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Supporting Material for Experiment 3
Fluid Mechanics: Fundamentals and Applications, Yunus A. Cengel, John M. Cimbala
PRESSURE AND FLUID STATICS
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PRESSURE
Pressure: A normal force exerted by a fluid per unit area
21 Pa 1 N/m pascal
5
2 2 4 2 4
1 bar 10 Pa 0.1 MPa 100 kPa
1 atm 101,325Pa 101.325 kPa 1.01325 bars
1 kgf/cm 9.807 N/cm 9.807 10 N/m 9.807 10 Pa
0.9807 bar
0.9679 atm
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Absolute pressure: The actual pressure at a given position. It is measured relative to absolute vacuum (i.e., absolute zero pressure). Gage pressure: The difference between the absolute pressure and the local atmospheric pressure. Most pressure-measuring devices are calibrated to read zero in the atmosphere, and so they indicate gage pressure. Vacuum pressures: Pressures below atmospheric pressure.
gage abs atm
vac atm abs
P P P
P P P
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Pressure at a Point
Pressure is the compressive force per unit area but it is not a vector.
Pressure at any point in a fluid is the same in all directions.
Pressure has magnitude but not a specific direction, and thus it is a scalar quantity.
Pressure is a scalar quantity, not a vector; the pressure at a point in a fluid is the same in all directions.
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Variation of Pressure with Depth
atm gageorP P gh P gh
2
2 1 1 P P P g dz When the variation of density with elevation is known
The pressure of a fluid at rest increases with depth (as a result of added weight).
Free-body diagram of a rectangular fluid element in equilibrium.
When density is constant
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The pressure is the same at all points on a horizontal plane in a given fluid regardless of geometry, provided that the points are interconnected by the same fluid.
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When analyzing hydrostatic forces on submerged surfaces, the atmospheric pressure can be subtracted for simplicity when it acts on both sides of the structure.
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HYDROSTATIC FORCES ON SUBMERGED PLANE SURFACES
The pressure at the centroid of a surface is equivalent to the average pressure on the surface.
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The resultant force acting on a plane surface is equal to the
product of the pressure at the centroid of the surface and the
surface area, and its line of action passes through the
center of pressure.
R CF P A
,xx C P C
C
I y y
y A Ixx,C is the second moment of area
(area moment of inertia) about the x-axis passing by the centroid of the surface
Center of pressure:
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The centroid and the centroidal moments of inertia for some common geometries.
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Pressure acts normal to the surface, and the hydrostatic forces acting on a flat plate of any shape form a volume whose base is the plate area and whose length is the linearly varying pressure.
This virtual pressure prism has an interesting physical interpretation: its volume is equal to the magnitude of the resultant hydrostatic force acting on the plate since FR = PdA, and the line of action of this force passes through the centroid of this homogeneous prism.
The projection of the centroid on the plate is the pressure center.
Therefore, with the concept of pressure prism, the problem of describing the resultant hydrostatic force on a plane surface reduces to finding the volume and the two coordinates of the centroid of this pressure prism.
The hydrostatic forces acting on a plane surface form a pressure prism whose base (left face) is the surface and whose length is the pressure.
©McGraw-Hill Education. Acts 2/3 on the way down
Special Case: Partially Submerged Rectangular Plate
Average pressure:
Area:
Resultant force:
Center of Pressure: ,xx C
P C C
I y y
y A
Key concept: The force on a rectangular plate with the top edge even with the liquid surface, acts 2/3 of the way down.
𝑦
©McGraw-Hill Education. Acts 2/3 on the way down
Special Case: Fully submerged Rectangular Plate
Average pressure:
Area:
Resultant force:
Center of Pressure: ,xx C
P C C
I y y
y A
Key concept: The force on a rectangular plate fully submerged equals the volume of the pressure prism acting on the surface, and the center of pressure is located at the centroid of this pressure prism. It can be calculated using concepts from Statics.
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HYDROSTATIC FORCES ON SUBMERGED CURVED SURFACES
2 2 R H VF F F /V HF F
Determination of the hydrostatic force acting on a submerged curved surface.
:
:
H x
V y
Horizontal force component on curved surface
Vertical force component on curved surface
F F
F F W
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When a curved surface is above the liquid, the weight of the liquid and the vertical component of the hydrostatic force act in the opposite directions.
The hydrostatic force acting on a circular surface always passes through the center of the circle since the pressure forces are normal to the surface and they all pass through the center.