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Significant dimensionless groups in fluid mechanics

Forces encountered in flowing fluids include those due to

inertia, viscosity, pressure, gravity, surface tension and

compressibility

Ratio of any two forces will be dimensionless

Inertia force =ma V2L2

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Inertia forces are important in fluid mechanics problems

The ratio of inertia force to each of the other forces listedabove leads to five fundamental dimensionless groups

Reynolds no.

Re after the name of the scientist who first developed it and is

thus proportional to the magnitude ratio of inertia force to

viscous force

He studied the transition between laminar and turbulent flow

regimes in a tube Re is the criterion by which the flow regime may be

determined

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Later experiments have shown that Re is a key parameter for

other flow cases as well, thus in general

WhereL is the characteristic length descriptive of the flow

geometry

Euler no. Euler is credited with being first to recognize the role of

pressure in fluid motion

The Euler no. is the ratio of pressure forces to inertia forces

Euler no. is often called thepressure coefficient, Cp

(p is the local pressure minus freestream pressure)

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Froude no.

This is significant for flows with free stream effects

Squaring the Froudes no. gives,

Which may interpreted as the ratio of inertia forces to viscous

forces.

The length L is the characteristic length descriptive of the flow

field In case of open channel flow, L is water depth;

Fr < l indicate subcritical flow, Fr > 1 indicate super critical

flow

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The dimensionless termsRe andEu represent the criteria of

dynamic similarity for the flows which are affected only by

viscous, pressure and inertia forces. Such instances, forexample, are

the full flow of fluid in a completely closed conduit,

flow of air past a low-speed aircraft and

the flow of water past a submarine deeply submerged to produceno waves on the surface.

Hence, for a complete dynamic similarity to exist between the

prototype and the model for this class of flows, the Reynolds

number, Re and Euler number, Eu have to be same for the two

(prototype and model). Thus

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Weber no.

Is the ratio of inertia forces to surface tension forces

(named after the German naval architect Moritz Weber who

first suggested the use of this term as a relevant parameter)

Mach no.

Where V is the flow speed and c is the local sonic speed

Key parameter that characterizes compressibility effects in a

flow

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The Mach no. may be written

It can be interpreted as the ratio of inertia forces to

compressibility forces

For truly incompressible flow (under some conditions even

liquids are quite compressible), c= so that M=0

The effects of compressibility become important when theMach no. exceeds 0.33

The situation arises in the flow of air past high speed aircraft,

missiles, propellers and rotary compressors

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