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2-1 Fluid as a continuum

Continuum means that properties vary smoothly from one point to another, fluid properties such as (density, temperature, velocity, pressure) are continuous function of position & time.

,

Continuum means no voids.

Specific gravity (S.G): is defined as the ratio of the materials density to the maximum density of water.

Specific weight (ɣ): is defined as the weight per unit volume.

2-2 Velocity filed

Steady flow term is used when properties at every point in flow field do not change with time

In mathematics:

Where ( ) represent any fluid properties

One, two and three dimensional flow:-

Flow classified as 1D, 2D, 3D depending on the number of space coordinates required to specify velocity field:

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⇒steady, 2D

2. ⇒steady, 1D

Uniform flow at given cross section the velocity is constant across that section (don’t change with position).

Uniform flow field is used to describe a flow in which the (v) is constant (independent of the space coordinates).

Sometimes we want usual representation of the flow, this provided by path lines, time lines, streak line and stream lines.

1. Timeline: a line that form by joining / marking a number of adjacent fluid particles at constant instant of time.

2. Path line: a line traced out by moving fluid particle.3. Streak line: a line joining fluid particles that passes through affixed location

in space.4. Stream lines: Lines that are tangent to the direction of the flow (they are

tangent to the velocity vector), no flow can cross streamline. For steady flow, path line stream line and streak line are identical

Example:

The velocity field is given by the following equation:

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Find:

(a) The equation for the stream line.(b) The equation of the stream line at point (2, 8).(c) Velocity of the particle at point (2, 8).(d) The particle passing through point (2,8)at t=0, find the location of the

Particle at time 6.(e) Velocity of that particle at t=6.(f) Path line equation at (2, 8).

Solution:

(a) ⇒ ⇒

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(b)

(c)

(d) Using Lagragian method

,

⇒

, where At The location is (12.1, 1.32)

(e)

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(F)

2-3 Stress field

Each fluid particle can experience two kinds of forces:

(a) Surface force :( acting on area)

Example: pressure, friction

Generated by contact with other fluid particles or solid surfaces.

(b) body force :( acting on volume)

Example: gravity, electromagnetic

The stress at a point is subjected by:-

First subscript indicates the normal to the surface.

Second subscript indicates the direction of the shear.

2-4 Viscosity

Where stresses do came from?

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For fluid shear stress arise due to fluid viscous flow.

For fluid at rest there are no shear stresses. rate of deformation (shear rate)

df: the applied forcedu: small change in velocity

Relation between shear rate and shear stresses:

1. Newtonian fluid: is a fluid in which shear stresses is directly proportional to the shear rate.

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Where ( is the dynamic viscosity ( )

Most fluids are Newtonian like (air, gasoline, oils, water, and blood)

2. Non_newtonian fluid: is a fluid in which shear stresses is not directly proportional to the shear rate.

Example: blood, honey, mastered, toothpaste.

2-6 Description and classification of fluid motion

Viscous and invicid flow:-

What is the nature of the drag force of the air in the ball?

1. Due to friction of air as it moves over the ball.

F (drag force)

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2. Pressure builds up in the front of the ball.

Can we predict ahead of time the relative importance of viscous force due to pressure build up?

Yes, by computing Reynolds number

Where:

V: reference velocity of the flow.

D: characteristic length or size of the flow.

Absolute viscosity of the fluid.

: Density of the fluid.

If Re is small, viscous effect will be important (dominant), on the other hand

If Re is large, viscous effect will be negligible.

Example:

Basic equation:

Case (1): ball in air

F(Lift force)

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Viscous effect in this case can be neglected.

Case (2): dust particle in air

⇒viscous effect in this case is significant.

Conclusion:

The flow is considered to be friction dominated not only based on the viscosity of the fluid, but the complete flow system.

Laminar and turbulent flows:-

Laminar flow: is one in which fluid particles move in smooth layer.

Turbulent flow: is one in which there is a rapid mixing between fluid particles as they move due to three dimensional random velocity fluctuation.

Flow of water in the pipe is unwanted turbulent but unavoidable; because it generates more resistance to the flow.

If Re ≤2300 ⇒ turbulent.

Re> 2300 ⇒ laminar.

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Example: water moves in a pipe

Turbulent in the flow of the blood:

Through blood vessel is wanted because it allows random mixing for exchanging

and other nutrients.

For 1D laminar flow

However for turbulent flows there is no simple relation is valid, so we depend heavily in experimental data and semi empirical theories.

Compressible and incompressible flow:-

Incompressible flow is a flow in which variation in density is negligible

(Liquids except at high pressure)

Compressible flow a flow in which variation in density is not negligible. (Gases except at low velocity)

Notes:

(1) Gas flow considered compressible (generally) and liquid flow considered incompressible.

(2) Gas flow at relatively low speed is compared to the speed of sound might be incompressible.

(3) Mach number :

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If (M<0.3) gas flow incompressible ⇒

Example:

Consider flow in parallel plates, the lower plate is fixed, upper plate moving with a constant speed. Find the shear stress at both plates assuming (d) is very small

Sol:

Governing equation

Assumptions:

(1)Linear velocity distribution.(2)Steady flow.

(3) Constant.

Where a and b obtained from boundary conditions.⇒ No slip

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⇒⇒

For direction:At the upper plate the shear stress act in the negative x direction.

At the lower plate the shear stress act in the positive x direction.

Example:A pair of parallel disk is used is to measure the viscosity of the liquid, the upper disk rotates at high H from the lower, obtain an algebraic expression for the torque required to turn the disk??Solution:

, ,

A and B obtained from boundary conditions:

⇒ ⇒

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But

⇒