2.2 Bernoulli's Principle and Continuity Equation

Bernoulli's Principle and the Continuity Equation are key to understanding fluid behavior in motion. They explain how pressure, velocity, and fluid properties interact, forming the basis for analyzing airflow around aircraft wings and through various systems.

These concepts are crucial for grasping how fluids behave in different situations. They help us predict and control fluid flow, which is essential for designing everything from airplane wings to water pipes and even understanding blood flow in our bodies.

Bernoulli's Equation Components

Pressure Components and Their Relationships

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• Bernoulli's Equation expresses the relationship between pressure, velocity, and elevation in a flowing fluid: $P + \frac{1}{2}\rho v^2 + \rho gh = constant$
• Static Pressure represents the pressure exerted by a stationary fluid on its surroundings, measured perpendicular to the flow direction
• Dynamic Pressure quantifies the kinetic energy per unit volume of a moving fluid, calculated as $\frac{1}{2}\rho v^2$
• Total Pressure sums static and dynamic pressures, remaining constant along a streamline in steady, inviscid, incompressible flow

Applications of Bernoulli's Principle

• Explains lift generation on aircraft wings (faster-moving air above the wing creates lower pressure)
• Demonstrates how fluid velocity increases as it flows through a constriction (venturi effect)
• Illustrates the principle behind carburetors in internal combustion engines (fuel drawn into the airstream)
• Describes the behavior of flow in pipes, including pressure drops and velocity changes

Limitations and Assumptions

• Assumes steady, inviscid, and incompressible flow along a streamline
• Neglects viscous effects and heat transfer
• May not accurately describe flow with significant energy losses or in highly turbulent conditions
• Requires modification for compressible flows (high-speed aerodynamics)

Continuity Equation Variables

Mass Conservation Principle

• Continuity Equation expresses mass conservation in fluid flow: $\dot{m} = \rho_1A_1v_1 = \rho_2A_2v_2$
• Mass Flow Rate remains constant for steady, incompressible flow through a system
• Velocity changes inversely with cross-sectional area in incompressible flow (faster in constrictions, slower in expansions)
• Cross-sectional Area variations affect fluid velocity and pressure (narrower sections increase velocity, decrease pressure)

Applications in Fluid Systems

• Designs efficient piping systems by predicting flow rates and pressure changes
• Optimizes nozzle shapes for jet engines and rocket propulsion
• Analyzes blood flow through varying vessel diameters in the circulatory system
• Determines appropriate pipe sizes for water distribution networks

Practical Considerations and Extensions

• Accounts for density changes in compressible flows (gases at high speeds or under significant pressure variations)
• Incorporates friction losses and turbulence effects in real-world applications
• Extends to multiphase flows (mixtures of liquids and gases) with appropriate modifications
• Applies to both steady-state and time-dependent flow scenarios with suitable adaptations