💧Fluid Mechanics Unit 3 – Pressure and Fluid Forces

Key Concepts and Definitions

• Fluid a substance that deforms continuously under shear stress (liquids and gases)
• Pressure force per unit area, typically measured in pascals (Pa) or pounds per square inch (psi)
• Hydrostatic pressure pressure exerted by a fluid at rest due to its weight
  • Depends on fluid density, acceleration due to gravity, and depth
• Buoyancy upward force exerted by a fluid on an object immersed in it
• Archimedes' principle states that the buoyant force is equal to the weight of the fluid displaced by the object
• Gauge pressure pressure relative to atmospheric pressure
• Absolute pressure sum of gauge pressure and atmospheric pressure

Pressure in Fluids

• Pressure in a fluid increases with depth due to the weight of the fluid above
• Pressure at a point in a fluid acts equally in all directions
• Pascal's law states that pressure applied to a confined fluid is transmitted undiminished throughout the fluid
  • Enables hydraulic systems (brakes, lifts, and presses)
• Pressure in a fluid depends on the fluid's density and the acceleration due to gravity
• Atmospheric pressure pressure exerted by the Earth's atmosphere (approximately 101,325 Pa or 14.7 psi at sea level)
• Fluid pressure can be expressed as either gauge pressure or absolute pressure

Hydrostatic Pressure

• Hydrostatic pressure is the pressure exerted by a fluid at rest due to its weight
• Calculated using the formula: $p = \rho gh$
  • $p$ is the hydrostatic pressure
  • $\rho$ is the fluid density
  • $g$ is the acceleration due to gravity
  • $h$ is the depth below the fluid surface
• Hydrostatic pressure increases linearly with depth
• Independent of the shape of the container
• Hydrostatic pressure acts perpendicular to any submerged surface

Buoyancy and Archimedes' Principle

• Buoyancy is the upward force exerted by a fluid on an object immersed in it
• Archimedes' principle states that the buoyant force is equal to the weight of the fluid displaced by the object
  • Mathematically: $F_b = \rho gV$, where $F_b$ is the buoyant force, $\rho$ is the fluid density, $g$ is the acceleration due to gravity, and $V$ is the volume of fluid displaced
• An object will float if its weight is less than the buoyant force acting on it
• An object will sink if its weight is greater than the buoyant force
• Neutral buoyancy occurs when an object's weight equals the buoyant force, causing it to remain suspended in the fluid

Fluid Forces on Submerged Surfaces

• Fluid pressure exerts a force on any surface submerged in the fluid
• The force acting on a submerged surface depends on the pressure distribution and the surface area
• Hydrostatic force on a submerged surface is calculated using the formula: $F = pA$
  • $F$ is the force
  • $p$ is the pressure at the centroid of the surface
  • $A$ is the area of the surface
• The point of application of the hydrostatic force is called the center of pressure
  • Located below the centroid of the surface for a vertically oriented surface
• Fluid forces on curved surfaces can be resolved into horizontal and vertical components

Pressure Measurement Techniques

• Manometers devices that measure pressure using a column of liquid (usually mercury or water)
  • Pressure is determined by the height difference of the liquid column
• Bourdon gauges mechanical devices that measure pressure using the deformation of a curved tube
  • Pressure causes the tube to straighten, moving a pointer on a calibrated scale
• Pressure transducers convert pressure into an electrical signal
  • Types include strain gauges, capacitive, and piezoelectric transducers
• Barometers measure atmospheric pressure
  • Mercury barometers use a column of mercury to measure pressure
  • Aneroid barometers use the deformation of a sealed capsule to measure pressure

Applications in Engineering

• Hydraulic systems use fluid pressure to transmit force (brakes, lifts, and presses)
• Hydrostatic bearings use fluid pressure to support rotating shafts with minimal friction
• Submersibles and underwater structures must be designed to withstand hydrostatic pressure
• Fluid forces on dams, tanks, and pipelines must be considered in their design
• Pressure measurements are crucial in monitoring and controlling various industrial processes (oil and gas, chemical processing, and power generation)
• Aerodynamic and hydrodynamic forces on vehicles (aircraft, ships, and automobiles) are influenced by fluid pressure distribution

Problem-Solving Strategies

• Identify the type of fluid (liquid or gas) and its properties (density, viscosity)
• Determine the relevant pressure (gauge, absolute, or hydrostatic)
• Sketch the problem, labeling key dimensions and forces
• Apply the appropriate equations and principles (hydrostatic pressure, Archimedes' principle, or fluid force calculations)
  • Use $p = \rho gh$ for hydrostatic pressure
  • Use $F_b = \rho gV$ for buoyant force
  • Use $F = pA$ for fluid force on a submerged surface
• Consider the direction of forces and pressure gradients
• Double-check units and ensure consistency throughout the problem
• Verify that the solution makes physical sense and satisfies any given constraints