ce review

FLUID MECHANICS

REVIEWREVIEW 2nd semester, SY 2014-20152nd semester, SY 2014-2015

Classification of matter 2Classification of matter 2

• Solids– tightly packed, usually in a regular pattern– retains a fixed volume and shape – not easily compressible – doesn’t easily flow

Liquids close together with no regular arrangement assumes the shape of the part of the container

which it occupies not easily compressible, flow easilyflow easily

Gas well separated with no regular arrangement assumes the shape of the part of the container easily compressible, flow easilyflow easily

Images from:http://www.chem.purdue.edu/gchelp/liquids/character.html

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What’s in store for us? 3What’s in store for us? 3

Fluid statics

• Density

• Pressure

• Buoyancy

Fluid dynamics

• Continuity equation

• Bernoulli’s equation

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Properties of fluids 4Properties of fluids 4

FLUID PROPERTIES

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Density, ρ (rho)

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Density

M

V

Units :

1 kg/m3 = 10-3 g/cm3

Material Density, kg/m3

Air (1 atm, 200 C) 1.20

Water (1 atm, 40 C) 1.000 x 103

Ice 0.917 x 103

Blood 1.060 x 103

Seawater 1.024 x 103

Styrofoam 1 x 102

Gold 19.3 x 103

• Density may vary from point to point

• Higher ρ sinks under lower ρ

• Solids and liquids: ρ independent of T & P

Gases: strongly dependent on T & P

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Specific weight (unit weight), γ (gamma)

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Unit weight of Water (1 atm, 40 C) 9810 N/m3


Specific gravity/relative density

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Specific gravity/relative density

• Specific gravity is dimensionless.

S > 1 object sinks under water

S < 1 object floats over water

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Ideal Gas Law

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Properties of fluids 11Properties of fluids 11REVIEWREVIEW


Viscosity of Water (200 C) 10-3 N.s/m2


Temperature Dependency

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• The effect of temperature on viscosity is different for liquids and gases.

• The viscosity of liquids decreases as the temperature increases, whereas the viscosity of gases increases with increasing temperature; this trend is also true for kinematic viscosity


Kinematic Viscosity of Water (200 C) 10-6 m2/s


Bulk modulus of elasticity, Ev

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Vapor pressure

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• The pressure at which a liquid will vaporize, or boil, at a given temperature.

• Vapor pressure increases with temperature.


Problem 1.

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• Calculate the density and specific weight of nitrogen at an absolute pressure of 1 MPa and a temperature of 40°C. (Gas constant, R, for nitrogen = 297 J/kg·K)

• Answer: ρ = 10.75 kg/m3γ = 105.4 N/m3


Problem 2.

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• A reservoir of carbon tetrachloride (CCl4) has a mass of 500 kg and a volume of 0.35 m3. Find the carbon tetrachloride’s weight, mass density, specific weight, and specific gravity. (γCCl4=15.57)

• Answer: W = 4905 N ρ = 1587 kg/m3γ = 15.57 kN/m3 s = 1.59

Fluid Statics 20Fluid Statics 20

Principles of Hydrostatics

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Fluid Pressure 21Fluid Pressure 21

Pressure

A

Fp

Useful Units :

1 Pa = 1 N/m2

1 atm = 101,325 Pa = 760 Torr =1,013 mbar

• Fluid exerts a force at each point on the surface of an object in

contact with it.

• Force is perpendicular to the object surface

• Pressure has no preferred direction (scalar)

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Fluid pressure 22Fluid pressure 22

At equilibrium, the pressure in a fluid of uniform density depends depends only on the depthonly on the depth, NOT THE SHAPE, of the container.

hpp 12

p2 = pressure at some depth h2

p1 = pressure at some depth h1γ = unit weight of the fluid

h = difference in depth between h2 and h1.

* With the assumption that g is uniform all throughout the fluid.

• Pressure below > Pressure above

• Pressure is the same at all points at the same depth of the fluid.

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For a homogeneoushomogeneous fluid in an open container, the pressure is the same at a given depth independent of the container’s shape.

p(y)

y

Differences in fluid pressure at the same elevation will arise only if the densities are different.

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Example:• A U tube contain immiscible liquids of

density 1 and 2. Compare the densities of the liquids.

1

2d

dhgpp 10

At the bottom, both liquids have the same pressure.At the top, both are in equilibrium with the atmosphere.At the interface, both have the same pressure as well.So from the Pressure-Depth relation:

ghpp 20

h

dhgpghp 1020

dhh 12 12 h

dh 12

hdh

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Pressure applied to fluid 26Pressure applied to fluid 26

Pascal’s Principle

“PPressure applied to an enclosed fluid is transmitted

undiminished to every portion of the fluid and to the walls of the

containing vessel.”

By adding more weight at the top, the pressure also increases proportionally within the fluid.

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Pascal’s principle 27Pascal’s principle 27

Application: Hydraulic lift

Small applied force, F1

A1

1

1

A

Fp

p is transmitted through the larger piston

11

22 F

A

AF

Larger than F1

2

2

A

Fp

1

1

A

F

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Problem 3.

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A hydraulic jack has the dimensions shown. If one exerts a force F of 100 N on the handle of the jack, what load, F2, can the jack support? Neglect lifter weight.

