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Fluid Mechanics EM 208

- Chapter 3
- Fluid Statics

Contents

- The Basic Equation of Fluid Statics
- The Standard Atmosphere
- Pressure Variation in a Static Fluid
- Hydraulic Systems
- Hydrostatic Force on Submerged Surfaces
- Buoyancy and Stability
- Fluids in Rigid-Body Motion (on the Web)
- Summary and Useful Equations

The Basic Equation of Fluid Statics

- The first objective of this chapter is to obtain

an equation for computing the pressure field in a

static fluid. - We will deduce what we already know from everyday

experience, that the pressure increases with

depth. - To do this, we apply Newtons second law to a

differential fluid element of mass dm with sides

dx, dy, and dz, as shown

The Basic Equation of Fluid Statics

The Basic Equation of Fluid Statics

Body Forces

- two general types of forces maybe applied to a

fluid - body forces
- and surface forces.
- The only body force that must be considered in

most engineering problems is due to gravity

Surface Forces

The basic equation of Fluid statics

The basic equation of Fluid statics

The basic equation of Fluid statics

The basic equation of Fluid statics

The basic equation of Fluid statics

- pressure values must be stated with respect to a

reference level. - If the reference level is a vacuum, pressures are

termed absolute - Most pressure gages indicate a pressure

differencethe difference between the measured

pressure and the ambient level (usually

atmospheric pressure) - Pressure levels measured with respect to

atmospheric pressure are termed gage pressures

Absolute and gage pressures, showing reference

levels

- For example, a tire gage might indicate 30 psi

the absolute pressure would be about 44.7 psi. - Absolute pressures must be used in all

calculations with the ideal gas equation or other

equations of state

The Standard Atmosphere

Pressure Variation in a Static Fluid

- Above Equation indicates that the pressure

difference between two points in a static

incompressible fluid can be determined by

measuring the elevation difference between the

two points. - Devices used for this purpose are called

manometers

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Most important equation in Fluid statics

computations

Step 1 Assign letters to all points

Step 1 Assign letters to all points

Step 2 Write governing equation for each of two

point set

Step 1 Assign letters to all points

Step 3 Obtained as follows

Step 1 Assign letters to all points

Step 4 Solve it all

Gases

- The density of gases generally depends on

pressure and temperature. The ideal gas equation

of state, - and the pressure variation, in a gas whose

temperature varies linearly with elevation, is

given by

TT0-mz for linear variation of Temp with height

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Review of Centroid

Introduction

- The earth exerts a gravitational force on each of

the particles forming a body. These forces can

be replace by a single equivalent force equal to

the weight of the body and applied at the center

of gravity for the body.

- The centroid of an area is analogous to the

center of gravity of a body. The concept of the

first moment of an area is used to locate the

centroid.

- Determination of the area of a surface of

revolution and the volume of a body of revolution

are accomplished with the Theorems of

Pappus-Guldinus.

Center of Gravity of a 2D Body

Centroids and First Moments of Areas and Lines

First Moments of Areas and Lines

- The first moment of an area with respect to a

line of symmetry is zero.

- If an area possesses a line of symmetry, its

centroid lies on that axis

- The centroid of the area coincides with the

center of symmetry.

Centroids of Common Shapes of Areas

Hydraulic Force on Submerged surfaces

- We must know Magnitude of Force, Direction of

Force and Line of Action of Force

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where pc is the absolute pressure in the liquid

at the location of the centroid of area A.

Our next task is to determine (x, y), the

location of the resultant force

- Lets first obtain y by recognizing that the

moment of the resultant force about the x axis

must be equal to the moment due to the

distributed pressure force

Our next task is to determine (x, y), the

location of the resultant force

Our next task is to determine (x, y), the

location of the resultant force

Our next task is to determine (x, y), the

location of the resultant force

- Equation 3.11a is the integral equation for

computing the location y of the resultant force

- Eq. 3.11b is a useful algebraic form for

computing y when we are interested in the

resultant force on the submerged side of the

surface - Eq. 3.11c is for computing y when we are

interested in the net force for the case when the

same p0 acts at the free surface and on the other

side of the submerged surface

Similarly we could get the X of the location of

the resultant force

- Equation 3.11a is the integral equation for

computing the location y of the resultant force

- Eq. 3.11b is a useful algebraic form for

computing y when we are interested in the

resultant force on the submerged side of the

surface - Eq. 3.11c is for computing y when we are

interested in the net force for the case when the

same p0 acts at the free surface and on the other

side of the submerged surface

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For resultant Force

For Location of Force

Using Algebraic equations

Using Algebraic equations

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Summary

- We reviewed basic concepts including
- Deriving basic equation in vector form
- Applying this equation to determine pressure

variation in a static fluid - Incompressible fluids pressure increases

uniformly with depth - Gases pressure decreases non uniformly with

elevation - We studied gage and absolute pressures
- Use of manometers and barometers
- Analysis of fluid force magnitude and location in

submerged plane and curved surfaces - Principle of Buoyancy

Useful equations

Useful equations

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