Name
Standard symbol
Definition
Named after
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}}}}
Archimedes fluid mechanics (motion of fluids due to density differences)
Atwood number A
A
=
ρ
1
−
ρ
2
ρ
1
+
ρ
2
{\displaystyle \mathrm {A} ={\frac {\rho _{1}-\rho _{2}}{\rho _{1}+\rho _{2}}}}
George Atwood [citation needed 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 }}}
Ralph Bagnold
Granular flow (grain collision stresses to viscous fluid stresses)
Bejan number Be
B
e
=
Δ
P
L
2
μ
α
{\displaystyle \mathrm {Be} ={\frac {\Delta PL^{2}}{\mu \alpha }}}
Adrian Bejan fluid mechanics (dimensionless pressure drop along a channel )[ 4]
Bingham number Bm
B
m
=
τ
y
L
μ
V
{\displaystyle \mathrm {Bm} ={\frac {\tau _{y}L}{\mu V}}}
Eugene C. Bingham fluid mechanics , rheology (ratio of yield stress to viscous stress)[ 5]
Biot number Bi
B
i
=
h
L
C
k
b
{\displaystyle \mathrm {Bi} ={\frac {hL_{C}}{k_{b}}}}
Jean-Baptiste Biot 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}}}
Frank C. Blake (1892–1926)
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 }}}
Wilfrid Noel Bond geology , fluid mechanics , porous media (buoyant versus capillary forces, similar to the Eötvös number )[ 6]
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})}}}
Henri Brinkman heat transfer , fluid mechanics (conduction from a wall to a viscous fluid )
Burger number
Bu
B
u
=
(
R
o
F
r
)
2
{\displaystyle \mathrm {Bu} =\left({\dfrac {\mathrm {Ro} }{\mathrm {Fr} }}\right)^{2}}
Alewyn P. Burger (1927–2003)
meteorology , oceanography (density stratification versus Earth's rotation )
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 }}}
Lloyd E. Brownell and Donald L. Katz
fluid mechanics (combination of capillary number and Bond number )[ 7]
Capillary number Ca
C
a
=
μ
V
γ
{\displaystyle \mathrm {Ca} ={\frac {\mu V}{\gamma }}}
porous media , fluid mechanics (viscous forces versus surface tension )
Cauchy number Ca
C
a
=
ρ
u
2
K
{\displaystyle \mathrm {Ca} ={\frac {\rho u^{2}}{K}}}
Augustin-Louis Cauchy compressible flows (inertia forces versus compressibility force)
Cavitation number Ca
C
a
=
p
−
p
v
1
2
ρ
v
2
{\displaystyle \mathrm {Ca} ={\frac {p-p_{\mathrm {v} }}{{\frac {1}{2}}\rho v^{2}}}}
multiphase flow (hydrodynamic cavitation , pressure over dynamic pressure )
Chandrasekhar number C
C
=
B
2
L
2
μ
o
μ
D
M
{\displaystyle \mathrm {C} ={\frac {B^{2}L^{2}}{\mu _{o}\mu D_{M}}}}
Subrahmanyan Chandrasekhar hydromagnetics (Lorentz force versus viscosity )
Colburn J factors J M , J H , J D
Allan Philip Colburn (1904–1955)
turbulence ; heat , mass , and momentum transfer (dimensionless transfer coefficients)
Damkohler number Da
D
a
=
k
τ
{\displaystyle \mathrm {Da} =k\tau }
Gerhard Damköhler chemistry (reaction time scales vs. residence time)
Darcy friction factor C f or f D
Henry Darcy fluid mechanics (fraction of pressure losses due to friction in a pipe ; four times the Fanning friction factor )
Darcy number
Da
D
a
=
k
d
2
{\displaystyle \mathrm {Da} ={\frac {k}{d^{2}}}}
Henry Darcy
Fluid dynamics (permeability of the medium versus its cross-sectional area in porous media )
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}}
William Reginald Dean 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} }}}}
Deborah 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)
Dukhin number
Du
D
u
=
κ
σ
K
m
a
.
