All dimensions less qunatities.

1. FLUID MECHANICS LAB
GROUP DISCUSSION TOPIC-
SIGNIFICANCE OF
DIMENSIONLESS NUMBERS

2. S.No
.
Name Standa
rd
Symbol
Formula Application
1. Archimedes
Number
Ar Ar=
re2
fr
=
gl3ρ(ρs − ρ)
μ2
Ratio of gravitational forces to viscous forces
Used in design of tubular chemical process .
2. Atwood
Number
A A=
(ρ1 − ρ2)
(ρ1 + ρ2)
Used in the study of hydrodynamic instabilities in
density stratified flows.
3. Biot Number Bi Bi=
hl
ks
=
conductive resistance in solid
convective resistance in thermal boundary layer
Ratio of internal thermal resistance of a solid to
boundary layer thermal resistance
Use in extended surface heat transfer
calculations,transient conduction problems in a
lumped parameter solution.
4. Weber
Number
We We =
ρu2
l
σ
=
inertial force
surface tension force
It is used in analyzing formation of droplet
And in capillary studies.

3. 5. Brinkman
Number
Br Br=
μu2
k(tw − to)
Ratio between heat produced by viscous
dissipation and heat transported by molecular
conduction. I.E., The ratio of viscous heat
generation to external heating.
Brinkman number is related to heat conduction from
a wall to a flowing viscous fluid. It is commonly used
in polymer processing.
6. Capillary
Number
Ca Ca=
we
re
=
uμ
ς
Ratio of viscous force to surface tension force
It represents the relative effect of viscous drag forces
versus surface tension forces acting across an
interface between a liquid and a gas, or between
two immiscible liquids.
7. Cauchy
Number
Ca Ca= m2
=
ρu2
k
Ratio between inertial and the compressibility
force (elastic force) in a flow
Used in the study of compressible flows. Useful for
analyzing fluid flow dynamics problems where
compressibility is a significant factor.
8. Dean
Number
De De=
re
(r h)1 2
Reynolds number times centrifugal force over
inertial force
Dean number deals with the stability of two-
dimensional flows in a curved channel with mean
radius rand width 2h.In the study of flow in
curved pipes and channels.

4. 9. Deborah
Number
De De =
t
tp
=
relaxation time
characteristic time scale
Commonly used in rheology to characterize how
"fluid" a material is. The smaller the de, the more the
fluid the material appears.
10. Eckert
Number
Ec Ec=
u2
cpδt
Kinetic energy of flow relative to boundary layer
enthalpy difference
Eckert number represents the kinetic energy of the
flow relative to the boundary layer enthalpy
difference.
11. Ekman
Number
Ek Ek=
μ
ρςl2 =
viscous force
coriolis force
The Ekman number gives a measure of the rate at
which stresses at a boundary (e.g., Wind-induced
stresses at the ocean surface) are communicated to
the fluid interior.
12. EÖtvÖs
Number
Eo Eo=
we
fr
=
δρgl2
ς
Ratio of gravitational force to surface force
It is used in analyzing the behavior of
Bubbles.
13. Euler
Number
Eu Eu=
δp
ρu2 =
pressure force
inertial force
It is used to characterize energy losses in a
Flow.

5. 14. Fourier
Number
Fo Fo=
αt
l2 =
rate of heat conduction
rate of thermal energy stored
Fourier number represents the dimensionless time. It
may be interpreted as the ratio of current time to
time to reach steady-state. Used along with Biot
number to solve transient state heat transfer
problems.
15. Froude
Number
Fr Fr =
u2
gl
=
inertial force
gravitational force
It is used to analyze nature of flow where free surface
is present. E.G. Weirs, channels, spillways. It is used to
determine the resistance of an body which is
submerged partially moving along with water.
16. Galileo
Number
Ga Ga=
re2
fr
=
ρ2gl3
μ2
Reynolds number times gravitational force over
viscous force
The Galilei number is used in viscous flow and
thermal expansion calculations, for example to
describe fluid film flow over walls.
17. Graetz
Number
Gz Gz=
di pe
l
=
udi
ν
Ratio of thermal capacity to convective heat
transfer
This number is useful in determining the thermally
developing flow entrance length in ducts.
18. Grashof
Number
Gr Gr=
gβ(thot − tref)l3
ν2 =
buoyancy force
viscous force
It is used in flow analysis in natural convection.

6. 19. Hagen
Number
Hg Hg= −
dp
dx
ρl3
μ2
It is the forced flow equivalent of Grashof number.
20. Jakob
Number
Ja Ja=
cp(tw − tsat)
hfg
Ratio of sensible to latent energy absorbed
during liquid-vapour phase change
It is used in analyzing flow pattern in
Process of boiling.
21. Knudsen
Number
Kn Kn=
λ
l
=
length of mean free path
characteristic length of system
It is useful in determining whether
Continuum mechanics formulation can be applied or
not in a system.
22. Laplace
Number
La La=
re2
we
=
ρσl
μ2
Ratio of surface tension to the momentum-
transport (especially dissipation) inside a fluid.
Used in the characterization of free surface fluid
dynamics.
23. Lewis
Number
Le Le=
α
dab
=
thermal diffusivity
mass diffusivity
Fluid flow with simultaneous heat & mass transfer by
convection. It is also ratio of Schmidt number to
Prandtl number. It is used to analyze fluid flow with
simultaneous heat and mass transfer by convection.

