🍏Principles of Physics I Unit 13 – Fluid Mechanics

Key Concepts and Definitions

• Fluids include liquids and gases that continuously deform under shear stress
• Density $(\rho)$ is the mass per unit volume of a substance, measured in $kg/m^3$
• Pressure $(P)$ is the force per unit area, typically measured in pascals $(Pa)$ or $N/m^2$
• Buoyancy is the upward force exerted by a fluid on an object immersed in it
  • Governed by Archimedes' principle, which states that the buoyant force is equal to the weight of the fluid displaced by the object
• Viscosity is a measure of a fluid's resistance to flow or deformation
  • Measured by the coefficient of viscosity $(\mu)$, which has units of $Pa \cdot s$
• Laminar flow occurs when fluid moves in parallel layers without mixing, characterized by low Reynolds numbers
• Turbulent flow is characterized by chaotic and irregular motion, with high Reynolds numbers
• Bernoulli's principle relates pressure, velocity, and elevation in a flowing fluid, assuming constant density and no viscosity

Properties of Fluids

• Fluids are substances that continually deform under applied shear stress
  • Liquids have a definite volume but take the shape of their container
  • Gases expand to fill their container and are easily compressible
• Density is a fundamental property of fluids, defined as mass per unit volume
  • Water has a density of $1000 \, kg/m^3$ at standard temperature and pressure
• Specific gravity compares the density of a substance to that of a reference substance (usually water for liquids and air for gases)
• Pressure is the force per unit area acting on a surface
  • Atmospheric pressure is the force exerted by the weight of the atmosphere (101,325 Pa at sea level)
• Compressibility is the ability of a fluid to change its volume under pressure
  • Liquids are generally considered incompressible, while gases are highly compressible
• Surface tension is the result of cohesive forces between liquid molecules at the surface
  • Causes phenomena such as capillary action and the formation of droplets
• Viscosity is a fluid's resistance to flow, arising from internal friction between molecules
  • Honey has a higher viscosity than water, making it flow more slowly

Fluid Statics and Pressure

• Fluid statics deals with fluids at rest and the forces they exert on surfaces
• Pressure at a point in a static fluid depends on the depth and the fluid's density
  • Pressure increases linearly with depth: $P = \rho gh$, where $h$ is the depth below the surface
• Pascal's principle states that pressure applied to an enclosed fluid is transmitted undiminished to every part of the fluid and the walls of the container
  • Hydraulic systems (car brakes) use this principle to multiply force
• Gauge pressure is the pressure relative to atmospheric pressure, while absolute pressure is the total pressure (gauge + atmospheric)
• Hydrostatic pressure is the pressure exerted by a fluid at rest due to its weight
  • In a tank of water, hydrostatic pressure increases with depth
• Buoyancy is the upward force exerted by a fluid on an immersed object
  • Archimedes' principle: the buoyant force equals the weight of the displaced fluid
  • Objects with a density less than the fluid will float (ice on water)

Fluid Dynamics and Flow

• Fluid dynamics studies the motion and behavior of fluids
• Streamlines are imaginary lines that trace the path of fluid particles
  • In steady flow, streamlines are parallel and do not intersect
• Laminar flow occurs when fluid moves in parallel layers without mixing
  • Characterized by low Reynolds numbers (ratio of inertial to viscous forces)
  • Laminar flow in pipes exhibits a parabolic velocity profile
• Turbulent flow is characterized by chaotic, irregular motion and mixing
  • Occurs at high Reynolds numbers and is common in many real-world situations (airflow around a car)
• Continuity equation states that the mass flow rate is constant for an incompressible fluid
  • $A_1v_1 = A_2v_2$, where $A$ is the cross-sectional area and $v$ is the velocity
• Ideal fluids are inviscid (no viscosity), incompressible, and have steady flow
  • Real fluids have viscosity, can be compressible, and may exhibit unsteady flow
• Pressure drops along a pipe due to viscous effects and friction
  • Pressure loss is proportional to the pipe length, fluid velocity, and viscosity

