FLUID MECHANICS IIA – MFMV201 – FLUID STATICS
Hydrostatic Equilibrium and Pressure
Hydrostatic Equilibrium
A fluid is in hydrostatic equilibrium when
each fluid particle is in force equilibrium,
with the net force due to pressure
balancing the particle’s weight.
fig_03_02
Pressure intensity
Pressure intensity or simply pressure, is the force exerted by a fluid per unit
area.
𝑃=
𝐹
𝐴
𝑃𝑎
Units: Pascals, N/m2, bar (x105 Pa)
Pressure at a point: (Proof) page 63-64
Pascal’s principle: When there is an increase in pressure at any point in a
confined fluid, there is an equal increase at every other point in the
container.
If you can proof P1 = P2 = P3, then pressure at a given point in a static fluid
acts equally in all directions.
Pressure at a point: (Proof)
Forces acting on the prism:
𝐹𝐴𝐵 = 𝑃1 × 𝐴𝐵 × 𝑠
𝐹𝐵𝐶 = 𝑃2 × 𝐵𝐶 × 𝑠
𝐹𝐴𝐶 = 𝑃3 × 𝐴𝐶 × 𝑠
Consider vertical and horizontal forces:
Fluid is in equilibrium hence;
σ 𝐹 ↕ = 0 𝑎𝑛𝑑 σ 𝐹 ⟷ = 0
Vertical: 𝐹𝐴𝐵 = 𝐹𝐴𝐶 cos 𝜃
𝑃1 × 𝐴𝐶 cos 𝜃 × 𝑠 = 𝑃3 × 𝐴𝐶 × 𝑠 × cos 𝜃
𝑃1 = 𝑃3
Horizontal: 𝐹𝐵𝐶 = 𝐹𝐴𝐶 sin 𝜃
𝑃2 × 𝐴𝐶 sin 𝜃 × 𝑠 = 𝑃3 × 𝐴𝐶 × 𝑠 × sin 𝜃
𝑃2 = 𝑃3
∴ 𝑃1 = 𝑃2 = 𝑃3
cos 𝜃 =
𝐴𝐵
∴ 𝐴𝐵 = 𝐴𝐶 cos 𝜃
𝐴𝐶
𝐵𝐶
sin 𝜃 =
∴ 𝐵𝐶 = 𝐴𝐶 sin 𝜃
𝐴𝐶
Categories of Pressure (page 62)
• Atmospheric Pressure: The pressure exerted by the weight of the
atmosphere. At sea level (101.325 [kPa])
• Gauge Pressure: The pressure measured above or below atmospheric
pressure
• Vacuum: A perfect vacuum is a completely empty space in which the
pressure is zero.
• Absolute Pressure: Pressure measured above the absolute zero
(vacuum)
𝑷𝒂𝒃𝒔 = 𝑷𝒈𝒂𝒖𝒈𝒆 + 𝑷𝒂𝒕𝒎
Categories of Pressure
fig_03_04
Pressure and depth (page 66-67)
Pressure and depth
Pressure at a depth in a fluid is directly related to the weight of the column
of fluid above that point.
In equilibrium therefore:
𝐹𝑜𝑟𝑐𝑒𝑏𝑎𝑠𝑒 = 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑐𝑜𝑙𝑢𝑚𝑛 𝑜𝑓 𝑙𝑖𝑞𝑢𝑖𝑑
𝑃×𝐴=𝛾×𝐴×ℎ -
𝑷 = 𝜸𝒉
𝑷 = 𝝆𝒈𝒉
Since the same relation
applies wherever the column
is taken:
The intensity of pressure is the same at all points in the same horizontal
plane in a liquid at rest. [same liquid/same level]
Pressure head
Pressure Head
The pressure, P, at a point in a fluid can be expressed in terms of the
height, h, of the column of the fluid which causes the pressure, or which
would cause an equal pressure if the actual pressure is applied by
other means.
