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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 flow along a : P+12ρv2+ρgh=constantP + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}
    • PP represents the pressure, which is the force per unit area exerted by the fluid on its surroundings ()
    • ρ\rho denotes the of the fluid, defined as the mass per unit volume
    • vv signifies the velocity of the fluid, describing both the speed and direction of fluid flow
    • gg is the , which has a constant value of approximately 9.81 m/s29.81 \text{ m/s}^2 on Earth
    • hh 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 (PP), (12ρv2\frac{1}{2}\rho v^2), and (ρgh\rho gh)
    • The kinetic energy term (12ρv2\frac{1}{2}\rho v^2) is also known as

Applications of 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
  • : Narrowing a pipe increases fluid velocity and decreases pressure, utilized in carburetors to mix air and fuel and in aspirators to create suction
  • : Measure fluid velocity by comparing the static pressure and the dynamic pressure, commonly used in airspeed indicators for aircraft
  • generation: Faster airflow over the top of an 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 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 , 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. : Fluid properties (velocity, pressure, density) do not change with time at any given point in the flow
    2. : 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
  • and 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 is the study of fluid motion and its interactions with surfaces and other fluids
  • The is a fundamental principle in fluid dynamics that relates the flow rate of a fluid through different cross-sectional areas
  • occurs when fluid particles move in smooth, parallel layers without mixing, which is essential for the application of Bernoulli's equation

Key Terms to Review (35)

