Equation of Bernoulli. Static and dynamic pressure

The liquid incompressible and not having internal friction or viscosity is called ideal liquid.

Current of a liquid is called stationary or established if speeds of particles of liquid in each point of a stream in due course do not change. The established current characterizes lines of a current - the imagined lines conterminous with trajectories of particles. The part of a stream of the liquid, limited from different directions lines of a current, forms a tube of a current or a jet. We shall allocate a tube of a current so narrow; it is possible to consider speeds of particles V are identical in its any section S perpendicular axes of a tube, on all section.

Then the volume ΔV of a liquid proceeding through any section of a tube in unit of time remains to constants, as movement of particles in a liquid occurs only along an axis of a tube: ΔV =vS=const. This parity is called a condition of indissolubility of a jet.

From here follows, as for a real liquid at the established current on a pipe of variable section quantity Q of the liquid proceeding in unit of time through any section of a pipe, remains to constants (Q = const) and average speeds of current in various sections of a pipe inversely proportional to the areas of these sections:

Let's allocate in a stream of an ideal liquid a tube of a current, and in it - small enough volume of a liquid ΔV in mass Δm, which at current of a liquid moves from position A to position B.

As volume is small, it is possible to consider, that all particles of a liquid in it are in equal conditions: in position A have pressure speed and are at height _h1 from a zero level; in position B - accordingly . Sections of a tube of a current accordingly S₁ and S₂.

The liquid which is being under pressure, possesses internal potential energy (energy of pressure), and can make work. This energy W_p is measured by product of pressure upon volume V of a liquid

In this case moving of mass of liquid occurs under action of a difference of forces of pressure in sections S₁ and S₂. Work made at it А_р is equal to differences potential energies in points A and B: . This work is spent for work on overcoming gravity and on change of kinetic energy of mass liquid

Hence,

Positions A and B are chosen any way, therefore it is possible to approve, that in any place of a tube of a current the condition is kept

Having divided this equation on , we shall receive

Where — density of a liquid.

It is equation Bernoulli.All members of the equation, have dimension of pressure and refer to: static: hydrostatic: - dynamic. Then equation Bernoulli it is possible to formulate so:Full pressure equal to the sum of static, hydrostatic and dynamic pressure, remains in size of a constant in any cross-section section of a stream at stationary current of an ideal liquid. Hydrostatic pressure remains to constants and can be carried in the right part of the equation which thus becomes for a horizontal tube of a current

Static pressure causes potential energy of a liquid (energy of pressure), dynamic pressure - kinetic.

From this equation the conclusion, which is called the rule Bernoulli follows:

Static pressure of a nonviscous liquid at current on a horizontal pipe increases there where its speed decreases and on the contrary.

It is possible to show it, for example, passing the tinted water on enough wide pipe of variable section and comparing levels of a liquid in manometrical tubes on various sites of a pipe. It is possible to pick up conditions, at which pressure of a liquid (or gas) in the narrowed site of a pipe becomes less atmospheric and then the jet in this place can have soaking up an effect. Such effect is applied in the water-jet pump. The water-jet pump is silent and hygienic. Therefore it is often applied in a medical practice, for example at removal liquids from a cavity of pleura.

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