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

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Ratios of the forces are dimensionless numbers. Therefore we can say that for the dynamic similarity between the model and the prototype, dimensionless numbers should be equal for the model and the prototype.

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As we know that Froude number is basically the ratio of inertia force and gravity force, therefore a fluid flow situation where gravity forces are alone predominant, models will be designed on the basis of Froude model law for the dynamic similarity between the model and the prototype.
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Now we
will go ahead to understand the basic concept of model laws or similarity laws in
the subject of fluid mechanics with the help of this post.

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**Model laws or similarity laws**

For the
dynamic similarity between the model and the prototype, ratio of corresponding
forces acting on corresponding points in the model and the prototype should be
same.

Ratios of the forces are dimensionless numbers. Therefore we can say that for the dynamic similarity between the model and the prototype, dimensionless numbers should be equal for the model and the prototype.

However,
it is quite difficult to satisfy the condition that all the dimensionless
numbers should be equal for the model and the prototype.

However for
practical problems, it is observed that one force will be most significant as
compared to others and that force is considered as predominant force. Therefore
for dynamic similarity, predominant force will be considered in practical
problems.

Therefore,
models are designed on the basis of ratio of force which is dominating in the phenomenon.

Hence, we can
define the model laws or similarity laws as the law on which models are designed
for the dynamic similarity.

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**There are following types of model
laws**

Reynold’s
Model law

Froude Model
law

Euler
Model law

Weber
Model law

Mach Model
law

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**Reynold’s Model law**

Reynold’s
model law could be defined as a model law or similarity law where models are
designed on the basis of Reynold’s numbers.

According
to the Reynold’s model law, for the dynamic similarity between the model and the
prototype, Reynold’s number should be equal for the model and the prototype.

In simple,
we can say that Reynold’s number for the model must be equal to the Reynold’s number
for the prototype.

As we know
that Reynold’s number is basically the ratio of inertia force and viscous
force, therefore a fluid flow situation where viscous forces are alone
predominant, models will be designed on the basis of Reynold’s model law for the
dynamic similarity between the model and the prototype.

Image: Reynold’s
model law

Where,

V

_{m}= Velocity of the fluid in the model
L

_{m}= Length of the model
ν

_{m}= Kinematic viscosity of the fluid in the model
V

_{P}= Velocity of the fluid in the prototype
L

_{P}= Length of the prototype
ν

_{P}= Kinematic viscosity of the fluid in the prototype####
**Models based on the Reynold’s model
law**

Pipe flow

Resistance
experienced by submarines, airplanes etc.

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**Froude Model law**

Froude model
law could be defined as a model law or similarity law where models are designed
on the basis of Froude numbers.

According
to the Froude model law, for the dynamic similarity between the model and the prototype,
Froude number should be equal for the model and the prototype.

In simple,
we can say that Froude number for the model must be equal to the Froude number for
the prototype.

As we know that Froude number is basically the ratio of inertia force and gravity force, therefore a fluid flow situation where gravity forces are alone predominant, models will be designed on the basis of Froude model law for the dynamic similarity between the model and the prototype.

Image: Froude
model law

Where,

V

_{m}= Velocity of the fluid in the model
L

_{m}= Length of the model
g

_{m}= Acceleration due to gravity at a place where model is tested
V

_{P}= Velocity of the fluid in the prototype
L

_{P}= Length of the prototype
g

_{P}= Acceleration due to gravity at a place where prototype is tested####
**Models based on the Froude model law**

Free
surface flows such as flow over spillways, weirs, sluices, channels etc,

Flow of
jet from an orifice or from a nozzle,

Where
waves are likely to be formed on surface

Where
fluids of different densities flow over one another

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**Euler’s Model law**

Euler’s model
law could be defined as a model law or similarity law where models are designed
on the basis of Euler’s numbers.

According
to the Euler’s model law, for the dynamic similarity between the model and the prototype,
Euler’s number should be equal for the model and the prototype.

In simple,
we can say that Euler’s number for the model must be equal to the Euler’s number
for the prototype.

As we know
that Euler’s number is basically the ratio of pressure force and inertia force,
therefore a fluid flow situation where pressure forces are alone predominant,
models will be designed on the basis of Euler’s model law for the dynamic
similarity between the model and the prototype.

Image: Euler’s
model law

Where,

V

_{m}= Velocity of the fluid in the model
P

_{m}= Pressure of fluid in the model
ρ

_{m}= Density of the fluid in the model
V

_{P}= Velocity of the fluid in the prototype
P

_{P}= Pressure of fluid in the prototype
ρ

_{P}= Density of the fluid in the prototype####
**Models based on the Euler’s model
law**

Euler’s model
law will be applicable for a fluid flow situation where flow is taking place in
a closed pipe, in which case turbulence will be fully developed so that viscous
forces will be negligible and gravity force and surface tension force will be
absent.

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**Weber Model law**

Weber model
law could be defined as a model law or similarity law where models are designed
on the basis of Weber numbers.

According
to the Weber model law, for the dynamic similarity between the model and the prototype,
Weber number should be equal for the model and the prototype.

In simple,
we can say that Weber number for the model must be equal to the Weber number for
the prototype.

As we know
that Weber number is basically the ratio of inertia force and surface tension
force, therefore a fluid flow situation where surface tension forces are alone
predominant, models will be designed on the basis of Weber model law for the
dynamic similarity between the model and the prototype.

Image: Weber
model law

Where,

V

_{m}= Velocity of the fluid in the model
σ

_{m}= Surface tension force in the model
ρ

_{m}= Density of the fluid in the model
L

_{m}= Length of surface in the model
V

_{P}= Velocity of the fluid in the prototype
σ

_{P}= Surface tension force in the prototype
ρ

_{P}= Density of the fluid in the prototype
L

_{P}= Length of surface in the prototype####
**Models based on the Weber model law**

Capillary
rise in narrow passage

Capillary
movement of water in soil

Capillary waves
in channels

Flow over
weirs for small heads

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**Mach Model law**

Mach model
law could be defined as a model law or similarity law where models are designed
on the basis of Mach numbers.

According
to the Mach model law, for the dynamic similarity between the model and the prototype,
Mach number should be equal for the model and the prototype.

In simple,
we can say that Mach number for the model must be equal to the Mach number for
the prototype.

As we know
that Mach number is basically the ratio of inertia force and Elastic force,
therefore a fluid flow situation where elastic forces are alone predominant,
models will be designed on the basis of Mach model law for the dynamic
similarity between the model and the prototype.

Image: Mach
model law

Where,

V

_{m}= Velocity of the fluid in the model
K

_{m}= Elastic stress for model
ρ

_{m}= Density of the fluid in the model
V

_{P}= Velocity of the fluid in the prototype
K

_{P}= Elastic stress for prototype
ρ

_{P}= Density of the fluid in the prototype####
**Models based on the Mach model law**

Water
hammer problems

Under
water testing of torpedoes

Aerodynamic
testing

Flow of
aeroplane and projectile through air at supersonic speed

We will discuss another important topic i.e. Euler's Equation of motions in the subject of fluid mechanics in our next post.

Do you
have any suggestions? Please write in comment box.

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**Reference:**

Fluid
mechanics, By R. K. Bansal

Image Courtesy: Google