MEE2026
FLUID MECHANICS
− Chapter III −
Fluid Statics
Dokuz Eylül University
MA Ezan, A Erek, E Alptekin
Department of Mechanical Engineering
makina.deu.edu.tr
/school/deumak
Fluid Mechanics − Chapter III: Fluid Statics
Ezan, Erek, Alptekin
Content of the Course
Chapter 1: Foundations
Chapter 2: Fluid Properties
Chapter 3: Fluid Statics
Chapter 4 / Part I: Fluid in Rigid Body Motion
Chapter 4 / Part II: Flowing Fluids – Fluid Kinematics
Chapter 4 / Part III: Bernoulli Equation
Chapter 5: Control Volume Approach & Continuity
Chapter 6: Momentum Equation
Chapter 7: The Energy Equation
Chapter 8: Dimensional Analysis & Similitude
Chapter 9: Flow in Conduits
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Objectives of Chapter III
Describing the Pressure
Absolute pressure, Gage pressure
Pressure Variation with Elevation
Hydrostatic Differential Equation, Piezometric Pressure / Head
Measuring Pressure
Barometer, Bourdon-Gage, Piezometer, Manometer, Transducers
Predicting Forces on PLANE Surfaces
Resultant Force, Centroid, Center of Pressure
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Pressure – Definition
Pressure is defined as a normal force exerted by a fluid per unit area
Pressure is a scalar quantity, not a vector; the pressure at a point in a fluid is
the same in all directions
The pressure at a point in a fluid
at rest is independent of direction
Blaise Pascal (1623– 1662)
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Pressure at a Point – Pascal’s Law
“the pressure at a point in a fluid at rest, or in motion, is independent of direction as long as
there are no shearing stresses present”
WHY?
PROOF
y-direction
z-direction
from geometry
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Pressure at a Point – Pascal’s Law
“the pressure at a point in a fluid at rest, or in motion, is independent of direction as long as
there are no shearing stresses present”
WHY?
PROOF
y-direction
z-direction
Considering a point δx, δy, and δz approach to zero, so that
or,
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Gage Pressure vs. Absolute Pressure
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Hydraulic Machinery
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PROBLEM 1 – Describing the Pressure
The Crosby gage tester shown in the figure is used to calibrate or to test pressure
gages. When the weights and the piston together weigh 140 N, the gage being
tested indicates 200 kPa.
If the piston diameter is 30 mm, what percentage of error exists in the gage?
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Pressure Variation inside a Fluid – Derivation
Pressure Variation in a Fluid at Rest
Since p depends only on z
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Pressure Variation inside a Fluid – Incompressible
Incompressible Fluid
The quantity p/γ is a length called the pressure head of the fluid
HEAD: The height of a column of fluid of specific weight - γ - required to give a pressure
difference of (p1 - p2)
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Exercise 1 – Comparing Pressures Different Sections
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PROBLEM 2 – Absolute vs. Gage Pressure
Consider the two tanks shown in Figure. Both tanks are pressurized, with one
tank partially filled with gasoline at 20°C and the other tank filled with air at
20°C. The measured gauge pressures at A and B are 80 kPa and 130 kPa,
respectively, when the atmospheric pressure is 100 kPa.
What is the absolute pressure at A and B?
Estimate the gauge pressures at C and D.
If atmospheric pressure changes to 80 kPa, what will be the gauge pressures
at A, B, C, and D?
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Pressure Variation inside a Fluid – Compressible
Pressure variation inside a fluid is defined as follows;
Let's consider Ideal-Gases as compressible fluid. The specific weight of an Ideal-Gas
could be defined in terms of density and temperature as,
Combine equations to obtain:
Considering iso-thermal condition, integration yields,
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Pressure Variation inside a Fluid – Liquid vs. Gas
Water
Air
Pressure variation along depth
Pressure variation along height
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Pressure Variation inside a Fluid – Compressible
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Pressure Variation inside a Fluid – Compressible
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Standard Sea-level Conditions of Air
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PROBLEM 3 – Pressure variation of air
In 2010, the Burj Khalifa skyscraper, was completed and opened in the United
Arab Emirates. The final height of the building, which had remained a secret
until completion, is 828 m.
Estimate the ratio of the pressure at the top of the
building to the pressure at its base, assuming the air to
be at a common temperature of 15°C.
Compare the pressure calculated in part (a) with that
obtained by assuming the air to be incompressible with
values for air at standard sea level conditions.
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Measuring Pressure – Barometer
Evangelista Torricelli
(1608– 1647)
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Measuring Pressure – Barometer
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Measuring Pressure – Manometer
Manometers use vertical or inclined liquid columns to measure pressure
Simple
U-Tube Manometer
Piezometer Tube
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Measuring Pressure – Manometer
Manometers use vertical or inclined liquid columns to measure pressure
Differential U-tube Manometer
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Inclined-tube Manometer
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Measuring Pressure – Bourdon Gage & Transducers
Bourdon Gage
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PROBLEM 4 – Manometer
A closed tank contains compressed air and oil (SGoil = 0.90) as is shown in Fig. A
U-tube manometer using mercury (SGHg = 13.6) is connected to the tank as
shown. The column heights are h1 = 36 in. (0.9144 m), h2 = 6 in. (0.1524 m), and
h3 = 9 in. (0.2286 m). Determine the pressure reading (in kPa) of the gage.
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PROBLEM 5 – Manometer
The ratio of container diameter to tube diameter is 10. When air in the container
is at atmospheric pressure, the free surface in the tube is at position 1. When the
container is pressurized, the liquid in the tube moves 1 m up the tube from
position 1 to position 2. What is the container pressure that causes this
deflection?
The liquid specific weight is 7850 N/m3.
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Forces on Plane Surfaces (Panels)
UNIFORM PRESSURE DISTRIBUTION
For a uniform pressure distribution, the CP (Center of Pressure)
is located at the centroid of area of the panel…
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Forces on Plane Surfaces (Panels)
HYDROSTATIC PRESSURE DISTRIBUTION
Notice that the CP is located below the centroid of area
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Forces on Plane Surfaces (Panels)
HYDROSTATIC PRESSURE DISTRIBUTION
Magnitude of
Resultant Hydrostatic
Force
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Forces on Plane Surfaces (Panels)
HYDROSTATIC PRESSURE DISTRIBUTION
Line of Action of the
Resultant Force
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Forces on Plane Surfaces (Panels)
CENTROIDS AND MOMENTS OF INERTIA OF PLANE AREAS
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PROBLEM 6 – Gate
A rectangular gate is hinged at the water line, as shown. The gate is 1.2 m high
and 1.8 m wide. The specific weight of water is 9810 N/m3. Find the necessary
force (in kN) applied at the bottom of the gate to keep it closed.
0.3 m
1.2 m
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PROBLEM 7 – Gate
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.
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Buoyancy – Definition
BUOYANT FORCE
The buoyant force equals the weight of liquid that would be needed to occupy the
volume
This volume is called the displaced volume
A body partially submerged in
a liquid.
Fluid Mechanics − Chapter III: Fluid Statics
A body immersed in a liquid.
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PROBLEM 8 – Hydrometer
If the hydrometer shown sinks 5.3 cm in water (15°C) and the stem sinks 6.3 cm
in oil (z = 6.3 cm), what is the specific gravity of the oil?
Note that the bulb volume is 1.0 cm3, and the stem area is 0.1 cm2.
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PROBLEM 9 – Buoyancy
Determine the minimum volume of concrete (γ = 23.6 kN/m3) needed to keep
the gate (1 m wide) in a closed position, with l = 2 m.
Note that the hinge is at the bottom of the gate.
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