Fluid Mechanics Chapter 3 Fluid Statics

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Fluid Mechanics EM 208

• Chapter 3
• Fluid Statics

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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

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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

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The Basic Equation of Fluid Statics
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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
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The basic equation of Fluid statics
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The basic equation of Fluid statics
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The basic equation of Fluid statics
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The basic equation of Fluid statics
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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

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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

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The Standard Atmosphere
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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
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Step 1 Assign letters to all points
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Step 1 Assign letters to all points
Step 2 Write governing equation for each of two
point set
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Step 1 Assign letters to all points
Step 3 Obtained as follows
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Step 1 Assign letters to all points
Step 4 Solve it all
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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

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TT0-mz for linear variation of Temp with height
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Review of Centroid
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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.

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Center of Gravity of a 2D Body
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Centroids and First Moments of Areas and Lines
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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.

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Centroids of Common Shapes of Areas
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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.
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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

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Our next task is to determine (x, y), the
location of the resultant force
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Our next task is to determine (x, y), the
location of the resultant force
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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

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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
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For Location of Force
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Using Algebraic equations
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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

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Useful equations
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Useful equations