Acceleration Due to Gravity: Acceleration due to gravity, often denoted as 'g', is the acceleration experienced by an object due to the Earth's gravitational pull. This constant acceleration affects the motion of objects near the Earth's surface, influencing various physical phenomena such as free fall, mass, weight, and gravitational fields.
Airfoil: An airfoil is a specially shaped surface, such as the wing of an aircraft or the blade of a propeller, that is designed to generate lift when moving through a fluid, such as air. The unique shape of an airfoil creates a difference in pressure between the upper and lower surfaces, resulting in a net upward force that counteracts the weight of the aircraft and allows it to fly.
Bernoulli: Bernoulli's Equation describes the relationship between pressure, velocity, and elevation in a moving fluid. It is derived from the principle of conservation of energy for flowing fluids.
Bernoulli’s equation: Bernoulli's equation describes the relationship between the pressure, velocity, and elevation in a moving fluid. It is derived from the conservation of energy principle for incompressible, non-viscous fluids.
Bernoulli's Equation: Bernoulli's equation is a fundamental principle in fluid dynamics that describes the relationship between pressure, flow speed, and elevation in a flowing fluid. It states that as the speed of a fluid increases, the pressure within the fluid decreases, and vice versa.
Bernoulli’s principle: Bernoulli's principle states that in a steady flow, the sum of the pressure energy, kinetic energy, and potential energy per unit volume is constant along a streamline. It implies that an increase in fluid speed results in a decrease in pressure or potential energy.
Bernoulli's Principle: Bernoulli's principle is a fundamental concept in fluid dynamics that describes the relationship between the pressure, speed, and elevation in a flowing fluid. It states that as the speed of a fluid increases, the pressure within the fluid decreases, and vice versa.
Cavitation: Cavitation is the formation and subsequent collapse of small vapor-filled cavities or bubbles within a liquid, often in areas of rapid or turbulent flow. This phenomenon can have significant impacts on the performance and lifespan of various fluid systems, including those found in rocket propulsion and Bernoulli's principle.
Conservation of Energy: The conservation of energy principle states that energy cannot be created or destroyed, only transformed from one form to another. This fundamental concept links various phenomena, illustrating how mechanical, kinetic, and potential energies interconvert while keeping the total energy constant in a closed system.
Continuity Equation: The continuity equation is a fundamental principle in fluid dynamics that describes the conservation of mass within a fluid flow. It states that the rate of change of mass within a given volume is equal to the net flow of mass into or out of that volume.
Daniel Bernoulli: Daniel Bernoulli was a Swiss mathematician and physicist who made significant contributions to the field of fluid dynamics. He is best known for his work on the relationship between pressure, velocity, and elevation in flowing fluids, which is now known as Bernoulli's Equation.
Density: Density is a fundamental physical property that describes the mass per unit volume of a substance. It is a measure of how much matter is packed into a given space and is a crucial concept in understanding the behavior of fluids, solids, and gases across various physics topics.
Dynamic Pressure: Dynamic pressure is the pressure exerted by a moving fluid, such as air or water, on a surface. It is a measure of the kinetic energy per unit volume of the fluid and is directly proportional to the fluid's density and the square of its velocity.
Elevation: Elevation refers to the height or vertical distance of an object or location above a reference point, typically the mean sea level. It is a crucial parameter in various fields, including physics, engineering, and geography, as it directly influences various physical phenomena and processes.
Entrainment: Entrainment is the process by which a fluid flow captures and carries along additional fluid from its surroundings. This phenomenon often occurs in turbulent flows and can significantly affect the behavior and properties of the fluid stream.
Flow Separation: Flow separation refers to the phenomenon where a fluid flow, such as air or water, detaches from a solid surface due to adverse pressure gradients or other factors, leading to the formation of a recirculation zone or wake region. This concept is crucial in understanding the behavior of fluids in various engineering applications, particularly in the context of Bernoulli's equation and the study of viscosity and turbulence.
Fluid Dynamics: Fluid dynamics is the study of the motion and behavior of fluids, including both liquids and gases. It is a fundamental branch of physics that explores the principles governing the flow, pressure, and other properties of fluids, and how they interact with their surroundings.
Incompressible Fluid: An incompressible fluid is a fluid that does not undergo significant changes in volume when subjected to pressure. This means that the density of the fluid remains constant regardless of the applied pressure. Incompressible fluids are an important concept in the study of fluid dynamics, as they simplify the mathematical analysis of fluid behavior.
Inviscid Fluid: An inviscid fluid is an idealized model of a fluid that has no viscosity, meaning it has no internal friction or resistance to flow. This concept is important in the context of Bernoulli's equation, as it simplifies the analysis of fluid dynamics by eliminating the effects of viscosity on the fluid's behavior.
Kinetic energy: Kinetic energy is the energy possessed by an object due to its motion. It depends on the mass and velocity of the object.
Laminar flow: Laminar flow is a type of fluid motion characterized by smooth, parallel layers that do not mix. It typically occurs at low velocities and with high-viscosity fluids.
Laminar Flow: Laminar flow is a smooth, orderly, and parallel flow of a fluid, where the fluid particles move in distinct, non-intersecting paths. This type of fluid flow is characterized by a high degree of organization and predictability, in contrast to turbulent flow.
Lift: Lift is the upward force that acts on an object, such as an aircraft wing or a boat's sail, as it moves through a fluid like air or water. This force is generated by the difference in pressure between the upper and lower surfaces of the object, which is described by Bernoulli's principle.
Linear mass density: Linear mass density is the measure of mass per unit length of a one-dimensional object, such as a string or rod. It is typically denoted by the symbol $\lambda$ and expressed in units of kg/m.
Pitot Tubes: Pitot tubes are a type of instrument used to measure fluid flow velocity, particularly in aerodynamics and fluid mechanics. They work by measuring the difference between the static pressure and the total or stagnation pressure of a fluid, which can then be used to calculate the fluid's velocity.
Potential Energy: Potential energy is the stored energy possessed by an object due to its position or state, which can be converted into kinetic energy or other forms of energy when the object is moved or transformed. This term is central to understanding various physical phenomena and the conservation of energy.
Prandtl tube: A Prandtl tube, also known as a Pitot tube, is a device used to measure fluid flow velocity by converting the kinetic energy in a fluid flow to potential energy. It is commonly used in aerodynamics and hydrodynamics for measuring airspeed and water speed.
Pressure: Pressure is a measure of the force applied per unit area, representing the amount of force exerted on a surface or object. This concept is fundamental in understanding various physical phenomena and principles, including mass and weight, hydraulic systems, fluid dynamics, sound propagation, and shock waves.
Static Pressure: Static pressure is the pressure exerted by a fluid, such as a liquid or gas, that is at rest or not flowing. It is the pressure that would be measured by a pressure gauge placed within the fluid, without disturbing the flow. Static pressure is a fundamental concept in the study of fluid dynamics and is closely related to Bernoulli's equation.
Steady Flow: Steady flow, also known as laminar flow, is a type of fluid flow in which the fluid particles move in parallel layers, with no disruption or turbulence between the layers. This type of flow is characterized by a smooth, predictable movement of the fluid, with a constant velocity and direction at any given point in the system.
Streamline: Streamlining refers to the design of an object's shape to minimize the resistance or drag experienced when moving through a fluid, such as air or water. It is a fundamental concept in fluid dynamics that is particularly important in the context of Bernoulli's Equation.
Turbulent flow: Turbulent flow is a type of fluid motion characterized by chaotic changes in pressure and flow velocity. It typically occurs at high flow rates or with complex geometries.
Turbulent Flow: Turbulent flow is a complex and chaotic pattern of fluid motion characterized by irregular and unpredictable changes in velocity and pressure. It is in contrast to laminar flow, where the fluid moves in smooth, parallel layers. Turbulent flow plays a crucial role in various topics in physics, including drag force, fluid dynamics, Bernoulli's equation, and viscosity.
Velocity: Velocity is a vector quantity that describes the rate of change of an object's position with respect to time. It includes both the speed and the direction of an object's motion, making it a more complete description of an object's movement compared to just speed alone.
Venturi Effect: The Venturi effect is a principle in fluid dynamics that describes the decrease in fluid pressure that occurs when a fluid flows through a constricted section of a pipe or channel. This decrease in pressure is accompanied by an increase in the fluid's velocity, as dictated by the conservation of mass and the Bernoulli's principle.
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Glossary
Glossary
14.7 Viscosity and Turbulence