohydrodynamics (ratio of magnetic advection to magnetic diffusion )
Manning roughness coefficient n open channel flow (flow driven by gravity )[ 16]
Marangoni number Mg
M
g
=
−
d
σ
d
T
L
Δ
T
η
α
{\displaystyle \mathrm {Mg} =-{\frac {\mathrm {d} \sigma }{\mathrm {d} T}}{\frac {L\Delta T}{\eta \alpha }}}
fluid mechanics (Marangoni flow ; thermal surface tension forces over viscous forces)
Markstein number
M
{\displaystyle {\mathcal {M}}}
M
=
L
b
δ
L
{\displaystyle {\mathcal {M}}={\frac {{\mathcal {L}}_{b}}{\delta _{L}}}}
fluid dynamics , combustion (turbulent combustion flames)
Morton number Mo
M
o
=
g
μ
c
4
Δ
ρ
ρ
c
2
σ
3
{\displaystyle \mathrm {Mo} ={\frac {g\mu _{c}^{4}\,\Delta \rho }{\rho _{c}^{2}\sigma ^{3}}}}
fluid dynamics (determination of bubble /drop shape)
Nusselt number Nu
N
u
d
=
h
d
k
{\displaystyle \mathrm {Nu} _{d}={\frac {hd}{k}}}
heat transfer (forced convection ; ratio of convective to conductive heat transfer)
Ohnesorge number Oh
O
h
=
μ
ρ
σ
L
=
W
e
R
e
{\displaystyle \mathrm {Oh} ={\frac {\mu }{\sqrt {\rho \sigma L}}}={\frac {\sqrt {\mathrm {We} }}{\mathrm {Re} }}}
fluid dynamics (atomization of liquids, Marangoni flow )
Péclet number Pe
P
e
d
=
d
u
ρ
c
p
k
=
R
e
d
P
r
{\displaystyle \mathrm {Pe} _{d}={\frac {du\rho c_{p}}{k}}=\mathrm {Re} _{d}\,\mathrm {Pr} }
heat transfer (advection –diffusion problems; total momentum transfer to molecular heat transfer)
Péclet number Pe
P
e
d
=
d
u
D
=
R
e
d
S
c
{\displaystyle \mathrm {Pe} _{d}={\frac {du}{D}}=\mathrm {Re} _{d}\,\mathrm {Sc} }
mass transfer (advection –diffusion problems; total momentum transfer to diffusive mass transfer)
Prandtl number Pr
P
r
=
ν
α
=
c
p
μ
k
{\displaystyle \mathrm {Pr} ={\frac {\nu }{\alpha }}={\frac {c_{p}\mu }{k}}}
heat transfer (ratio of viscous diffusion rate over thermal diffusion rate)
Pressure coefficient CP
C
p
=
p
−
p
∞
1
2
ρ
∞
V
∞
2
{\displaystyle C_{p}={p-p_{\infty } \over {\frac {1}{2}}\rho _{\infty }V_{\infty }^{2}}}
aerodynamics , hydrodynamics (pressure experienced at a point on an airfoil ; dimensionless pressure variable)
Rayleigh number Ra
R
a
x
=
g
β
ν
α
(
T
s
−
T
∞
)
x
3
{\displaystyle \mathrm {Ra} _{x}={\frac {g\beta }{\nu \alpha }}(T_{s}-T_{\infty })x^{3}}
heat transfer (buoyancy versus viscous forces in free convection )
Reynolds number Re
R
e
L
=
v
L
ρ
μ
{\displaystyle \mathrm {Re} _{L}={\frac {vL\rho }{\mu }}}
fluid mechanics (ratio of fluid inertial and viscous forces)[ 1]
Richardson number Ri
R
i
=
g
h
u
2
=
1
F
r
2
{\displaystyle \mathrm {Ri} ={\frac {gh}{u^{2}}}={\frac {1}{\mathrm {Fr} ^{2}}}}
fluid dynamics (effect of buoyancy on flow stability; ratio of potential over kinetic energy )[ 17]
Roshko number Ro
R
o
=
f
L
2
ν
=
S
t
R
e
{\displaystyle \mathrm {Ro} ={fL^{2} \over \nu }=\mathrm {St} \,\mathrm {Re} }
fluid dynamics (oscillating flow, vortex shedding )
Schmidt number Sc
S
c
D
=
ν
D
{\displaystyle \mathrm {Sc} _{D}={\frac {\nu }{D}}}
mass transfer (viscous over molecular diffusion rate)[ 18]
Shape factor H
H
=
δ
∗
θ
