We were discussing the basic concept of streamline and equipotential line, dimensional homogeneity, Buckingham pi theorem, difference between model and prototype and basic principle of similitude i.e. types of similarity in the subject of fluid mechanics, in our recent posts.

In order to secure the qualitative behavior of a physical problem, we will have to understand the concept of dimensional analysis. Dimension analysis plays a very important role in determining the fluid behavior in various engineering applications such as heat transfer, fluid mechanics, Geology, chemical engineering, aeronautical engineering etc.

We will discuss here the various dimensionless numbers in this post and simultaneously we will also note here the importance of dimensionless number in various engineering applications as listed above.

Let us see what is a dimensionless number?

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In order to secure the qualitative behavior of a physical problem, we will have to understand the concept of dimensional analysis. Dimension analysis plays a very important role in determining the fluid behavior in various engineering applications such as heat transfer, fluid mechanics, Geology, chemical engineering, aeronautical engineering etc.

We will discuss here the various dimensionless numbers in this post and simultaneously we will also note here the importance of dimensionless number in various engineering applications as listed above.

Let us see what is a dimensionless number?

Dimensionless numbers could also be expressed as non
dimensional parameters because these are basically secured from ratio of one
force to another force and therefore such dimensionless numbers will not have
any unit. There are various important dimensionless numbers that we will
discuss here one by one.

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**Reynolds
number**

Reynolds number of a flowing fluid could be defined
as the ratio of inertia force and viscous or friction force. We will be able to
determine the type of flow i.e. laminar flow, transient flow or turbulent flow
on the basis of Reynolds number.

- Fluid flow will be laminar, if Re < 2300
- Fluid flow will be transient, if Reynolds number is in between 2300 and 4000
- Fluid flow will be Turbulent, if Re > 4000

Reynolds number will be expressed by a symbol Re and
will be defined mathematically as mentioned here.

Re = Inertia force / Viscous force

Reynolds Number (Re) = ρ V L / μ

Where,

ρ is density of flowing fluid, V is velocity , L is
characteristic length and μ is dynamic viscosity of fluid.

If fluid is flowing through a pipe or duct,
Characteristics length L will be replaced by hydraulic diameter D. Therefore Reynolds
number for a flowing fluid through a pipe could be expressed as mentioned her

Reynolds Number (Re) = ρ V D / μ

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**Froude
number **

Froude number of a flowing fluid is also a
dimensionless number which is basically defined as the square root of ratio of the
inertia force to gravity force and it is used in engineering applications in
order to express the influence of gravity over motion of fluid.

Froude number will be expressed by a symbol Fr and
will be defined mathematically as mentioned here.

Fr = [Inertia force / gravity force]

^{ 1/2}
Fr = V/ (L g)

^{1/2}
Where,

V is velocity, L is characteristic length and g is
acceleration due to gravity

Let us see the three cases of Froude number as
mentioned here

- Fr<1, Subcritical flow
- Fr>1, Supercritical flow
- Fr= 1, Critical flow

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**Euler
number**

Euler number of a flowing fluid is also a
dimensionless number which is basically defined as the ratio of pressure forces
to inertia forces and Euler number is quite important in determining the fluid
flow dynamic problem where difference of pressure between two points are very
important.

Euler number will be expressed by a symbol E

_{U}and will be defined as mentioned here
E

_{U}= P/ ρV^{2}
Where,

V is velocity of fluid flow, P is characteristic pressure
and ρ is density of fluid

Generally, Pressure difference between two points
are used in determining the Euler number, therefore we can express the above
equation of Euler number in term of following pattern

E

_{U}= ΔP/ ρV^{2}###
**Weber
number**

Weber number of a flowing fluid is also a
dimensionless number which indicates the ratio of inertia force to surface
tension force. Weber number is quite important in determining the dominant
energy between kinetic energy and surface tension energy. Weber number could
also be used successfully in determining the thin film problems and also in
problems of formation of droplets.

Weber number will be expressed as mentioned here

We = ρV

^{2}L/ σ
Where,

V is velocity of fluid flow, L is characteristic length,
σ is surface tension and ρ is density of fluid.

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**Mach
Numbers**

Mach number is also a very important dimensionless
number which is widely used in fluid flow dynamic problems where compressibility
plays a very important role. Mach number of flowing fluid will be defined as
the square root of ratio of the inertia force to elastic force and we can write
it as mentioned here.

M = [Inertia force / Elastic force]

^{ 1/2}
M = V/ (K/ρ)

^{ 1/2}
In some higher speed fluid flow problems, density
will be dependent over pressure and hence effect of compressibility will be
quite important in such fluid flow dynamic problems.

Mach number, in such fluid flow dynamic problems, will
be used to determine the type of flow i.e. compressible or incompressible flow
and will be expressed as mentioned here

M= V/C

C= (K/ρ)

^{ 1/2}
Where,

V is velocity of fluid flow, C is speed of sound and
K is elastic stress

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