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Bernoulli Equation

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9.2Bernoulli Equation


In these revision notes for Bernoulli Equation, we cover the following key points:

  • What is an ideal fluid?
  • How does the flowing speed relate to the thickness of the tube?
  • What does the Bernoulli equation say about flowing fluids?
  • What are some special cases of Bernoulli Equation use?
  • How to calculate the flowing speed of a liquid from a hole?

Bernoulli Equation Revision Notes

The mathematical apparatus used to study the behaviour of real fluid is too complicated. Therefore we use a simplified concept that helps us study fluids with a satisfactory approximation. This concept is known as "Ideal Fluid" and obviously, it contains some restrictions compared to real fluids. They are:

  1. 1- Steady flow. It is a kind of flow, in which the velocity of the fluid at a particular fixed point does not change with time.
  2. 2- Incompressible flow. Real fluids are slightly compressible. However, since this compression is very small, we neglect it when dealing with ideal gases and thus, we consider them as incompressible.
  3. 3- Non-viscous flow. Ideal fluids are considered as non-viscous as they do not make any resistance to the flow.
  4. 4- Non-rotational flow. This means fluid particles do not rotate during the flow but they move only forward.

The equation of continuity for ideal fluids states "the cross sectional area of a flowing fluid and its flowing speed are inversely proportional to each other." The equation of continuity for two situations 1 and 2, is

A1 × v1 = A2 × v2

where A stands for cross sectional area of the tube and v for the flowing speed.

Bernoulli Equation gives the relationship between static and dynamic pressure exerted in a flowing fluid. Its states that

Total Pressure = Static Pressure + Dynamic Pressure = Constant

or

P + ρ × v2/2 + ρ × g × h = constant

If we consider two situations 1 and 2, the Bernoulli equation is written as

P1 + ρ × v21/2 + ρ × g × h1 = P2 + ρ × v22/2 + ρ × g × h2

It is not for sure that all terms of the Bernoulli equation are to be used in a given situation. Some terms may cancel out according to the situation involved. Therefore, usually we obtain simplified versions of Bernoulli equations when solving problems related to hydrodynamics.

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