Lecture 3 MAAE 2300 Fluid Mechanics - Hydrostatics

Lecture 3 focuses on the fundamentals of fluid pressure under static conditions, known as hydrostatics. It establishes that in a stationary fluid, pressure is isotropic (the same in all directions at a point) and acts as a normal stress. The key principle is that while pressure remains constant in any horizontal plane, it increases linearly with depth due to the fluid's weight, defined by the relationship dp = -γ dz. The lecture also introduces important pressure definitions: absolute pressure (measured from a perfect vacuum), gauge pressure (relative to atmospheric pressure), and vacuum pressure. These concepts are solidified with examples, such as calculating the pressure at the bottom of a lake, demonstrating the practical application of the hydrostatic equation.

1. MAAE 2300: Fluid Mechanics I
Hydrostatics
Carleton University -Fall2025 MAAE 2300 -Fluid Mechanics I 1

2. Hydrostatic Condition -Motivation
•Motivation:
•Many fluid problems do not
involve motion. For example:
water tanks and reservoirs.
•When a fluid’s velocity is zero,
the fluid is considered to bein
a hydrostatic condition.
•Many civil engineering
problems fall under this
hydrostatic category.
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3. Hydrostatic Fluid Pressure
•Considering any plane
through a fluid element at
rest, it is always
experiencing a normal
stress, which we called
the fluid pressure ݌.
•It is taken positive for
compression by common
convention.
Carleton University -Fall2025 MAAE 2300 -Fluid Mechanics I 3
݌
௫
݌
௭
Differential length
݌
௫
݌
௭
ݔ
ݕ
݌
௫
݌
௭
݌
௦

4. Hydrostatic Fluid Pressure
෍ܨ
௫=0 ⇒
݌
௫ܾΔݖ − ݌
௡ܾΔݏsinߠ=0
෍ܨ
௭=0 ⇒
݌
௭ܾΔݔ − ݌
௡ܾΔݏcosߠ−
1
2
ߩܾ݃ΔݔΔݖ=0
Δݏsinߠ=Δݖ Δݏcosߠ= Δݔ
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5. Hydrostatic Fluid Pressure
݌
௫=݌
௡
݌
௭=݌
௡+
ଵ
ଶ
ఘ௚୼௭
Two important principles of
hydrostatic:
1)There is no pressure change in the
horizontal direction.
2)There is a vertical change in
pressure that is proportional to the
density, gravity and depth change.
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6. Hydrostatic Fluid Pressure (Pascal’s Relation)
ߠis arbitrary
Δݖ⟶0(element becomes a point)
⇒
݌
௫=݌
௭=݌
௡=݌
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7. Hydrostatic Fluid Pressure
•The pressure is a single valued property at a point that is
independent of any direction⇒it is not a vector quantity!
•It is a thermodynamic property of the fluid like temperature
and density. It is not a force!
•A body or surface in contact with a static fluid experiences a
force due to this pressure.
•Pascal’s Law:A pressure change occurring anywhere in a
confined incompressible fluid is transmitted throughout the
fluid such that the same change occurs everywhere.
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8. Absolute Pressure, Gauge Pressure and Vacuum
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݌
ீ௔௨௚௘=݌
஺௕௦௢௟௨௧௘−݌
௔௧௠௢௦௣௛௘௥௘
݌
௏௔௖௨௨௠=݌
௔௧௠௢௦௣௛௘௥௘−݌
஺௕௦௢௟௨௧௘
Differential/Relative
Pressure

9. Static Pressure Variation
෍ܨ
௬=0
݌Δܣ−݌+
߲݌
߲ݕ
ΔݕΔܣ=0
߲݌=0
The same result will occur in the
x-direction, and since this change
in pressure is zero, it indicates
that the pressure remains
constant in the horizontal plane.
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ݕ
Δݕ
Δܣ

10. Static Pressure Variation
෍ܨ
௭=0
݌Δܣ−݌+
߲݌
߲ݕ
ΔݕΔܣ−ߛΔܣ Δݖ=0
݀݌=−ߛ݀ݖ ⇒ ݌
ଶ−݌
ଵ=−නߛ ݀ݖ
ଶ
ଵ
•Negative sign means pressure will
decrease as one moves upward
•Pressure is only function of z.
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ݖ
Δݖ
Δܣ
݌Δܣ
݌+
߲݌
߲ݖ
ΔݖΔܣ
ߛΔܣ Δݖ
Depends on density and
gravity distribution within
the fluid

11. Static Pressure Variation
•Assuming incompressible liquid with no gravity field
variation:
݌
ଶ−݌
ଵ=−ߛݖ
ଶ−ݖ
ଵ
or
ݖ
ଵ+
݌
ଵ
ߛ
=ݖ
ଶ+
݌
ଶ
ߛ
•In a continuously distributed uniform static fluid, the
pressure varies only with vertical distanceand it is
independent of the shape of the container. The pressure is
the same at all points on a given horizontal plane in the
sameuniform fluid.
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12. Static Pressure Variation
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݌
௔=݌
௕=݌
௖=݌
ௗ
݌
஺= ݌
஻=݌
஼≠݌
஽

13. Specific Weight of Some Common Liquids
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14. Hydrostatic Pressure Distribution in Oceans and
Atmosphere
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݌=݌
௔−ߛݖ
Be careful of the signs !
They all have tobe
reconciled.

15. Static Pressure Variation: Example
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16. Static Pressure Variation: Example
Water fills the pipe AB such that the
absolute pressure at A is 400 [kPa]. If
the atmospheric pressure is 101 [kPa],
then determine the resultant force the
water and surrounding air exert on
the cap at B. The inner diameter of the
pipe is 50 [mm].
Carleton University -Fall2025 MAAE 2300 -Fluid Mechanics I 16
A
B
0.3m
0.4m

17. Static Pressure Variation: Example
݌
஺−݌
஻=−ߛ
௪ݖ
஺− ݖ
஻
݌
஻=݌
஺+ߛ
௪ݖ
஺− ݖ
஻= ݌
஺+ߩ
௪݃ݖ
஺− ݖ
஻
݌
஻=400 kPa
+1000kgm
ଷ
⁄×9.81ms
ଶ
⁄×−0.3m
=397.06 kPa
ܨ
஻=(݌
஻−݌
௔௧௠)ߨݎ
ଶ
=397.06 kPa−101.0 kPa×ߨ0.025݉
ଶ
=581.3 [N]
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A
B
0.3m
0.4m

18. End of Lecture
Thank You
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