Bernoulli's Principle

• Bernoulli's principle relates pressure, velocity, and elevation in a flowing fluid
  • Assumes constant density, steady flow, and no viscosity
  • States that an increase in fluid velocity is accompanied by a decrease in pressure, and vice versa
• Bernoulli's equation: $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$
  • $P$ is the static pressure, $\frac{1}{2}\rho v^2$ is the dynamic pressure, and $\rho gh$ is the hydrostatic pressure
• Venturi effect is a consequence of Bernoulli's principle
  • A constriction in a pipe causes the fluid velocity to increase and the pressure to decrease
  • Used in carburetors to create a low-pressure region that draws fuel into the airstream
• Pitot tubes measure fluid velocity by comparing the stagnation pressure (at the tip) to the static pressure
  • Velocity is calculated using Bernoulli's equation: $v = \sqrt{\frac{2(P_0 - P)}{\rho}}$
• Lift on an airplane wing is a result of Bernoulli's principle
  • Airflow over the curved upper surface is faster than the lower surface, creating a pressure difference

Viscosity and Fluid Resistance

• Viscosity is a fluid's resistance to flow or deformation
  • Arises from internal friction between fluid molecules
  • Measured by the coefficient of viscosity $(\mu)$, which has units of $Pa \cdot s$
• Newton's law of viscosity relates shear stress $(\tau)$ to the velocity gradient $(\frac{dv}{dy})$
  • $\tau = \mu \frac{dv}{dy}$, where $\mu$ is the coefficient of viscosity
• Newtonian fluids have a constant viscosity that is independent of shear stress
  • Examples include water, air, and most common liquids
• Non-Newtonian fluids have a viscosity that depends on shear stress
  • Shear-thinning fluids (paint) become less viscous under stress, while shear-thickening fluids (cornstarch in water) become more viscous
• Reynolds number $(Re)$ is a dimensionless quantity that characterizes the flow regime
  • $Re = \frac{\rho vD}{\mu}$, where $D$ is a characteristic length (pipe diameter)
  • Low $Re$ indicates laminar flow, while high $Re$ suggests turbulent flow
• Viscous drag is the force resisting motion due to fluid viscosity
  • Depends on the object's shape, size, and velocity, as well as the fluid's density and viscosity
• Stokes' law gives the drag force on a spherical object in laminar flow
  • $F_D = 6\pi\mu rv$, where $r$ is the sphere's radius and $v$ is its velocity

Applications in Real-World Systems

• Fluid mechanics principles are essential in designing and analyzing various systems
• Hydraulic systems use incompressible fluids (oil) to transmit force
  • Based on Pascal's principle, they can multiply force for applications like car brakes and lifts
• Aerodynamics is the study of airflow around objects
  • Streamlining reduces drag by minimizing flow separation (cars, airplanes)
  • Lift generated by airfoils (wings) is a result of Bernoulli's principle and circulation
• Piping systems transport fluids for industrial, residential, and commercial use
  • Pressure drops due to viscous effects and friction must be considered in design
  • Pumps are used to overcome pressure losses and maintain flow
• Turbomachinery includes devices that transfer energy between a fluid and a rotor
  • Pumps and fans add energy to the fluid, while turbines extract energy from the fluid
  • Efficiency depends on factors like blade design, flow rate, and fluid properties
• Cardiovascular system can be modeled using fluid mechanics principles
  • Heart acts as a pump, blood vessels are like pipes, and valves ensure unidirectional flow
  • Atherosclerosis (plaque buildup) narrows arteries, increasing resistance to blood flow
• Weather and climate are influenced by atmospheric fluid dynamics
  • Pressure gradients drive wind patterns, while temperature differences create convection currents
  • Coriolis effect due to Earth's rotation affects large-scale fluid motion

Problem-Solving Strategies

• Identify the relevant fluid properties (density, viscosity) and flow characteristics (laminar/turbulent, steady/unsteady)
• Determine the appropriate governing equations and principles
  • Continuity equation for mass conservation
  • Bernoulli's equation for pressure-velocity relationships
  • Newton's law of viscosity for shear stress and viscous effects
• Sketch the problem and establish a coordinate system
  • Label known and unknown quantities, and identify boundary conditions
• Simplify the problem by making reasonable assumptions
  • Incompressible flow, steady-state conditions, or inviscid fluid, if appropriate
• Apply the governing equations and solve for the desired quantities
  • Use algebra, calculus, or numerical methods, depending on the complexity
• Check the units and the reasonableness of the answer
  • Verify that the solution is physically plausible and consistent with expectations
• Analyze the results and consider the implications
  • Identify trends, limitations, or potential improvements to the system or design
• Use dimensional analysis to check the consistency of equations and to derive relationships between variables
  • Buckingham Pi theorem states that any physically meaningful equation can be expressed in terms of dimensionless groups