{\displaystyle {\rm {Du}}={\frac {\kappa ^{\sigma }}{{\mathrm {K} _{m}}a}}.}
Stanislav and Andrei Dukhin
Fluid heterogeneous systems (surface conductivity to various electrokinetic and electroacoustic effects)
Eckert number Ec
E
c
=
V
2
c
p
Δ
T
{\displaystyle \mathrm {Ec} ={\frac {V^{2}}{c_{p}\Delta T}}}
Ernst R. G. Eckert 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 }}}
Vagn Walfrid Ekman
Geophysics (viscosity to Coriolis force ratio)
Eötvös number Eo
E
o
=
Δ
ρ
g
L
2
σ
{\displaystyle \mathrm {Eo} ={\frac {\Delta \rho \,g\,L^{2}}{\sigma }}}
Loránd Eötvös fluid mechanics (shape of bubbles or drops )
Ericksen number Er
E
r
=
μ
v
L
K
{\displaystyle \mathrm {Er} ={\frac {\mu vL}{K}}}
Jerald Ericksen 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}}}}
Leonhard Euler 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 )[ 8]
Fanning friction factor f
John T. Fanning fluid mechanics (fraction of pressure losses due to friction in a pipe ; 1/4th the Darcy friction factor )[ 9]
Froude number Fr
F
r
=
U
g
ℓ
{\displaystyle \mathrm {Fr} ={\frac {U}{\sqrt {g\ell }}}}
William Froude 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}}}}
Galileo Galilei 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}}
Henry Görtler [de ] fluid dynamics (boundary layer flow along a concave wall)
Goucher number [fr ]
Go
G
o
=
R
(
ρ
g
2
σ
)
1
/
2
{\displaystyle \mathrm {Go} =R\left({\frac {\rho g}{2\sigma }}\right)^{1/2}}
Frederick Shand Goucher (1888–1973)
fluid dynamics (wire coating problems)
Garcia-Atance number GA
G
A
=
p
−
p
v
ρ
a
L
{\displaystyle \mathrm {G_{A}} ={\frac {p-p_{v}}{\rho aL}}}
Gonzalo Garcia-Atance Fatjo
phase change (ultrasonic cavitation onset, ratio of pressures over pressure due to acceleration)
Graetz number Gz
G
z
=
D
H
L
R
e
P
r
{\displaystyle \mathrm {Gz} ={D_{H} \over L}\mathrm {Re} \,\mathrm {Pr} }
Leo Graetz 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}}}}
Franz Grashof heat transfer , natural convection (ratio of the buoyancy to viscous force)
Hartmann number Ha
H
a
=
B
L
(
σ
ρ
ν
)
1
2
{\displaystyle \mathrm {Ha} =BL\left({\frac {\sigma }{\rho \nu }}\right)^{\frac {1}{2}}}
Julius Hartmann (1881–1951)
magnetohydrodynamics (ratio of Lorentz to viscous forces)
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}}}}
Gotthilf Hagen heat transfer (ratio of the buoyancy to viscous force in forced convection )
Iribarren number Ir
I
r
=
tan
α
H
/
L
0
{\displaystyle \mathrm {Ir} ={\frac {\tan \alpha }{\sqrt {H/L_{0}}}}}
Ramón Iribarren wave mechanics (breaking surface gravity waves on a slope)
Jakob number Ja
J
a
=
c
p
,
f
(
T
w
−
T
s
a
t
)
h
f
g
{\displaystyle \mathrm {Ja} ={\frac {c_{p,f}(T_{w}-T_{sat})}{h_{fg}}}}
Max Jakob heat transfer (ratio of sensible heat to latent heat during phase changes )
Jesus number
Je
J
e
=
σ
L
M
g
{\displaystyle \mathrm {Je} ={\frac {\sigma \,L}{M\,g}}}
Jesus
Surface tension (ratio of surface tension and weight)
Karlovitz number Ka
K
a
=
k
t
c
{\displaystyle \mathrm {Ka} =kt_{c}}
Béla Karlovitz turbulent combustion (characteristic flow time times flame stretch rate)
Kapitza number Ka
K
a
=
σ
ρ
(
g
sin
β
)
1
/
3
ν
4