𝑃 = 𝜌𝑔ℎ [𝑃𝑎]
ℎ=
𝑃
𝑃
=
𝜌𝑔
𝛾
𝑚𝑓𝑙𝑢𝑖𝑑
• Pressure head, h, is given in meters of “x” fluid. The fluid must be
specified.
Simple Hydraulic Machine
Hydraulic Machine
A hydraulic machine uses a fluid to transmit a force or energy to assist in the
performance of a human task. (e.g. car braking systems, forklifts, power
steering units and aircraft controls)
The hydraulic jack
Mechanical advantage is defined as the ratio of output force to input force.
2
𝐷𝑙𝑜𝑎𝑑
𝐿𝑜𝑎𝑑
𝐴𝑟𝑒𝑎𝑙𝑜𝑎𝑑
𝑀𝐴 =
=
= 2
𝐸𝑓𝑓𝑜𝑟𝑡
𝐴𝑟𝑒𝑎𝑒𝑓𝑓𝑜𝑟𝑡
𝐷𝑒𝑓𝑓𝑜𝑟𝑡
Simple Hydraulic Machine
The hydraulic jack
Consider the hydraulic jack shown below. Calculate the load (kg) that can be
lifted.
Simple Hydraulic Machine
The hydraulic jack
Consider the hydraulic jack shown below. Calculate the load that can be
lifted.
Pressure Measurement Devices
Barometers
An instrument that is used to measure atmospheric pressure is called a
barometer. The simplest type is the mercury barometer.
Aneroid barometer
Sealed (partially evacuated) elastic bellows detects atmospheric changes
and deforms, causing a mechanical lever to deflect against a scale
indicating local atmospheric pressure
𝑃𝑎𝑡𝑚 = 𝛾𝐻𝑔 ℎ = 760 𝑚𝑚𝐻𝑔
Pressure Measurement Devices
Bourdon-Tube Gauge
• Measures pressure by sensing the deflection of
an elliptical cross-sectional tube, which is bent
into an arc.
• At atmospheric pressure the tube is not deflected
and reads zero.
• As pressure is increased in
the tube, it straightens,
moving a pointer to read
positive gauge pressure
fig_03_11
Pressure Measurement Devices
Pressure Transducers
• A device that converts pressure into an electrical signal
• Examples:
• Strain gauge type
• Capacitive pressure transducer
• Piezoelectric transducer (uses a quartz crystal)
• Digital data-logging.
fig_03_15
Pressure Measurement Devices
Piezometer (Pressure tube)
• A vertical tube, usually transparent, in which a liquid rises in response to
a positive gauge pressure.
• Pressure in the pipe pushes the column to a height, h.
•
The gauge pressure at the center of the pipe can be calculated using the
formula:
𝑃 = 𝛾𝑓𝑙𝑢𝑖𝑑 ℎ
fig_03_12
Pressure Measurement Devices
Manometer
• A manometer is a device used for measuring
pressure by raising and lowering a column of
liquid.
• The pressure is determined by the height
difference of the liquid.
• Types:
• U-tube
• Differential manometer
• Inclined manometer (increases
measurement sensitivity)
• Consider the manometer top right:
𝑃2 = 𝑃𝑎𝑡𝑚 + 𝛾𝐻𝑔 Δh
𝑃3 = 𝑃4 + 𝛾𝑓𝑙𝑢𝑖𝑑 × 𝑙
𝑃2 = 𝑃3 - same liquid, same level
fig_03_13
fig_03_14
Lecture Summary
• Hydrostatic equilibrium and pressure intensity
• Pressure at a point (Pascal’s principle)
• Categories of pressure
• Pressure and depth
• Pressure head
• Simple Hydraulic Machine
• Pressure measurement devices
• Manometers
Next Lecture
• Hydrostatics – Pressure and Pressure measurement (exercise)
Homework
Textbook page 95 -102
• 3.2, 3.3, 3.5, 3.6
• 3.8, 3.9, 3.11, 3.13, 3.14, 3.15, 3.16, 3.17,3.21, 3.23,3.30
• 3.45, 3.46, 3.48, 3.51, 3.54, 3.59