🌊College Physics II – Mechanics, Sound, Oscillations, and Waves Unit 14 – Fluid Mechanics

Key Concepts and Definitions

• Fluids include both liquids and gases that continuously deform under shear stress
• Density ($\rho$) is the mass per unit volume of a substance, expressed as $\rho = \frac{m}{V}$
• Pressure ($P$) is the force per unit area, calculated using the formula $P = \frac{F}{A}$
• Buoyancy is the upward force exerted by a fluid on an object immersed in it
• Archimedes' principle states that the buoyant force on an object is equal to the weight of the fluid displaced by the object
• Fluid dynamics studies the motion and behavior of fluids, including their velocity, pressure, and density
• Viscosity is a measure of a fluid's resistance to flow or deformation
• Laminar flow occurs when fluid moves in parallel layers without mixing, while turbulent flow is characterized by chaotic and irregular motion

Properties of Fluids

• Fluids are substances that can flow and take the shape of their container, such as water, air, and oil
• Fluids are characterized by their ability to deform continuously under shear stress, which is a force applied parallel to the surface
• Fluids have a definite volume but no fixed shape, allowing them to conform to the boundaries of their container
• The density of a fluid determines its mass per unit volume and affects its behavior under various conditions
  • Water has a density of approximately 1,000 kg/m³ at standard temperature and pressure
  • Air has a much lower density, typically around 1.225 kg/m³ at sea level
• Fluids can be compressed to varying degrees, with gases being more compressible than liquids
• The viscosity of a fluid is a measure of its resistance to flow and is influenced by factors such as temperature and pressure
  • Honey has a higher viscosity than water, making it flow more slowly
• Surface tension is a property that allows fluids to resist external forces due to cohesive forces between molecules at the surface

Fluid Statics and Pressure

• Fluid statics deals with the study of fluids at rest and the forces they exert on surfaces
• Pressure in a fluid is the force per unit area exerted perpendicular to a surface
• The pressure at a point in a fluid depends on the depth, density of the fluid, and acceleration due to gravity
• Pascal's principle states that pressure applied to a confined fluid is transmitted undiminished in all directions and acts with equal force on equal areas
  • This principle is used in hydraulic systems, such as car brakes and lifts
• Hydrostatic pressure is the pressure exerted by a fluid at rest due to its weight and is given by $P = \rho gh$, where $h$ is the depth below the surface
• Gauge pressure is the pressure relative to atmospheric pressure, while absolute pressure is the sum of gauge pressure and atmospheric pressure
• The pressure at a given depth in a fluid is the same in all directions, as long as the fluid is at rest and not accelerating

Buoyancy and Archimedes' Principle

• Buoyancy is the upward force exerted by a fluid on an object immersed in it, which opposes the object's weight
• Archimedes' principle states that the buoyant force on an object is equal to the weight of the fluid displaced by the object
  • This principle explains why objects appear to lose weight when submerged in a fluid
• The buoyant force depends on the density of the fluid, the volume of the displaced fluid, and the acceleration due to gravity
• An object will float if the buoyant force is greater than its weight, sink if its weight is greater than the buoyant force, or remain neutrally buoyant if the two forces are equal
  • A helium balloon floats because the buoyant force of the air displaced is greater than the weight of the balloon and its contents
• The apparent weight of an object in a fluid is the actual weight minus the buoyant force
• The center of buoyancy is the point where the buoyant force acts on an object and is located at the centroid of the displaced fluid volume
• Stability of floating objects depends on the relative positions of the center of gravity and the center of buoyancy

Fluid Dynamics and Flow

• Fluid dynamics studies the motion and behavior of fluids, including their velocity, pressure, and density
• Streamlines are imaginary lines that represent the path of fluid particles in a steady flow
• Laminar flow occurs when fluid moves in parallel layers without mixing, resulting in a smooth and predictable motion
  • Blood flow in small blood vessels is typically laminar
• Turbulent flow is characterized by chaotic and irregular motion, with fluid particles moving in random directions
  • Airflow around an airplane wing can become turbulent at high speeds
• The Reynolds number (Re) is a dimensionless quantity that predicts the transition from laminar to turbulent flow, given by $Re = \frac{\rho vD}{\mu}$, where $v$ is velocity, $D$ is a characteristic length, and $\mu$ is dynamic viscosity
• Continuity equation states that the mass flow rate in a steady flow remains constant, expressed as $\rho_1 A_1 v_1 = \rho_2 A_2 v_2$, where $A$ is the cross-sectional area
• Viscosity is a measure of a fluid's resistance to flow or deformation and is caused by the friction between fluid layers
  • The viscosity of motor oil helps to lubricate engine components and prevent wear