/
3
{\displaystyle \mathrm {Ka} ={\frac {\sigma }{\rho (g\sin \beta )^{1/3}\nu ^{4/3}}}}
Pyotr Kapitsa fluid mechanics (thin film of liquid flows down inclined surfaces)
Keulegan–Carpenter number KC
K
C
=
V
T
L
{\displaystyle \mathrm {K_{C}} ={\frac {V\,T}{L}}}
Garbis H. Keulegan (1890–1989) and Lloyd H. Carpenter
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}}}
Martin Knudsen 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}}}}
Samson Kutateladze fluid mechanics (counter-current two-phase flow )[ 10]
Laplace number La
L
a
=
σ
ρ
L
μ
2
{\displaystyle \mathrm {La} ={\frac {\sigma \rho L}{\mu ^{2}}}}
Pierre-Simon Laplace 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} }}}
Warren K. Lewis 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 }}}}}
R. W. Lockhart and Raymond C. Martinelli
two-phase flow (flow of wet gases ; liquid fraction)[ 11]
Mach number M or Ma
M
=
v
v
s
o
u
n
d
{\displaystyle \mathrm {M} ={\frac {v}{v_{\mathrm {sound} }}}}
Ernst Mach gas dynamics (compressible flow ; dimensionless velocity )
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 }}}
Carlo Marangoni fluid mechanics (Marangoni flow ; thermal surface tension forces over viscous forces)
Markstein number Ma
M
a
=
L
b
l
f
{\displaystyle \mathrm {Ma} ={\frac {L_{b}}{l_{f}}}}
George H. Markstein turbulence , combustion (Markstein length to laminar flame thickness)
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}}}}
Rose Morton fluid dynamics (determination of bubble /drop shape)
Nusselt number Nu
N
u
=
h
d
k
{\displaystyle \mathrm {Nu} ={\frac {hd}{k}}}
Wilhelm Nusselt 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} }}}
Wolfgang von Ohnesorge
fluid dynamics (atomization of liquids, Marangoni flow )
Péclet number Pe
P
e
=
L
u
D
{\displaystyle \mathrm {Pe} ={\frac {Lu}{D}}}
P
e
=
L
u
α
{\displaystyle \mathrm {Pe} ={\frac {Lu}{\alpha }}}
Jean Claude Eugène Péclet fluid mechanics (ratio of advective transport rate over molecular diffusive transport rate), heat transfer (ratio of advective transport rate over thermal diffusive transport rate)
Prandtl number Pr
P
r
=
ν
α
=
c
p
μ
k
{\displaystyle \mathrm {Pr} ={\frac {\nu }{\alpha }}={\frac {c_{p}\mu }{k}}}
Ludwig Prandtl 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}}
John William Strutt, 3rd Baron Rayleigh heat transfer (buoyancy versus viscous forces in free convection )
Reynolds number Re
R
e
=
U
L
ρ
μ
=
U
L
ν
{\displaystyle \mathrm {Re} ={\frac {UL\rho }{\mu }}={\frac {UL}{\nu }}}
Osborne Reynolds fluid mechanics (ratio of fluid inertial and viscous forces)[ 5]
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}}}}
Lewis Fry Richardson fluid dynamics (effect of buoyancy on flow stability; ratio of potential over kinetic energy )[ 12]
Roshko number Ro
R
o
=
f
L
2
ν
=
S
t
R
e
{\displaystyle \mathrm {Ro} ={fL^{2} \over \nu }=\mathrm {St} \,\mathrm {Re} }
Anatol Roshko fluid dynamics (oscillating flow, vortex shedding )
Rossby number Ro
Ro
=
U
L
f
,
{\displaystyle {\text{Ro}}={\frac {U}{Lf}},}
Carl-Gustaf Rossby fluid flow (geophysics , ratio of inertial force to Coriolis force )
Rouse number
P
P
=
w
s
κ
u
∗
