Fluid Mechanics Essentials: Pressure, Viscosity, and Hydrostatic Principles

Fundamental Concepts in Fluid Mechanics

Pressure and Phase Changes

• Evaporation: The constant movement of molecules on the surface of a liquid, allowing them to escape into the atmosphere.
• Boiling: Occurs when the pressure to which a liquid is subjected equals its vapor pressure.
• Vapor Pressure: The partial pressure exerted by molecules in the gaseous state within a container when occupying a space above a liquid.
• Saturation Pressure: The state where there is a dynamic balance between the number of molecules that evaporate and condense.
• Cavitation: A phenomenon where, if the local pressure (p) in a liquid drops to or below its vapor pressure, the liquid rapidly evaporates, forming vapor bubbles. These bubbles then move and collapse violently within the fluid.
  • Implications: Cavitation causes significant erosion of metal parts in pumps and turbines, particularly in low-pressure zones.

Understanding Pressure

• Pressure: The effect caused by a force acting perpendicularly on a surface.
• Equation for Incompressible Hydrostatic Fluid: In a fluid at rest, the pressure difference between two points is given by p₂ - p₁ = ρgh, where ρ is fluid density, g is acceleration due to gravity, and h is the vertical height difference.
  • In a fluid at rest, all points at the same horizontal plane have the same pressure.
  • The free surface of a liquid in equilibrium (often visualized as a piezometric map) indicates points of equal pressure.
  • The pressure of a liquid column is equal to its base pressure: p = ρgh.
• Atmospheric Pressure: The pressure exerted by the air that surrounds us.
  • Common approximate values: 101.3 kPa, 760 mmHg, 1.033 atm.
• Absolute Pressure: Pressures measured relative to a perfect vacuum (absolute zero pressure).
• Relative Pressure (Gauge Pressure): Pressures measured relative to the surrounding atmospheric pressure.
  • Measured using instruments like manometers, vacuometers, and barometers.
  • Relationship: Absolute Pressure = Gauge Pressure + Atmospheric Pressure.

Fundamental Principles of Fluid Statics

• Pascal's Principle: The pressure exerted on any point of an incompressible fluid at rest is transmitted undiminished in all directions throughout the fluid.
  • Formula for hydraulic systems: F₂ / A₂ = F₁ / A₁ (where F is force and A is area).
• Archimedes' Principle: Any body partially or totally submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces.
  • Formula: F_buoyant = ρ_fluid · g · V_displaced (where ρ_fluid is fluid density, g is acceleration due to gravity, and V_displaced is the volume of displaced fluid).

Properties of Fluids

Hydrostatics and Fluid Classification

• Fluid Mechanics - Hydrostatics: The branch of fluid mechanics that studies fluids at static equilibrium (at rest).
• Properties of Incompressible Fluids: These fluids lack a fixed shape and adhere to Pascal's Principle.
• Gases: Compressible fluids that also lack a fixed shape.

Density and Elasticity

• Specific Density (Mass Density): The mass per unit volume of a substance.
  • Formula: ρ = m/V.
• Relative Density: The ratio of a substance's density to the density of distilled water at 4°C.
• Specific Gravity (Weight Density): The weight per unit volume of a substance.
  • Formula: γ = W/V.
• Volumetric Elastic Modulus (Bulk Modulus): Expresses the elasticity of a fluid, representing the relationship between a change in pressure and the resulting fractional change in volume.
  • Formula: E = -ΔP / (ΔV / V).

Viscosity and Flow Characteristics

• Viscosity: A fundamental property of fluids that quantifies their resistance to flow or movement.
• Velocity Gradient: The rate of change of velocity with respect to the distance perpendicular to the flow, typically observed between two parallel plates where fluid occupies the space.
  • Formula: dV/dy (or ΔV/ΔY).
• Dynamic Viscosity (Absolute Viscosity): A coefficient (μ) that relates the shear stress (τ) to the velocity gradient. The shear force (F) is proportional to the plate's surface area (A) and the velocity gradient.
  • Formula: F = μ · A · (ΔV/ΔY).
  • Units: [Pa·s] or [Poise].
• Kinematic Viscosity: The ratio of dynamic viscosity to density.
  • Units: [m²/s] or [Stoke].
• Kinematic Viscosity Measurement Practices: Common methods include:
  • Engler Degrees: The ratio of time required for 200 cm³ of liquid to flow through a standardized orifice compared to the same volume of water at 20°C (often measured at 50°C or 20°C).
  • Redwood Viscometer
  • Saybolt/SAE Viscometer
• Viscosity Index (VI): An empirical number indicating how much a fluid's viscosity changes with temperature. A high VI signifies that the viscosity varies little with temperature.
• Fluidity Point (Pour Point): The lowest temperature at which a liquid can still flow under specified conditions.
• Surface Tension: The force (F) per unit length (L) acting parallel to the surface, causing a tension effect on the liquid's surface.
  • Formula: σ = F/L.