Fluid Statics

Why it matters

Fluid statics is essential in engineering because it helps in understanding how fluids behave when they are not in motion. This knowledge is crucial for designing dams, hydraulic systems, and even in predicting weather patterns.

Key ideas

• Pressure in a Fluid: Pressure at a point in a fluid at rest is the same in all directions. This is known as isotropic pressure.
• Pascal's Law: A change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and to the walls of its container.
• Hydrostatic Pressure: The pressure exerted by a fluid at equilibrium due to the force of gravity. It increases with depth.
• Manometry: The technique of measuring pressure using columns of liquid in a U-tube or other devices.
• Buoyancy: The upward force exerted by a fluid that opposes the weight of an object immersed in it.

Pressure datum

In a static constant-density fluid, p = p_surface + ρgh. With z positive upward, dp/dz = -ρg. The expression ρgh alone is the pressure increase below the reference surface. In a manometer, add ρg times each downward height and subtract it for each upward height, using each segment’s density.

Formulas

• Hydrostatic Pressure: P = ρ·g·h
  • P: Pressure (Pa)
  • ρ: Density of the fluid (kg/m³)
  • g: Acceleration due to gravity (9.81 m/s²)
  • h: Height of the fluid column (m)
• Hydrostatic pressure difference: ΔP = ρ·g·Δh when Δh is the positive downward depth change in a constant-density fluid. This is a hydrostatic relation, not the statement of Pascal’s law.
  • ΔP: Change in pressure (Pa)
  • Δh: Change in height (m)
• Buoyant Force: F_b = ρ·V·g
  • F_b: Buoyant force (N)
  • V: Volume of the fluid displaced (m³)

Worked example

Problem: Calculate gauge pressure at a depth of 10 meters below the free surface of water open to the atmosphere.

Given:

• Density of water, ρ = 1000 kg/m³
• Depth, h = 10 m
• Acceleration due to gravity, g = 9.81 m/s²

1. Use the hydrostatic pressure formula: P = ρ·g·h
2. Substitute the given values: P = 1000 kg/m³ · 9.81 m/s² · 10 m
3. Calculate: P = 98100 Pa

Answer: The gauge pressure is 98100 Pa = 98.1 kPa. Absolute pressure is atmospheric pressure plus 98.1 kPa.

Common mistakes

• Confusing gauge pressure with absolute pressure.
• Forgetting to convert units, especially when dealing with different systems of measurement.
• Misapplying Pascal's Law by not considering the entire fluid system.

For GATE ME

Questions often involve calculating pressures at various depths, understanding buoyancy, and applying Pascal's Law. Practice problems involving manometers and pressure differences in fluids.

Quick check

1. What is the pressure at the surface of a fluid?
2. How does pressure change with depth in a fluid?
3. What principle explains the buoyant force?

Answers: 1. Atmospheric pressure only for a free surface open to the atmosphere; in a closed vessel use the gas pressure above it, 2. Increases, 3. Archimedes' principle.