fluid mechanics chapter 2 - e-learning.kku.ac.th

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Fluid Mechanics

Asst.Prof.Denpong Soodphakdee

Department of Mechanical Engineering

Chapter 2Fluid Statics

ผศ.ดร.เด่นพงษ์ สุดภักดีภาควิชาวิศวกรรมเครื่องกล คณะวิศวกรรมศาสตร์ มหาวิทยาลัยขอนแก่น [2]

Fluid Statics

Important assumption is there is no shearing stress present in a fluid at rest

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Basic Equation for Pressure Field


Pressure Variation in a Fluid at rest

For a fluid at rest a = 0, then

or in component form

or

k̂ 0p

0 0p p p

x y z

dp

dz

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Pressure Variation in a Fluid at rest --- Incompressible Fluid

Integrate previous equation

yield

2 2

1 1

p z

p zdp dz

2 1 2 1 1 2 2 1 or p p z z p p z z



1 2p p h

1 2or p h p

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The pressure different between two points can be specified by the distance, h since

The pressure head, h, is the height of a column of fluid that would give the specified pressure difference.

1 2p ph



In a work that fluid is expose to atmosphere at free surface, it is convenient to use free surface pressure as a reference pressure, p0

0p h p

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Standard Atmosphere

Temperature 288.15 K

Pressure 101.33 kpa (abs)

Density 1.225 kg/m3

Specific weight 12.014 N/m3

Viscosity 1.789 × 10-5 N.s/m2


Variation of Temperature with Altitude

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Measurement of Pressure

Pressure is designate as either absolute pressure or gage pressure


Manometry

Manometers use vertical or inclined liquid columns to measure pressure

1 1Ap h 2 2 1 1Ap h h

Pie

zom

ete

r T

ub

e

U-T

ub

e M

an

om

ete

r

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Manometry

Manometer are often used to measure the difference in pressure between two points

1 1 2 2 3 3

2 2 3 3 1 1

or

A B

A B

p h h h p

p p h h h


Inclined-Tube Manometer

Inclined-tube manometer can be used to measure small pressure differences accurately

2 2 2

2

sin or sin

A BA B

p pp p

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Pressure Gage


Hydrostatic Force on a Plane Surface

When a surface submerged in a fluid, forces develop on the surface due to fluid.Resultant force is important in the design of storage tanks, ships, dams, and other hydraulic structure.For fluid at rest, the force is perpendicular to the surface since there are no shearing stresses present.For horizontal surface, the magnitude of resultant force is simply, FR=pA.

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dF pdA hdA

sin

sin

RA

A

A

F hdA

y dA

ydA

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cA

ydA y A is the

first moment of area.

where yc is the ycoordinate of the centroid.

sin

or

R c

R c

F Ay

F h A



hc is the vertical distance from the fluid surface to the centroid of the area.The magnitude of the resultant force is depend only on the specific weight of the fluid, the total area, and the depth of the centriod of the area. ORThe magnitude of the resultant force is equal to the pressure at the centriod of the area multiply by the total area.

The line of action of the resultant force is NOTpass through the centriod of the area.

R cF h A

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The y coordinate of the resultant force can be determined by summation of moments around the x-axis.

2

2

2

sin

sin

since sin therefore

R RA A

A

R c

AR

c

F y ydF y dA

y dA

F Ay

y dAy

y A



The integral term is the second moment of the area (moment of inertia), Ix, thus

xR

c

Iy

y A

2by parallel axis theorem, x xc cI I Ay

xcR c

c

Iy y

y A

Similarly for xR,

sin

therefore

R RA

xyAR

c c

F x xydA

xydA Ix

y A y A

by parallel axis theorem

xyc

R c

c

Ix x

y A

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Pressure Prism

The pressure prism is a geometric representation of the hydrostatic force on a plane surface.

When we draw pressure variation on depth, h, acting on area, A, the volume is the pressure prism.

The magnitude of the resultant fluid force is equal to the volume of the pressure prism and passes through its centroid.

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Pressure Prism

1

volume2 2

R

hF h bh A


Pressure Prism

1 2RF F F 1 1 2 2R AF y F y F y

Magnitude: Line of action:

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Pressure Prism

Pressure prism for inclined surface

The resultant fluid force acting on a submerged area is affected by the pressure at free surface.


Hydrostatic Force on a Curved Surface

The development of a free-body diagram of a suitable volume of fluid can be used to determine the resultant fluid force acting on a curved surface.

Although the resultant fluid force can be determined by integration, as was done for plane surface, this is generally a rather tedious process and no simple general formulas can be developed for nonplanar surface.

Movies/V2_4.mov

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Hydrostatic Force on a Curved Surface

An alternative approach is the consideration of the equilibrium of the fluid volume enclosed by the curved surface of interest and the horizontal and vertical projectionof this surface.

2

1

2 2

H

V

R H V

F F

F F

F F F

W


Buoyancy

2 1BF F F W 2 1 2 1where F F h h A

2 1 2 1BF h h A h h A V

BF V

Archemedes’ principlestates that the buoyant force has a magnitude equal to the weight of the fluid displaced by the body and is directed vertically upward.

Movies/V2_5.mov

Movies/V2_6.mov

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Stability

Submerged or floating bodies can be either in a stable or unstable position.

The Stability of a body can be determined by considering what happens when it is displaced from its equilibrium position.

Submerged body


Stability

The determination of the stability of submerged or floating bodies can be difficult since the analysis depends in a complicated fashion on the particular geometry and weight distribution of the body.

Floating body

Movies/V2_7.mov

fluid mechanics chapter 2 - e-learning.kku.ac.th

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