14.6 Bernoulli’s Equation

Bernoulli's_Equation_0### is a cornerstone of fluid dynamics, linking pressure, velocity, and elevation in steady, incompressible flows. It's the math behind why planes fly and how water flows through pipes. Understanding this principle helps us grasp the behavior of fluids in motion.

This equation has wide-ranging applications, from designing airplane wings to creating efficient plumbing systems. However, it's important to note its limitations, as real-world fluids often behave differently due to factors like viscosity and turbulence.

Bernoulli's Equation

Components of Bernoulli's equation

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• Bernoulli's equation relates pressure, velocity, and elevation in a steady, incompressible, and inviscid fluid flow along a streamline: $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$
  • $P$ represents the pressure, which is the force per unit area exerted by the fluid on its surroundings (static pressure)
  • $\rho$ denotes the density of the fluid, defined as the mass per unit volume
  • $v$ signifies the velocity of the fluid, describing both the speed and direction of fluid flow
  • $g$ is the acceleration due to gravity, which has a constant value of approximately $9.81 \text{ m/s}^2$ on Earth
  • $h$ represents the height of the fluid above a reference level, indicating the elevation relative to a chosen datum (sea level, ground level)
• Each term in the equation represents a form of energy per unit volume: pressure energy ($P$), kinetic energy ($\frac{1}{2}\rho v^2$), and potential energy ($\rho gh$)
  • The kinetic energy term ($\frac{1}{2}\rho v^2$) is also known as dynamic pressure

Applications of Bernoulli's principle

• Bernoulli's principle, derived from Bernoulli's equation by assuming constant elevation and density, states that an increase in fluid velocity leads to a decrease in pressure and vice versa
• Venturi effect: Narrowing a pipe increases fluid velocity and decreases pressure, utilized in carburetors to mix air and fuel and in aspirators to create suction
• Pitot tubes: Measure fluid velocity by comparing the static pressure and the dynamic pressure, commonly used in airspeed indicators for aircraft
• Lift generation: Faster airflow over the top of an airfoil compared to the bottom creates lower pressure above the wing, resulting in an upward lift force that enables flight in airplanes and birds

Bernoulli's equation vs energy conservation

• Bernoulli's equation is a statement of the conservation of energy principle applied to fluid flow
• The constant in Bernoulli's equation represents the total energy per unit volume along a streamline
• In the absence of energy losses due to factors like friction, the total energy remains constant as the fluid moves along the streamline
• Changes in pressure, velocity, or elevation along the streamline result in the conversion of energy from one form to another while maintaining a constant total energy

Real-world examples of Bernoulli's principle

• Lift in airplane wings:
  • The airfoil shape of a wing causes the air flowing over the top surface to move faster than the air beneath the wing
  • According to Bernoulli's principle, the faster airflow above the wing results in lower pressure compared to the higher pressure below the wing
  • The pressure difference creates an upward lift force that enables the airplane to fly
• Flow of water through pipes:
  • Constrictions or narrowing in pipes lead to increased water velocity and decreased pressure at those points
  • Pressure drops caused by high-velocity flow can result in cavitation, where bubbles form and collapse, potentially damaging the pipes
  • Understanding Bernoulli's principle helps engineers design efficient piping systems and avoid flow-related issues (leaks, pipe bursts)

Limitations of Bernoulli's equation

• Bernoulli's equation relies on several assumptions:
  1. Steady flow: Fluid properties (velocity, pressure, density) do not change with time at any given point in the flow
  2. Incompressible fluid: The density of the fluid remains constant throughout the flow, typically valid for liquids and gases at low speeds
  3. Inviscid fluid: The fluid has no viscosity, meaning there are no viscous forces or friction within the fluid or between the fluid and the boundaries
  4. Flow along a streamline: The equation applies to a specific path followed by fluid particles, known as a streamline
• Real fluids have viscosity, which leads to energy losses and pressure drops due to friction, causing Bernoulli's equation to overestimate velocity and underestimate pressure in practical scenarios
• Compressible fluids, such as gases at high speeds, experience significant density changes, violating the incompressible flow assumption
• Turbulent flow and flow separation can occur in real-world situations, invalidating the steady and streamline flow assumptions
• Despite its limitations, Bernoulli's equation provides valuable insights and reasonable approximations for many fluid flow problems encountered in engineering and everyday life (water hoses, wind instruments)

Fluid Dynamics and Related Concepts

• Fluid dynamics is the study of fluid motion and its interactions with surfaces and other fluids
• The continuity equation is a fundamental principle in fluid dynamics that relates the flow rate of a fluid through different cross-sectional areas
• Laminar flow occurs when fluid particles move in smooth, parallel layers without mixing, which is essential for the application of Bernoulli's equation