Educational

Pressure in Fluids

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by

Saim Khalid
in

Pressure in fluids is a fundamental concept in fluid mechanics, governing how liquids and gases exert force on surfaces and interact with the environment. Understanding fluid pressure is crucial in engineering, hydraulics, aviation, meteorology, and medicine. This post provides a detailed explanation of pressure in fluids, its types, formulas, measurement devices, and real-life applications.

1. What is Pressure?

Pressure (P) is defined as the force exerted per unit area: P=FAP = \frac{F}{A}P=AF​

Where:

  • FFF = perpendicular force applied (N)
  • AAA = area over which the force acts (m²)

SI unit: Pascal (Pa) = 1 N/m²
Other units: atm (atmosphere), bar, mmHg, psi

Key Concept: Pressure acts perpendicular to the surface of a fluid or object immersed in a fluid.

2. Characteristics of Fluid Pressure

  1. Isotropic Nature: Pressure at a point in a fluid acts equally in all directions.
  2. Dependent on Depth: In a static fluid, pressure increases with depth.
  3. Independent of Shape: Pressure at a given depth is the same, regardless of container shape.
  4. Transmitted Uniformly: According to Pascal’s Law, pressure applied to a confined fluid transmits equally in all directions.

3. Types of Fluid Pressure

3.1 Atmospheric Pressure

  • Pressure exerted by the weight of air above a surface
  • Average at sea level: 101,325 Pa (~1 atm)
  • Measured with a barometer
  • Applications: Weather prediction, altimeters, vacuum systems

3.2 Gauge Pressure

  • Pressure relative to atmospheric pressure:

Pgauge=Pabsolute−PatmosphericP_\text{gauge} = P_\text{absolute} – P_\text{atmospheric}Pgauge​=Pabsolute​−Patmospheric​

  • Can be positive (pressurized systems) or negative (vacuum)

3.3 Absolute Pressure

  • Total pressure including atmospheric pressure:

Pabsolute=Pgauge+PatmosphericP_\text{absolute} = P_\text{gauge} + P_\text{atmospheric}Pabsolute​=Pgauge​+Patmospheric​

3.4 Hydrostatic Pressure

  • Pressure exerted by a fluid at rest:

P=P0+ρghP = P_0 + \rho g hP=P0​+ρgh

Where:

  • P0P_0P0​ = surface pressure (atmospheric)
  • ρ\rhoρ = fluid density
  • ggg = acceleration due to gravity
  • hhh = depth below fluid surface

Applications: Dams, submarines, deep-sea exploration.

4. Variation of Pressure with Depth

  • Key Principle: Pressure increases linearly with depth in incompressible fluids.
  • Derived from hydrostatic equilibrium:

dPdz=−ρg\frac{dP}{dz} = – \rho gdzdP​=−ρg

Integrating: P=P0+ρghP = P_0 + \rho g hP=P0​+ρgh

  • Example: At 10 m depth in water (ρ=1000 kg/m³\rho = 1000 \, \text{kg/m³}ρ=1000kg/m³):

P=101,325+1000⋅9.81⋅10≈199,425 PaP = 101,325 + 1000 \cdot 9.81 \cdot 10 \approx 199,425 \, \text{Pa}P=101,325+1000⋅9.81⋅10≈199,425Pa

5. Pascal’s Law

  • Statement: Pressure applied to a confined fluid is transmitted equally in all directions.
  • Applications: Hydraulic lifts, presses, brakes.

Formula: F1A1=F2A2\frac{F_1}{A_1} = \frac{F_2}{A_2}A1​F1​​=A2​F2​​

  • Small force on a small piston can produce a large force on a larger piston.

Example: Lifting a car using a hydraulic lift.

6. Measurement of Fluid Pressure

6.1 Manometers

  • U-tube, inclined, or differential
  • Measure pressure difference in fluids
  • Hydrostatic principle:

P=ρghP = \rho g hP=ρgh

6.2 Barometers

  • Measure atmospheric pressure
  • Mercury barometer: column of mercury balances atmospheric pressure

Patm=ρghP_\text{atm} = \rho g hPatm​=ρgh

  • Torricelli’s experiment demonstrates this principle

6.3 Pressure Gauges

  • Bourdon tube, diaphragm, or digital sensors
  • Measure gauge pressure in pipes, tanks, and industrial systems

7. Pressure in Liquids at Rest (Hydrostatics)

  • Hydrostatic pressure depends on density, depth, and gravity, not on container shape
  • Hydrostatic paradox: Pressure at a point depends only on depth, not fluid volume
  • Equation: P=P0+ρghP = P_0 + \rho g hP=P0​+ρgh

