fm notes problems unit ii

8/3/2019 FM Notes Problems UNIT II

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UNIT II FLUID MECHANICS

1

FLUID MECHANICS

AND

HYDRAULIC MACHINES

K.BALASUNDARAM,M.E.,

ASSISTANT PROFESSOR


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UNIT II

FLUID KINEMATICS AND FLUID DYNAMICS

Velocity and Acceleration - Classification of Flow - Continuity

Equation - Streamline, Streakline, Pathline - Potential Function and Stream

Function - Flow Net Analysis - Control Volume -Euler Equation - Bernoullie's

Equation - Darcy's Equation - Momentum Principle - Free and Forced Vortex

Motion.


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FLOW CHARACTERISTICS

The fluid motion is described by two methods. They are

1.)Lagrangian Method and2.) Eulerian Method.Lagrangian Method

In the Lagrangian method, a signal fluid particle is followed during its motion

and its velocity, acceleration, density, etc.

Eulerian Method.

In the Eulerian method, the velocity, acceleration, pressure, density etc. are

described at a point in flow field. The Eulerian method is commonly used in fluid

mechanics.


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TYPES OF FLUID FLOW

The fluid flow is classified as:

1.) Steady and unsteady flows2.) Uniform and non-uniform flows3.) Laminar and turbulent flows4.) Compressible and incompressible flows5.) Rotational and irrotational flows and6.) One, two and three- dimension flowsSteady flows

Steady flow is defined as that type of flow in which the fluid

characteristics like velocity, pressure, density, etc. at a point do not change with

time. Thus for steady flow,

Unsteady flow

Unsteady flow is that type of flow, in which the velocity, pressure of

density at a point changes with respect to time. For unsteady flow

0d V d p

od t d t

Uniform flow

Uniform flow is defined as that type of flow in which the velocity at any

given time does not change with respect to space


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0d V

d s

Non-uniform flow

Non-uniform flow is that type of flow in which the velocity at any given

times changes with respect to space. For non-uniform flow

0d V

d s

Laminar flows

Laminar flow is defined as that type of flow in which the fluid particles move

along well defined paths or stream line and all the stream-lines are straight andparallel .Thus the particles moves in laminas or layers gliding smoothly over the

adjacent layer. This type of flow is also called stream-line flow or viscous flow.

Turbulent flow

Turbulent flow is that type of flow in which the fluid particles moves in a zig -

zig way .Due to the movement of fluid particles in a zig- zag way, the eddies

formation takes place which are responsible for high energy loss.

REYNOLD NUMBER

The type of flow is determined by a non-dimensional number called the

Reynolds number.

Reynolds number =V D

Where

D = Diameter of pipe

V = Mean velocity of flow in pipe

= Kinematics viscosity of fluid.


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NOTE

If the Reynold number is less than 2000, the flow is called laminar

If the Reynold number is more than 4000, it is called turbulent flow

If the Reynold number lies between 2000 and 4000, the flow may be laminar or

turbulent.

Compressible flows

Compressible flow is that type of flow in which density of the fluid changes

from point to point or in other words the density is not constant for the fluid.

Compressible flow

tanC o n s t

Incompressible flow

Incompressible flow is that type of flow in which the density is constant for

the fluid flow. Liquids are generally incompressible while gases are compressible.

Incompressible flow

a nC o n s t

Rotational Flow.

Rotational flow is that type of flow in which the fluid particles while

flowing along stream-lines, also rotate about their own axis.

Irrotational Flows

If the fluid particles while flowing along stream-lines, do not rotate about

their own axis that type of flow is called irrotational flow.


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One, Two and Three-Dimensional flows.

One- dimensional flow is that type of flow in which the flow parameter

such as velocity is a function of time and space co-ordinate only, say x. For a

steady one-dimensional flow, the velocity is a function of one-space-co-ordinate

only. The variation of velocities in other two mutually perpendicular directions is

assumed negligible. Hence mathematically, for one-dimensional flow

u = f(x), v = 0 and w = 0

Where u, v and w are velocity components in x ,y and z directions respectively.

Two-dimensional flow is that type of flow in which the velocity is a function of

time and two rectangular space co-ordinates say x and y. For a steady two-

dimensional flow the velocity is a function of two space co-ordinates only. The

variation of velocity in the third direction is negligible. Thus, mathematically for

two-dimensional flow

u = f1 ( x, y ) v = f2 ( x, y ) and w = 0.

Three-dimensional flow is that type of flow in which the velocity is a function of

time and three mutually perpendicular directions. But for a steady three-

dimensional flow the fluid parameters are functions of three space co-

ordinates(x, y and z) only.

u=f1(x, y, z ) v= f2 (x, y, z ) w= f3 (x, y. z).

RATE OF FLOW OR DISCHARGE(Q)

It is defined as the quantity of a fluid flowing per second through a section

of a pipe or a channel for an incompressible fluid the rate of flow or discharge is

expressed as the volume of fluid flowing across the section per second.

Let A= Cross-sectional area of pipe

V = Velocity of fluid across the section


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Discharge

Q =AV (m3).

CONTINUNITY EQUATION

The equation based on the principle of conservation of mass is called

continuity equation .Thus for a fluid flowing through the pipe at all the cross-

section, the quantity of fluid per second is constant.

Consider two cross-section of a pipe as shown in fig.

Let

V1 = Velocity at cross section 1-1

1 = Density at section 1-1

A1 = Area of pipe at section 1-1

And V2 , 2 , A2 are corresponding values at section 2-2

According to law of conservation of mass

Rate of flow at section 1-1 = Rate of flow at section 2-2

1A1V1 = 2A 2V 2

This equation is applicable to the compressible as well as incompressible fluids

and is called continuity equation.


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If the fluid is incompressible, then

1 = 2 and continuity equation reduces to

A1V1 = A 2V 2


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THE MOMENTUM EQUATION

It is based on the law of conversation of momentum or on the momentum

principle,

It states that

the net force acting on a fluid mass is equal to the change in momentum of

flow per unit time in that direction. The force on a fluid mass m is given by the

Newtons second law of motion .

F = m a

Where

M= mass

A = acceleration

We know that


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a=v

t

F = mdv

dt

F = d mv

t

F . dt = d(mv).------ impulse momentum equation

Which is know as the impulse momentum equation and states that the impulse

of a force F acting on a fluid of mass m in a short interval of time it is equal to thechange of momentum d(mv) in the direction of force.


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fm notes problems unit ii

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