7. 24. Mach
Number
M M=
u
a
=
inertial force
elastic (compressibility) force
Ratio of the fluid velocity to the velocity of
sound in that medium.
It is used to analyze fluid flow where
Compressibility is a important factor.
25. Marangoni
Number
Ma Ma=
dσ
dt
lδt
μα
Marangoni number is the ratio of thermal surface
tension force to the viscous force.
26. Morton
Number
Mo Mo=
we3
fr re4 =
gμ4
δρσ3
Used together with the Eötvös number or bond
number to characterize the shape of bubbles or drops
moving in a surrounding fluid or continuous phase, c.
27. Nusselt
Number
Nu Nu=
hl
kf
Nusselt number represents the dimensionless
temperature gradient at the solid surface.
For heat transfer in pipes, the characteristic length is
the pipe diameter. It describe the enhancement of
heat transfer because of convection in comparison to
conduction.
28. Ohnesorge
Number
Oh Oh=
we1/2
re
=
μ
(ρσl)1/2
Ratio of viscous force to square root of product
of inertial and surface forces
Used to relate to free surface fluid dynamics such as
dispersion of liquids in gases and in spray technology.

8. 29. Peclet
Number
Pe Pe=
ul
α
=
inertia (convection)
diffusion
Dimensionless independent heat transfer
parameter
It is used in calculations including
Convective heat transfer rate.
30. Prandtl
Number
Pr Pr=
ν
α
=
momentum diffusivity
thermal diffusivity
It is used in many calculations involving heat transfer
in flowing fluids. It is used to describe thermal
Boundary layer.
31. Rayleigh
Number
Ra Ra= gr pr =
gβ(thot − tref)l3
να
=
buoyancy
viscous × rate of heat diffusion
Used in heat transfer and free convection
calculations. Its value is used to characterize laminar
to turbulent transition.
32. Reynolds
Number
Re Re=
ρul
μ
=
inertial force
viscous force
It is often used to determine if the flow is either
laminar or turbulent.
33. Richardson
Number
Ri Ri=
gr
re2 =
gβ(thot − tref)l
u2 =
buoyancy force
inertial force
Practical importance in weather forecasting and in
investigating density and turbidity currents in oceans,
lakes, and reservoirs.

9. 34. Rossby
Number
Ro Ro=
u
ωl
=
inertial force
coriolis force
It is commonly used in geophysical phenomena in
the oceans and atmosphere, where it characterizes
the importance of coriolis accelerations arising
from planetary rotation.
35. Rotating
Froude
Number
FrR Frr=
fr
ro2 =
ω2l
g
It is often used to analyze fluid flow problems where
there is a free surface. It is used to analyze nature of
flow where free surface is present. E.G. Weirs,
channels, spillways. It is used to determine the
resistance of an body which is submerged partially
moving along with water.
36. Schmidt
Number
Sc Sc= le pr =
ν
dab
=
momentum diffusivity
mass diffusivity
It characterizes mass transfer in a flowing fluid. It is
used in fluid flows where simultaneous mass and
momentum diffusion occurs.
37. Sherwood
Number
Sh Sh=
hml
dab
Itrepresents the dimensionless concentration
gradient at the solid surface. It is used in mass
transfer problems.
38. Stanton
Number
St St=
nu
re pr
=
h
ρucp
Itis the modified Nusselt number. It is used in analogy
between heat transfer and viscous transport in
boundary layers. It is used in heat transfer for forced
convection flows.

10. 39. Stefan
Number
St St=
cpdt
lm
=
specific heat
latent heat
Itis useful in the study of heat transfer during phase
change.
40. Stokes
Number
Stk Stk=
τuo
dc
=
stopping distance of a particle
characteristic dimension of the obstacle
Commonly used in particles suspended in fluid.
41. Strouhal
Number
(for
oscillatory
flow)
Sr St=
l
utref
=
inertia (local)
inertia (convection)
Proportional to reciprocal of vortex spacing
If tref is taken as the reciprocal of the circular
frequency ω of the system, then
St=
𝜔𝐿
𝑈
42. Taylor
Number
Ta Bi=
ρ2ωi
2l4
μ2
It is a dimensionless quantity that characterizes the
importance of centrifugal "forces" or so-called inertial
forces due to rotation of a fluid about an axis, relative
to viscous forces.
43. Womersley
Number
𝛂
Α = (π re st)1 2
= l
(ρω)1 2
μ1 2
Itis used in bio-fluid mechanics. It is a dimensionless
expression of the pulsatile flow frequency in relation
to the viscous effects.

11. BIBLIOGRAPHY
Wikipedia-
https://en.wikipedia.org/wiki/Dimensionless_numbers_in_fluid_mechanics
SlideShare- https://www.slideshare.net/anwesakar/dimensionless-number-
109675745 & https://www.slideshare.net/aqibtemuri/dimensionless-
number-65944837

12. THANK YOU