💨Fluid Dynamics Unit 1 – Fluid properties and statics

Key Concepts and Definitions

• Fluid dynamics studies the behavior of fluids (liquids and gases) at rest and in motion
• Fluids are substances that deform continuously under applied shear stress (no fixed shape)
• Fluid statics deals with fluids at rest, while fluid dynamics deals with fluids in motion
• Density $(\rho)$ is the mass per unit volume of a substance, expressed as $\rho = \frac{m}{V}$
  • Density varies with temperature and pressure (especially for gases)
• Specific weight $(\gamma)$ is the weight per unit volume of a substance, expressed as $\gamma = \frac{w}{V} = \rho g$
  • $g$ is the acceleration due to gravity (9.81 m/s² on Earth)
• Specific gravity $(SG)$ is the ratio of a substance's density to the density of a reference substance (usually water for liquids and air for gases)
• Viscosity $(\mu)$ is a measure of a fluid's resistance to deformation under shear stress
  • Dynamic viscosity $(\mu)$ is the ratio of shear stress to shear rate, expressed in units of Pa·s or N·s/m²
  • Kinematic viscosity $(\nu)$ is the ratio of dynamic viscosity to density, expressed in units of m²/s

Properties of Fluids

• Compressibility is the ability of a fluid to change its volume under pressure
  • Liquids are generally considered incompressible, while gases are compressible
• Surface tension is the tendency of a fluid to minimize its surface area due to cohesive forces between molecules
  • Capillary action (rise or fall of liquid in a narrow tube) is caused by surface tension
• Vapor pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its liquid or solid phase
  • Boiling occurs when the vapor pressure equals the ambient pressure
• Newtonian fluids have a linear relationship between shear stress and shear rate (constant viscosity)
  • Examples include water, air, and most common fluids
• Non-Newtonian fluids have a non-linear relationship between shear stress and shear rate (viscosity varies with shear rate)
  • Examples include blood, paint, and shampoo
• Ideal fluids are inviscid (no viscosity), incompressible, and have no surface tension
  • While no real fluids are ideal, many fluids can be approximated as ideal under certain conditions

Fluid Statics and Pressure

• Pressure $(p)$ is the force per unit area acting on a surface, expressed as $p = \frac{F}{A}$
  • Pressure is a scalar quantity and acts equally in all directions at a given point in a static fluid
• Hydrostatic pressure is the pressure exerted by a fluid at rest due to its weight
  • Hydrostatic pressure increases with depth $(h)$ in a fluid, expressed as $p = \rho g h$
• Gauge pressure is the pressure relative to atmospheric pressure, while absolute pressure is the total pressure (gauge + atmospheric)
• Pascal's law states that pressure applied to a confined fluid is transmitted undiminished in all directions
  • This principle is used in hydraulic systems (brakes, lifts, etc.)
• The hydrostatic paradox states that the pressure at a given depth in a fluid is independent of the shape of the container
• Atmospheric pressure is the pressure exerted by the Earth's atmosphere, which decreases with altitude
  • Standard atmospheric pressure is 101,325 Pa (1 atm) at sea level

Hydrostatic Forces

• Hydrostatic force is the force exerted by a fluid at rest on a submerged surface
• The magnitude of the hydrostatic force depends on the pressure distribution and the area of the surface
  • $F = \int p dA$, where $p$ is the pressure and $dA$ is the differential area element
• The center of pressure is the point on a submerged surface where the resultant hydrostatic force acts
  • For a rectangular vertical surface, the center of pressure is located below the centroid by a distance $\frac{I}{A \bar{y}}$, where $I$ is the second moment of area and $\bar{y}$ is the depth of the centroid
• Hydrostatic force on a curved surface can be resolved into horizontal and vertical components
  • The horizontal component is equal to the hydrostatic force on the projected area of the surface
  • The vertical component is equal to the weight of the fluid column above the surface
• Dams and tanks must be designed to withstand the hydrostatic forces acting on their walls
  • The force increases with depth, so the walls must be thicker at the bottom