Bernoulli's Equation

• Bernoulli's equation is a fundamental principle in fluid dynamics that relates pressure, velocity, and elevation in a steady, incompressible, and frictionless flow
• The equation is stated as $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$ along a streamline, where $P$ is pressure, $v$ is velocity, and $h$ is elevation
• The equation demonstrates that an increase in fluid velocity leads to a decrease in pressure, and vice versa
  • This principle is used in the design of aircraft wings to generate lift
• The Venturi effect is a consequence of Bernoulli's equation, where a constriction in a pipe leads to an increase in velocity and a decrease in pressure
  • Carburetors in engines use the Venturi effect to draw fuel into the airstream
• Pitot tubes, used to measure fluid velocity, rely on Bernoulli's equation by comparing the stagnation pressure and static pressure
• The equation assumes steady flow, incompressible fluid, and no friction, which are reasonable approximations in many real-world situations
• Bernoulli's equation can be modified to account for energy losses due to friction, turbulence, or other factors in real fluids

Applications in Real-World Systems

• Fluid mechanics principles are applied in various fields, such as aerospace, automotive, and biomedical engineering
• Aerodynamics utilizes fluid mechanics to design aircraft wings, control surfaces, and engines for optimal performance and efficiency
  • The shape of an airplane wing is designed to create a pressure difference and generate lift
• Hydrodynamics applies fluid mechanics to the study of water and other liquids in motion, such as in pipes, channels, and hydraulic systems
  • The design of ship hulls and propellers relies on hydrodynamic principles to minimize drag and improve efficiency
• Cardiovascular system can be modeled using fluid mechanics principles to understand blood flow, pressure, and the function of heart valves
  • Stenosis, a narrowing of blood vessels, can be analyzed using the continuity equation and Bernoulli's equation
• Meteorology uses fluid mechanics to study atmospheric circulation, weather patterns, and the formation of storms and hurricanes
• Fluid power systems, such as hydraulic presses and pneumatic tools, rely on the principles of fluid statics and dynamics to transmit force and motion
• Environmental engineering applies fluid mechanics to study the transport and dispersion of pollutants in air, water, and soil
  • The spread of oil spills in the ocean can be modeled using fluid dynamics principles

Common Problems and Solutions

• Pressure measurement: Manometers, pressure gauges, and transducers are used to measure pressure in fluids
  • A U-tube manometer measures the pressure difference between two points using a liquid column
• Flow rate measurement: Orifice plates, Venturi meters, and rotameters are used to measure the flow rate of fluids in pipes
  • An orifice plate creates a pressure drop that can be related to the flow rate using Bernoulli's equation
• Pipe flow: Friction losses in pipes can be calculated using the Darcy-Weisbach equation, which relates pressure drop to pipe diameter, length, and fluid properties
  • The Moody diagram is used to determine the friction factor based on the Reynolds number and pipe roughness
• Drag reduction: Streamlining shapes, such as airfoils and boat hulls, can reduce drag by minimizing flow separation and turbulence
  • Dimples on a golf ball create a thin turbulent boundary layer that reduces drag and increases lift
• Cavitation: Occurs when the local pressure in a fluid drops below the vapor pressure, leading to the formation of vapor bubbles and potential damage to surfaces
  • Cavitation can be prevented by maintaining sufficient pressure and avoiding sharp changes in flow geometry
• Turbulence: Can be mitigated by using flow straighteners, such as honeycomb structures or screens, to reduce swirl and non-uniformity in the flow
• Boundary layer control: Techniques such as suction, blowing, and vortex generators can be used to manipulate the boundary layer and delay flow separation
  • Vortex generators on aircraft wings create small vortices that energize the boundary layer and prevent stall at high angles of attack