{\displaystyle \mathrm {P} ={\frac {w_{s}}{\kappa u_{*}}}}
Hunter Rouse
Fluid dynamics (concentration profile of suspended sediment)
Schmidt number Sc
S
c
=
ν
D
{\displaystyle \mathrm {Sc} ={\frac {\nu }{D}}}
Ernst Heinrich Wilhelm Schmidt (1892–1975)
mass transfer (viscous over molecular diffusion rate)[ 13]
Scruton number
Sc
S
c
=
2
δ
s
m
e
ρ
b
ref
2
{\displaystyle \mathrm {Sc} ={\frac {2\delta _{s}m_{e}}{\rho b_{\text{ref}}^{2}}}}
Christopher 'Kit' Scruton
Fluid dynamics (vortex resonance)
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
=
K
L
D
{\displaystyle \mathrm {Sh} ={\frac {KL}{D}}}
Thomas Kilgore Sherwood mass transfer (forced convection ; ratio of convective to diffusive mass transport)
Shields parameter
θ
θ
=
τ
(
ρ
s
−
ρ
)
g
D
{\displaystyle \theta ={\frac {\tau }{(\rho _{s}-\rho )gD}}}
Albert F. Shields
Fluid dynamics (motion of sediment)
Sommerfeld number S
S
=
(
r
c
)
2
μ
N
P
{\displaystyle \mathrm {S} =\left({\frac {r}{c}}\right)^{2}{\frac {\mu N}{P}}}
Arnold Sommerfeld hydrodynamic lubrication (boundary lubrication )[ 14]
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} }}}
Thomas Ernest Stanton 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}}}}
Sir George Stokes, 1st Baronet particles suspensions (ratio of characteristic time of particle to time of flow)
Strouhal number St
S
t
=
f
L
U
{\displaystyle \mathrm {St} ={\frac {fL}{U}}}
Vincenc Strouhal Vortex shedding (ratio of characteristic oscillatory velocity to ambient flow velocity)
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} }}}
John Trevor Stuart 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}}}}
G. I. Taylor fluid dynamics (rotating fluid flows; inertial forces due to rotation of a fluid versus viscous forces )
Thoma number
σ
σ
=
N
P
S
H
h
p
u
m
p
{\displaystyle \mathrm {\sigma } ={\frac {\mathrm {NPSH} }{h_{\mathrm {pump} }}}}
Dieter Thoma (1881–1942)
multiphase flow (hydrodynamic cavitation , pressure over dynamic pressure )
Ursell number U
U
=
H
λ
2
h
3
{\displaystyle \mathrm {U} ={\frac {H\,\lambda ^{2}}{h^{3}}}}
Fritz Ursell wave mechanics (nonlinearity of surface gravity waves on a shallow fluid layer)
Wallis parameter j ∗
j
∗
=
R
(
ω
ρ
μ
)
1
2
{\displaystyle j^{*}=R\left({\frac {\omega \rho }{\mu }}\right)^{\frac {1}{2}}}
Graham B. Wallis
multiphase flows (nondimensional superficial velocity )[ 15]
Weber number We
W
e
=
ρ
v
2
l
σ
{\displaystyle \mathrm {We} ={\frac {\rho v^{2}l}{\sigma }}}
Moritz Weber multiphase flow (strongly curved surfaces; ratio of inertia to surface tension )
Weissenberg number Wi
W
i
=
γ
˙
λ
{\displaystyle \mathrm {Wi} ={\dot {\gamma }}\lambda }
Karl Weissenberg viscoelastic flows (shear rate times the relaxation time)[ 16]
Womersley number
α
{\displaystyle \alpha }
α
=
R
(
ω
ρ
μ
)
1
2
{\displaystyle \alpha =R\left({\frac {\omega \rho }{\mu }}\right)^{\frac {1}{2}}}
John R. Womersley biofluid mechanics (continuous and pulsating flows; ratio of pulsatile flow frequency to viscous effects )[ 17]
Zeldovich 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}}}}
Yakov Zeldovich fluid dynamics , Combustion (Measure of activation energy )
