8.1 Fluid Properties and Statics

Fluid mechanics is all about how liquids and gases behave. This section dives into the key properties that make fluids unique, like density and viscosity. It's like learning the personality traits of water, air, and other fluids.

We'll also explore how fluids exert pressure and force when at rest. This knowledge is super useful for designing everything from dams to floating ships. It's the foundation for understanding how fluids interact with their surroundings.

Key Fluid Properties

Density and Specific Weight

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• Density (ρ) measures mass per unit volume of fluid expressed in kg/m³ or slug/ft³
  • Affects fluid behavior and forces
  • Example: Water has a density of 1000 kg/m³ at 4°C
• Specific weight (γ) measures weight per unit volume of fluid in N/m³ or lbf/ft³
  • Related to density by acceleration due to gravity (g)
  • Calculated using the formula $γ = ρg$
• Temperature significantly impacts density
  • Most substances decrease in density as temperature increases (water is an exception between 0-4°C)

Viscosity and Surface Tension

• Viscosity (μ) quantifies fluid's resistance to flow and deformation
  • Dynamic viscosity expressed in Pa·s or lb·s/ft²
  • Kinematic viscosity (ν) represents ratio of dynamic viscosity to density
  • Example: Honey has a higher viscosity than water, flowing more slowly
• Surface tension (σ) measures fluid surface's resistance to external forces
  • Caused by cohesive forces between molecules
  • Typically measured in N/m or lbf/ft
  • Enables water striders to walk on water surface
• Temperature affects viscosity and surface tension
  • Generally, both properties decrease as temperature increases

Compressibility and Other Properties

• Compressibility measures volume change in fluid under pressure
  • Negligible for liquids but significant for gases
  • Example: Air in tires compresses under increased pressure, while water in hydraulic systems remains nearly incompressible
• Bulk modulus quantifies fluid's resistance to compression
  • Inverse of compressibility
  • Higher bulk modulus indicates lower compressibility
• Vapor pressure represents pressure at which liquid begins to vaporize
  • Increases with temperature
  • Critical in preventing cavitation in hydraulic systems

Hydrostatic Pressure and Force

Hydrostatic Pressure Principles

• Hydrostatic pressure results from fluid weight at rest
  • Increases linearly with depth according to $p = ρgh$
  • h represents depth, ρ is fluid density, g is gravitational acceleration
• Pascal's law states applied pressure transmits equally in all directions
  • Enables hydraulic systems like car brakes and elevators
• Total pressure at a point combines atmospheric and gauge pressure
  • Atmospheric pressure results from weight of air column above
  • Gauge pressure measures pressure relative to atmospheric pressure

Hydrostatic Force Calculations

• Hydrostatic force on submerged planar surface calculated by integrating pressure over area
  • Resultant force acts at center of pressure
  • Force magnitude: $F = ρghA$, where A is surface area
• Curved surfaces require separate horizontal and vertical force component calculations
  • Use projected areas for each component
  • Example: Dam design considers both horizontal and vertical forces on curved surfaces
• Pressure prisms visually represent pressure distribution on submerged surfaces
  • Aid in force calculation and visualization
  • Triangular for vertical surfaces, trapezoidal for inclined surfaces

Stability of Floating Objects

Buoyancy and Archimedes' Principle

• Archimedes' principle states buoyant force equals weight of displaced fluid
  • Applies to submerged or partially submerged objects
  • Buoyant force acts vertically upward through center of buoyancy
• Center of buoyancy represents centroid of displaced fluid volume
  • Position changes as object orientation changes
• Buoyant force calculation: $F_b = ρgV$, where V is displaced fluid volume
  • Example: A 1000 kg boat displacing 1 m³ of water experiences 9810 N buoyant force

Stability Analysis

• Stability determined by relative positions of center of gravity and metacenter
  • Metacenter intersects buoyant force vector and object's centerline
• Metacentric height measures distance between center of gravity and metacenter
  • Positive for stable equilibrium
  • Negative for unstable equilibrium
  • Zero for neutral equilibrium
• Submerged object stability achieved when center of buoyancy directly above center of gravity
• Reserve buoyancy represents volume between waterline and uppermost watertight deck
  • Crucial for vessel design and safety
  • Affects ship's ability to remain afloat when damaged

Manometers and Pressure Measurement

Manometer Types and Principles

• Manometers measure pressure differences using fluid column height
  • Simple manometers use single fluid
  • U-tube manometers compare two pressures
  • Inclined manometers increase sensitivity for small pressure differences
• Pressure difference calculated using $Δp = ρgh$
  • h represents height difference between fluid levels
• Multi-fluid manometers require considering density and height of each fluid section
  • Example: Mercury-water manometer uses density difference to measure larger pressure ranges

Pressure Measurement Devices

• Barometers measure atmospheric pressure
  • Typically use mercury due to high density
  • Standard atmospheric pressure: 101.325 kPa or 760 mmHg at sea level
• Pressure gauges measure relative to atmospheric pressure (gauge pressure)
  • Bourdon tubes use curved tube that straightens under pressure
  • Diaphragm gauges use flexible membrane displacement
• Absolute pressure combines gauge pressure and atmospheric pressure
  • Important distinction in pressure measurements and calculations
  • Example: Tire pressure often measured in gauge pressure, while vacuum systems use absolute pressure