Fluid Mechanics: Statics and Dynamics

• Fluid mechanics is the study of fluids and their behavior when at rest (fluid statics) and in motion (fluid dynamics).
• It encompasses liquids, gases, and plasmas and applies principles of mechanics to understand their properties and behavior.
• Fluid mechanics is crucial in many engineering disciplines, including mechanical, civil, chemical, and aerospace engineering, for designing systems involving fluid transport, energy conversion, and environmental control.

Key Concepts

• Density: Mass per unit volume, ρ = m/V (kg/m³).
• Specific volume: Volume per unit mass, v = 1/ρ (m³/kg).
• Specific weight: Weight per unit volume, γ = ρg (N/m³).
• Viscosity: A fluid's resistance to flow or deformation, a measure of its internal friction.
• Dynamic viscosity (μ): Also known as absolute viscosity, measured in Pascal-seconds (Pa·s) or Poise (P).
• Kinematic viscosity (ν): Ratio of dynamic viscosity to density, ν = μ/ρ, measured in m²/s or Stokes (St).
• Surface tension: The force that causes the surface of a liquid to contract and behave like a stretched membrane, due to cohesive forces between liquid molecules.
• Capillary action: The rise or fall of a liquid in a narrow tube due to surface tension and adhesive forces.

Fluid Statics

• Pressure: Force exerted per unit area by a fluid, P = F/A (Pa).
• Pressure at a depth: In a static fluid, pressure increases linearly with depth, P = ρgh, where h is the depth.
• Pascal's Law: Pressure applied to an enclosed fluid is transmitted undiminished to every point within the fluid.
• Manometry: Using liquid columns in tubes to measure pressure differences.
• Buoyancy: Upward force exerted by a fluid that opposes the weight of an immersed object.
• Archimedes' principle: Buoyant force is equal to the weight of the fluid displaced by the object.
• Stability of immersed and floating bodies: Determined by the relative positions of the center of gravity (CG) and the center of buoyancy (CB).

Fluid Dynamics

• Types of flow:
  • Laminar flow: Smooth, orderly fluid motion with layers sliding over each other.
  • Turbulent flow: Irregular, chaotic fluid motion with eddies and mixing.
  • Steady flow: Fluid properties at a point do not change with time.
  • Unsteady flow: Fluid properties at a point change with time.
  • Compressible flow: Density of the fluid changes significantly.
  • Incompressible flow: Density of the fluid remains nearly constant.
• Flow rate:
  • Volume flow rate (Q): Volume of fluid passing a point per unit time, Q = AV (m³/s).
  • Mass flow rate (ṁ): Mass of fluid passing a point per unit time, ṁ = ρAV (kg/s).
• Continuity equation: For steady flow, the mass flow rate is constant, A₁V₁ = A₂V₂ (for incompressible fluids).
• Bernoulli's equation: Relates pressure, velocity, and elevation in a steady, incompressible, inviscid flow: P + (1/2)ρV² + ρgh = constant.
• Energy equation: Considers energy losses due to friction and other factors: (P₁/ρg) + (V₁²/2g) + z₁ = (P₂/ρg) + (V₂²/2g) + z₂ + hL, where hL is the head loss.
• Momentum equation: Relates the sum of forces acting on a fluid element to the rate of change of momentum: ΣF = ṁ(V₂ - V₁).

Viscous Flow

• Navier-Stokes equations: A set of partial differential equations describing the motion of viscous fluids.
• These are complex and often solved numerically using computational fluid dynamics (CFD).
• Boundary layer: Thin layer near a solid surface where viscous effects are significant.
• Boundary layer separation: Occurs when the flow reverses direction near the surface, leading to increased drag and reduced lift.
• Drag: Force resisting the motion of a body through a fluid.
• Lift: Force perpendicular to the direction of motion, generated by an airfoil.
• Reynolds number (Re): Dimensionless number indicating the ratio of inertial forces to viscous forces, Re = (ρVL)/μ.
• Laminar flow occurs at low Reynolds numbers, while turbulent flow occurs at high Reynolds numbers.

Dimensional Analysis and Similitude

• Dimensional analysis: A method for reducing the number of variables needed in an experiment by using the dimensions of the variables.
• Buckingham Pi theorem: States that if there are n variables in a problem and k fundamental dimensions, then the problem can be reduced to n-k dimensionless groups.
• Similitude: The theory and art of predicting prototype performance from model observations.
• Geometric similarity: The model and prototype must have the same shape.
• Kinematic similarity: The velocity ratios must be the same.
• Dynamic similarity: The force ratios must be the same.

Open Channel Flow

• Open channel flow refers to the flow of liquids in conduits with a free surface (e.g., rivers, canals).
• Key parameters include channel geometry, flow depth, and channel slope.
• Hydraulic radius: Ratio of the cross-sectional area of flow to the wetted perimeter.
• Manning's equation: Empirical formula estimating the average velocity of a liquid flowing in an open channel: V = (1/n)R^(2/3)S^(1/2), where n is Manning's roughness coefficient, R is the hydraulic radius, and S is the channel slope.
• Specific energy: Energy per unit weight of fluid, used in analyzing open channel flow.
• Critical depth: Depth at which specific energy is minimum for a given flow rate.
• Hydraulic jump: Sudden transition from supercritical flow (high velocity, shallow depth) to subcritical flow (low velocity, deep depth).

Compressible Flow

• Compressible flow deals with fluids where density changes are significant, typically gases at high speeds.
• Mach number (M): Ratio of the flow velocity to the speed of sound, M = V/a.
• Subsonic flow: M < 1.
• Sonic flow: M = 1.
• Supersonic flow: M > 1.
• Shock waves: Abrupt changes in pressure, density, and temperature that occur in supersonic flow.
• Isentropic flow: Adiabatic and reversible flow, where entropy remains constant.
• Nozzles and diffusers: Devices used to accelerate or decelerate compressible flows.
• Converging-diverging nozzle: Used to accelerate a flow to supersonic speeds.

Measurement Techniques

• Pitot tube: Measures stagnation pressure.
• Static pressure tap: Measures static pressure.
• Venturi meter: Measures flow rate based on pressure drop.
• Orifice meter: Measures flow rate based on pressure drop through an orifice.
• Flow visualization: Techniques like dye injection or smoke to observe flow patterns.
• Hot-wire anemometry: Measures flow velocity based on heat transfer from a heated wire.
• Laser Doppler velocimetry (LDV): Measures flow velocity based on the Doppler shift of laser light.

Applications

• Pump and turbine design: Fluid mechanics principles are essential for designing efficient pumps and turbines.
• Pipe network analysis: Used to determine flow rates and pressure drops in pipe systems.
• Aerodynamics: Study of airflow around objects, crucial for aircraft and vehicle design.
• Hydrodynamics: Study of fluid motion in water, important for ship design and offshore structures.
• Microfluidics: Study of fluid behavior in micro-scale devices, used in biomedical and chemical applications.
• Meteorology and oceanography: Understanding atmospheric and oceanic flows.