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Fluid in Motion

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Kinematics of Fluid Mechanics

Fluid Motion

Flow Field

Stream lines

(Displacement, velocity, acceleration, ..etc) .

Flow

Non-viscous Flow Real Flow

(Ideal) (Real)

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Steady and Unsteady Flow

Steady

Unsteady

A steady flow is one in which the conditions (velocity, pressure and cross-section) may differfrom point to point but DO NOT change with time

If at any point in the fluid, the conditions change with time, the flow is described as unsteady

Fluid Valve

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Uniform and Non-uniform flow

Uniform flow: If the flow velocity is the same magnitude and direction at every point in the flow it is said to be uniform. That is, the flow conditions DO NOT change with position.

Non-uniform: If at a given instant, the velocity is not the same at every point the flow is non-uniform.

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Stream Line, Stream Tube and Pathline

vv

v

A stream line is a line that is everywhere tangent to the velocity vector at a given instant of time. A streamline is hence an instantaneous pattern.

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A tubular surface formed by streamlines along which the fluid flows is known as a stream tube, which is a tube whose walls are streamlines. Since the velocity is tangent to a streamline, no fluid can cross the walls of a stream tube.

A pathline is the actual path traversed by a given (marked) fluid particle.

Stream tube

Pathline

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One, Two and Three Dimensional Flows

Stream line, being mathematically line

Stream lines of flow picture are essentially straight and parallel.

One dimensional Flow:

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Two and Three Dimensional Flows:

Stream lines of flow pictures describe a flow field.

Two dimensional in a single plane ( Flow over weir and around wings ).

Flow over weir Flow over and around wings

Two Dimensional Flow

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Two axisymmetric Three dimensional

( Stream line surfaces and stream tubes )

Three dimensional in space.

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Velocity and acceleration:

Streamline

S

0

dS

v

a s

a r

r

c

Fixed point

For one dimensional flow:

dt

dSv

S : is the displacement in time dt.

as : is the tangent component of acceln.

Where:

2

2

s dt

Sda

dt

dS

dt

d

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Sd

dvva s

dt

dS

dS

dv

dt

dva s

and

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For generalization: y

x

u

v V

)x,y(

- For Cartesian Coordinates:

dt

dxu

u = u(x,y) and v = v(x,y)

dt

dua x and

dt

dyv

dt

dva y and

and for steady state condition:

dyy

udx

x

udu

and then

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ydy

vdx

xd

v

v

y

vv

x

vua y

Dividing the above equation by dt, we obtain:

y

uv

x

uua x

Dividing the above equation by dt, we obtain:

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Equation of Continuity

Control volume

A control volume is a finite region, chosen carefully by the analyst for a particular problem, with open boundaries through which mass, momentum, and energy are allowed to cross

conservation of mass

For steady flowMass entering per unit time = Mass leaving per unit time

Mass entering per unit time = Mass leaving per unit time + Change of mass in the control volume per unit time

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Equation of Continuity:

1- For One dimension:A2A1

Flow

ds2ds1

dt

dsAρ

dt

dsAρ 2

221

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Conservation of mass

Dividing by dt, we obtain:

222111 VAρVAρ

V1

V2

ρ1A1ds1= ρ2A2ds2

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.ConstVAρVAρm 222111

Where: is the mass flow rate.m

= ρAV

0ρAVd

mFor constant ρ: A1V1 = A2V2= Q

0V

dV

A

dA

ρ

dρ

Where: Q is the volume flow rate.

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Equation of Continuity in General:

dx dy

dz

x

z

y

ρu dydzρu dydz + (ρu dydz )dx

xIn flow

Out flow

For x-direction

Net flow ρu dydz - [ρu dydz + (ρu) dxdydz]x

= - [(ρu) dxdydz ]x

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For y-direction

Net flow = - [(ρv) dxdydz ]x

For z-direction

Net flow = - [ (ρw) dxdydz ]x

The sum of net flow

- [ (ρu) + (ρv) + (ρw ) ] dxdydzx

x

x

)1(

The rate of change

dzdydxtt

Volume.

t

m

)2(

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The positive rate of change = The sum of the net flow ie; Equation (1) = equation (2)

0z

ρw

y

ρv

x

ρu

t

ρ

0t

ρ

For three dimensional incompressible steady state

0z

w

y

v

x

u

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For two dimensional steady state

0y

v

x

u

For one dimensional steady state

0x

u

This represent the uniform flow, V = Constant and does not change

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Flow of an Incompressible Ideal Fluid

1- ρ = Constant.

2- Viscous forces = 0 ( In-viscous Flow ).

3- Frictionless.

4- Can apply Newton’s law ( F = mass x acceleration)

5- External forces = Effective forces.

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Example-1

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Example-2

If pipe 1 diameter = 50mm, mean velocity 2m/s, pipe 2 diameter 40mm takes 30% of total discharge and pipe 3 diameter 60mm. What are the values of discharge and mean velocity in each pipe?