Applications: Designing dams, silos, and pressure vessels

8. Buoyancy and Pressure

  • Buoyant force arises due to pressure difference between top and bottom of submerged object

FB=ρfluidgVdisplacedF_B = \rho_\text{fluid} g V_\text{displaced}FB​=ρfluid​gVdisplaced​

  • Key Concept: Pressure increases with depth → upward net force
  • Determines whether objects float or sink

Example: Ships, icebergs, submarines

9. Pressure in Gases

  • Ideal Gas Law: PV=nRTP V = n R TPV=nRT
  • Pressure arises from molecular collisions with container walls
  • Depends on temperature and volume

Applications: Pneumatic systems, airbags, pressurized tanks

10. Gauge vs Absolute vs Vacuum Pressure

  • Gauge Pressure: Measured relative to atmosphere
  • Absolute Pressure: Total pressure including atmosphere
  • Vacuum Pressure: Negative gauge pressure (below atmospheric)

Example: Car tire (gauge pressure 200 kPa, absolute 301 kPa)

11. Pressure in Moving Fluids

  • Governed by Bernoulli’s Principle:

P+12ρv2+ρgh=constant along streamlineP + \frac{1}{2} \rho v^2 + \rho g h = \text{constant along streamline}P+21​ρv2+ρgh=constant along streamline

  • Pressure can decrease where velocity increases
  • Applications: Airplane wings (lift), venturi meters, carburetors

12. Pressure in Pipes

  • Pipe flow: Pressure varies due to friction, elevation, and velocity
  • Darcy-Weisbach equation:

ΔP=fLDρv22\Delta P = f \frac{L}{D} \frac{\rho v^2}{2}ΔP=fDL​2ρv2​

Where fff = friction factor, LLL = pipe length, DDD = diameter

Applications: Water supply systems, chemical pipelines

13. Factors Affecting Fluid Pressure

  1. Depth: Deeper → higher hydrostatic pressure
  2. Density: Denser fluids → higher pressure
  3. Gravity: Stronger gravity → higher pressure
  4. Temperature: Affects gas pressure (ideal gas law)
  5. Flow velocity: Dynamic pressure adds to static pressure (Bernoulli)

14. Atmospheric Pressure and Its Effects

  • Atmospheric pressure decreases with altitude:

P=P0(1−LhT0)gMRLP = P_0 \left( 1 – \frac{L h}{T_0} \right)^{\frac{g M}{R L}}P=P0​(1−T0​Lh​)RLgM​

Where LLL = lapse rate, MMM = molar mass, RRR = gas constant

  • Influences weather patterns, aircraft design, breathing

15. Pressure in Hydraulic Systems

  • Hydraulic systems rely on fluid pressure transmission
  • Pascal’s Law allows small input force → large output force
  • Applications: Hydraulic brakes, lifts, presses, excavators

16. Pressure and Engineering Applications

  1. Dams: Hydrostatic pressure calculation prevents collapse
  2. Submarines: Pressure resistance required for deep-sea operation
  3. Pipelines: Design to withstand maximum pressure
  4. Aviation: Cabin pressure controlled for passenger comfort
  5. Medical Devices: Blood pressure measurement, IV fluids

17. Advanced Concepts in Fluid Pressure

  • Pressure Tensor: In anisotropic fluids or complex flows
  • Stagnation Pressure: Total pressure at a point in flow
  • Gauge vs Static vs Dynamic Pressure: Key in fluid mechanics experiments

Equation: Ptotal=Pstatic+PdynamicP_\text{total} = P_\text{static} + P_\text{dynamic}Ptotal​=Pstatic​+Pdynamic​

18. Examples and Problems

  1. Hydrostatic Pressure: Calculate pressure 5 m underwater:

P=101,325+1000⋅9.81⋅5≈150,375 PaP = 101,325 + 1000 \cdot 9.81 \cdot 5 \approx 150,375 \, \text{Pa}P=101,325+1000⋅9.81⋅5≈150,375Pa

  1. Hydraulic Lift: Input force 500 N, piston area 0.01 m², output piston 0.1 m²:

F2=F1A2A1=5000.10.01=5000 NF_2 = F_1 \frac{A_2}{A_1} = 500 \frac{0.1}{0.01} = 5000 \, \text{N}F2​=F1​A1​A2​​=5000.010.1​=5000N

  1. Gas Pressure: Air in tire 200 kPa gauge → absolute pressure:

Pabs=200+101.3≈301.3 kPaP_\text{abs} = 200 + 101.3 \approx 301.3 \, \text{kPa}Pabs​=200+101.3≈301.3kPa

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