Buoyancy and Stability

• Buoyancy is the upward force exerted by a fluid on a submerged or partially submerged object
• Archimedes' principle states that the buoyant force is equal to the weight of the fluid displaced by the object
  • $F_b = \rho g V_d$, where $V_d$ is the volume of the displaced fluid
• An object will float if its weight is less than the buoyant force, sink if its weight is greater, and be neutrally buoyant if they are equal
• The stability of a floating object depends on the relative positions of its center of gravity $(G)$ and center of buoyancy $(B)$
  • The center of buoyancy is the centroid of the displaced fluid volume
• A floating object is stable if $G$ is below $B$, unstable if $G$ is above $B$, and neutrally stable if $G$ and $B$ coincide
  • The metacenter $(M)$ is the point where the line of action of the buoyant force intersects the object's centerline when tilted
  • The metacentric height $(GM)$ is a measure of the object's stability, with larger values indicating greater stability
• Ships and boats must be designed with adequate stability to prevent capsizing
  • Adding weight below the center of gravity or increasing the beam (width) can improve stability

Fluid Measurements and Instruments

• Pressure measurement devices include manometers, bourdon tubes, and pressure transducers
  • Manometers measure pressure by balancing the fluid column against a known reference pressure (usually atmospheric)
  • Bourdon tubes measure pressure by the deformation of a curved tube, which is connected to a mechanical dial or electronic sensor
  • Pressure transducers convert pressure into an electrical signal using various techniques (piezoelectric, capacitive, etc.)
• Velocity measurement devices include pitot tubes, hot-wire anemometers, and laser doppler velocimeters
  • Pitot tubes measure velocity by comparing the stagnation pressure (at the tip) to the static pressure (at the side)
  • Hot-wire anemometers measure velocity by the cooling effect of the fluid on a heated wire
  • Laser doppler velocimeters measure velocity by the doppler shift of laser light scattered by particles in the fluid
• Flow measurement devices include orifice plates, venturi meters, and turbine meters
  • Orifice plates measure flow rate by the pressure drop across a narrow opening
  • Venturi meters measure flow rate by the pressure difference between a converging and diverging section
  • Turbine meters measure flow rate by the rotation speed of a turbine placed in the flow
• Calibration and uncertainty analysis are essential for ensuring the accuracy and reliability of fluid measurements
  • Calibration involves comparing the instrument's readings to a known standard
  • Uncertainty analysis involves estimating the possible errors in the measurements and their propagation through calculations

Applications in Engineering

• Aerodynamics is the study of the motion of air and its interaction with solid objects, such as aircraft wings and wind turbine blades
  • Lift and drag forces on airfoils are determined by the pressure distribution and boundary layer behavior
• Hydrodynamics is the study of the motion of liquids and their interaction with solid boundaries, such as ship hulls and hydraulic turbines
  • Cavitation (formation and collapse of vapor bubbles) can cause damage to propellers and turbines
• Pipe flow is the transport of fluids through closed conduits, such as water distribution networks and oil pipelines
  • Pressure drop and flow rate are determined by the pipe geometry, fluid properties, and boundary conditions (pumps, valves, etc.)
• Open channel flow is the flow of liquids with a free surface, such as rivers and canals
  • Flow depth and velocity are influenced by the channel slope, roughness, and cross-sectional shape
• Heat transfer often involves the flow of fluids, such as in heat exchangers and cooling systems
  • Convection (heat transfer by fluid motion) can be natural (buoyancy-driven) or forced (externally-driven)
• Environmental fluid mechanics deals with the flow of air and water in the natural environment, such as atmospheric circulation and ocean currents
  • Pollutant dispersion and mixing are important considerations for air and water quality management

Problem-Solving Techniques

• Identify the relevant fluid properties (density, viscosity, etc.) and flow conditions (pressure, velocity, etc.)
• Draw a clear and labeled sketch of the problem, including the coordinate system and boundary conditions
• List the known quantities and the desired unknown quantities
• Select the appropriate governing equations and simplify them based on reasonable assumptions
  • Conservation of mass (continuity equation)
  • Conservation of momentum (Navier-Stokes equations)
  • Conservation of energy (Bernoulli equation, energy equation)
• Solve the equations analytically (if possible) or numerically (using computational fluid dynamics)
  • Analytical solutions are exact but limited to simple geometries and boundary conditions
  • Numerical solutions are approximate but can handle complex geometries and boundary conditions
• Check the units and order of magnitude of the results to ensure they are physically reasonable
• Perform a sensitivity analysis to determine the influence of key parameters on the solution
• Validate the results with experimental data or benchmark solutions, if available
• Interpret the results in the context of the original problem and draw appropriate conclusions