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Bernoulli’s Equation

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Bernoulli’s equation is a fundamental principle in fluid mechanics that describes the behavior of a fluid as it moves through a pipe or over an obstacle. It is named after the Swiss mathematician Daniel Bernoulli who developed the equation in the 18th century. Bernoulli’s equation is widely used in engineering applications, including in the design of aircraft wings, wind turbines, and pipelines.

Bernoulli's Equation plays an essential role in the operation of water pipes.
Bernoulli’s Equation plays an essential role in the operation of water pipes.

The basic form of Bernoulli’s equation can be written as:

P + \dfrac{1}{2}\rho v^2 + \rho gh = \text{constant}

or

P_1+\dfrac{1}{2}\rho v_1^2+\rho g h_1 = P_2+\dfrac{1}{2}\rho v_2^2 +\rho gh_2

where P is the pressure of the fluid, \rho is the fluid density, v is the fluid velocity, g is the acceleration due to gravity, h is the height of the fluid above a reference point, and the constant term is the sum of the pressure, velocity, and height terms at any given point in the fluid (as demonstrated more clearly by the second version of the formula). The equation assumes that the fluid is incompressible and inviscid, and that the flow is steady and irrotational.

The first term in Bernoulli’s equation represents the static pressure of the fluid, which is the pressure exerted by the fluid on the walls of the container in which it is flowing. The second term represents the dynamic pressure of the fluid, which is the pressure due to the motion of the fluid. The third term represents the potential energy of the fluid due to its height above a reference point.

At its simplest level, Bernoulli’s equation describes the relationship between the pressure, velocity, and height of a fluid in a pipe or channel. It states that as the velocity of a fluid increases, the pressure decreases. Conversely, as the velocity of a fluid decreases, the pressure increases. This general principle regarding the conservation of energy for a laminar flow is known as the Bernoulli effect and can be observed in everyday situations, such as the flow of water through a hose.

At a more advanced level, Bernoulli’s equation can be generalized to include additional factors, such as viscosity and compressibility. This leads to the development of more complex equations, such as the Navier-Stokes equations, which are used to model fluid flow in a wide range of situations.

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2 responses to “Bernoulli’s Equation”

  1. Fluid Dynamics – Tru Physics

    […] Bernoulli’s equation is a crucial result in fluid dynamics that describes the conservation of energy in a fluid system. For steady, incompressible flow along a streamline, the equation is: […]

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  2. Venturi Effect – Tru Physics

    […] The Venturi effect is a direct consequence of the principle of conservation of energy applied to fluid flow, specifically the Bernoulli’s equation: […]

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