The law of conservation of mechanical energy for liquids and gases is a special kind formulated by Bernoulli. Let’s consider the conclusion of the law on the example of an ideal incompressible fluid. *The ideal liquid* is such liquid in which the internal friction can be ignored, *the incompressible liquid * is a liquid whose density is independent of pressure.

Let’s consider the liquid in which the velocity of each point is independent of time. Such fluid flow is called *the steady stream*. The line whose tangent at each point coincides with the direction of fluid flow velocity at this point is called *the current line*. The surface formed by the current lines drawn through all the points of a closed loop is called *the tube of the current. *Part of the fluid flow, limited by the tube of the current, is called *the jet*. A preliminary step to display the Bernoulli equation is the equation of continuity.

**The equation of continuity **

Let’s consider the elementary portion of the liquid jet, which is limited to two normal sections with squares and (fig.1). The flow velocity in cross sections and are, respectively и . In a steady stream of liquid mass flowing per time, through a cross-section of is equal to the weight of the liquid flowing in the same time through the section : . Given that , , we get:

(1).

Equation (1) is called *the equation of continuity.*

Fig.1. The tube of current

**Bernoulli’s principle **

To move the fluid present in the volume of to the volume of pressure forces do the work

(2)

The work of the pressure forces is equal to the increment of the total energy of the selected volume of the liquid:

(3)

Comparing equations (2) and (3), we receive after elementary transformations:

(4)

This equation is called *Bernoulli’s principle.*

If both sections of the current tube are at the same height, the equation becomes simpler form:

(5)

Visualization of (5) and the continuity equation.

(The information has been taken from the e-book «Fundamentals of classical physics” (author V.D. Shvets), which was published in 2007 by “1C-Multimedia Ukraine” publishing company. Copyright is reserved by certificate Number 55955 from 06.08.2014. More information can be found at https://ipood-kiev.academia.edu/ValentynaShvets/Books ).