Answer: F2=12.2 kN

Measuring pressure 29Measuring pressure 29

Pressure Measurements

5 scientific instruments for measuring pressure:

1.Barometer – A mercury barometer is made by inverting a mercury-filled tube in a container of mercury.

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2.Bourdon-tube gage - measures pressure by sensing the deflection of a coiled tube.

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3.Piezometer - is a vertical tube, usually transparent, in which a liquid rises in response to a positive gage pressure

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4.Manometer - often shaped like the letter “U,” is a device for measuring pressure by raising or lowering a column of liquid

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5.Transducer - is a device that converts pressure to an electrical signal. Modern factories and systems that involve flow processes are controlled automatically, and much of their operation involves sensing of pressure at critical points of the system

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Problem 4.

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Pressure measurements are always done with respect to the pressure of the surroundings.

Pabsolute = total pressure

Patm = atmospheric pressure

Pgage = pressure excess of atmospheric

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atmabsolutegauge PPP


Uniform Pressure Distribution• A plane surface or panel is a flat surface of arbitrary

shape. • A description of the pressure at all points along a

surface is called a pressure distribution. • When pressure is the same at every point, the

pressure distribution is called a uniform pressure distribution.

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Uniform Pressure Distribution

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Hydrostatic Pressure Distribution

- is produced by a fluid in hydrostatic equilibrium- is linear and that the arrows representing

pressure act normal to the surface

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Hydrostatic pressure 39Hydrostatic pressure 39

Problem 5.

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Determine the force acting on one side of a concrete form 2.44 m high and 1.22 m wide that is used for pouring a basement wall. The specific weight of concrete is 23.6 kN/m3.


Problem 6.

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An elliptical gate covers the end of a pipe 4 m in diameter. If the gate is hinged at the top, what normal force F is required to open the gate when water is 8 m deep above the top of the pipe and the pipe is open to the atmosphere on the other side? Neglect the weight of the gate.


Forces on Curved Surfaces

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Fx = FACFy=W+FCB


Problem 7.

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Surface AB is a circular arc with a radius of 2 m and a width of 1 m into the paper. The distance EB is 4 m. The fluid above surface AB is water, and atmospheric pressure prevails on the free surface of the water and on the bottom side of surface AB. Find the magnitude and line of action of the hydrostatic force acting on surface AB.

Buoyancy 43Buoyancy 43

Buoyancy• A buoyant force is defined as the upward

force that is produced on a body that is totally or partially submerged in a fluid when the fluid is in a gravity field.

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Buoyancy• Apparent weight loss of an object when totally/partially

immersed in a fluid

FB

Lower Pressure

Higher Pressure

mg

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Question:Based on the summation of forces, therefore, what makes an object sink, float or hover?

Sink(accelerate downwards)

Float(accelerate upwards)

Hover(stay at the same level)

objB WF

objB WF

objB WF

objf

objf

objf

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Example:What fraction of the iceberg afloat in seawater is visible from

the surface?

VVicebergiceberg = total volume of iceberg = total volume of iceberg

VVfluid,disp.fluid,disp. = equal to the submerged portion of the iceberg = equal to the submerged portion of the iceberg

= V= Vsubsub

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Buoyant Force equation

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• Hence, the buoyant force equals the weight of liquid that would be

needed to occupy the volume . This volume is called the displaced

volume.

• If the body is totally submerged, the displaced volume is the volume of

the body.

• If a body is partially submerged, the displaced volume is the portion of

the volume that is submerged.

• For a fluid of uniform density, the line of action of the buoyant force

passes through the centroid of the displaced volume.


Archimedes’ Principle

““WWhen a body is fully or partially submerged in a fluid, a hen a body is fully or partially submerged in a fluid, a buoyant force from the surrounding fluid acts on the buoyant force from the surrounding fluid acts on the body.” body.”

““TThe buoyant force is directed he buoyant force is directed UPWARDUPWARD and has a and has a magnitude equal to the magnitude equal to the WEIGHTWEIGHT of the of the displaced FLUIDdisplaced FLUID by the body.”by the body.”

• The line of action of FB passes through the CG of the displaced fluid, which doesn’t necessarily coincide with the CG of the submerged object

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Immersed Bodies• When a body is completely immersed in a liquid, its stability

depends on the relative positions of the center of gravity of the body and the centroid of the displaced volume of fluid, which is called the center of buoyancy.

• If the center of buoyancy is above the center of gravity, any tipping of the body produces a righting couple, and consequently, the body is stable.

• If the center of gravity is above the center of buoyancy, any tipping produces an increasing overturning moment, thus causing the body to turn through 180°.

• If the center of buoyancy and center of gravity are coincident, the body is neutrally stable—that is, it lacks a tendency for righting or for overturning

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Immersed Bodies• Conditions of Stability

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Floating Bodies• Consider the cross section of a ship. Here the center of gravity G is above

the center of buoyancy C. Therefore, at first glance it would appear that the ship is unstable and could flip over. However, notice the position of C and G after the ship has taken a small angle of heel.

• The center of gravity is in the same position, but the center of buoyancy has moved outward of the center of gravity, thus producing a righting moment. A ship having such characteristics is stable.

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