{\displaystyle H={\frac {\delta ^{*}}{\theta }}}
boundary layer flow (ratio of displacement thickness to momentum thickness)
Sherwood number Sh
S
h
D
=
K
L
D
{\displaystyle \mathrm {Sh} _{D}={\frac {KL}{D}}}
mass transfer (forced convection ; ratio of convective to diffusive mass transport)
Sommerfeld number S
S
=
(
r
c
)
2
μ
N
P
{\displaystyle \mathrm {S} =\left({\frac {r}{c}}\right)^{2}{\frac {\mu N}{P}}}
hydrodynamic lubrication (boundary lubrication )[ 19]
Stanton number St
S
t
=
h
c
p
ρ
V
=
N
u
R
e
P
r
{\displaystyle \mathrm {St} ={\frac {h}{c_{p}\rho V}}={\frac {\mathrm {Nu} }{\mathrm {Re} \,\mathrm {Pr} }}}
heat transfer and fluid dynamics (forced convection )
Stokes number Stk or Sk
S
t
k
=
τ
U
o
d
c
{\displaystyle \mathrm {Stk} ={\frac {\tau U_{o}}{d_{c}}}}
particles suspensions (ratio of characteristic time of particle to time of flow)
Strouhal number St or Sr
S
t
=
ω
L
v
{\displaystyle \mathrm {St} ={\omega L \over v}}
fluid dynamics (continuous and pulsating flow; nondimensional frequency )[ 20]
Stuart number N
N
=
B
2
L
c
σ
ρ
U
=
H
a
2
R
e
{\displaystyle \mathrm {N} ={\frac {B^{2}L_{c}\sigma }{\rho U}}={\frac {\mathrm {Ha} ^{2}}{\mathrm {Re} }}}
magnetohydrodynamics (ratio of electromagnetic to inertial forces)
Taylor number Ta
T
a
=
4
Ω
2
R
4
ν
2
{\displaystyle \mathrm {Ta} ={\frac {4\Omega ^{2}R^{4}}{\nu ^{2}}}}
fluid dynamics (rotating fluid flows; inertial forces due to rotation of a fluid versus viscous forces )
Ursell number U
U
=
H
λ
2
h
3
{\displaystyle \mathrm {U} ={\frac {H\,\lambda ^{2}}{h^{3}}}}
wave mechanics (nonlinearity of surface gravity waves on a shallow fluid layer)
Vadasz number Va
V
a
=
ϕ
P
r
D
a
{\displaystyle \mathrm {Va} ={\frac {\phi \,\mathrm {Pr} }{\mathrm {Da} }}}
porous media (governs the effects of porosity
ϕ
{\displaystyle \phi }
Prandtl number and the Darcy number on flow in a porous medium ) [ 21]
Wallis parameter j *
j
∗
=
R
(
ω
ρ
μ
)
1
2
{\displaystyle j^{*}=R\left({\frac {\omega \rho }{\mu }}\right)^{\frac {1}{2}}}
multiphase flows (nondimensional superficial velocity )[ 22]
Weber number We
W
e
=
ρ
v
2
l
σ
{\displaystyle \mathrm {We} ={\frac {\rho v^{2}l}{\sigma }}}
multiphase flow (strongly curved surfaces; ratio of inertia to surface tension )
Weissenberg number Wi
W
i
=
γ
˙
λ
{\displaystyle \mathrm {Wi} ={\dot {\gamma }}\lambda }
viscoelastic flows (shear rate times the relaxation time)[ 23]
Womersley number
α
{\displaystyle \alpha }
α
=
R
(
ω
ρ
μ
)
1
2
{\displaystyle \alpha =R\left({\frac {\omega \rho }{\mu }}\right)^{\frac {1}{2}}}
biofluid mechanics (continuous and pulsating flows; ratio of pulsatile flow frequency to viscous effects )[ 24]
Zel'dovich number
β
{\displaystyle \beta }
β
=
E
R
T
f
T
f
−
T
o
T
f
{\displaystyle \beta ={\frac {E}{RT_{f}}}{\frac {T_{f}-T_{o}}{T_{f}}}}
fluid dynamics , Combustion (Measure of activation energy )
![{\displaystyle {\frac {\operatorname {E} [(X-\mu _{X})(Y-\mu _{Y})]}{\sigma _{X}\sigma _{Y}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43151e1ba399138a436555ee061ca776afbfb690)
