## Bernoulli’s principle

January 2nd, 2012

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 - the  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 , called the jet. A preliminary step to display the Bernoulli equation is the equation of the continuity..

The equation of  continuity

Let’s consider the elementary portion of the liquid jet, which is limited to two normal sections with squares $S_1$ and $S_2$ (fig.1). The flow velocity in cross sections $S_1$ and $S_2$ are, respectively $v_1$ и $v_2$. In a steady stream of liquid mass flowing per time $dt$ through a cross section of  $S_1$ is equal to the weight of the liquid flowing in the same time through the section $S_2$: $m_1=m_2$. Given that $m_1= \rho v_1 S_1 dt$, $m_2 = \rho v_2 S_2 dt$,  we get:

$v_1S_1 = v_2S_2$            (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 $v_1S_1dt$ to the volume of $v_2S_2dt$ pressure forces do the work

$A=p_1S_1v_1dt-p_2S_2v_2dt=(p_1-p_2)\Delta V$   (2)

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

$\Delta E = (\frac {\rho\Delta Vv_2^2} {2} + \rho \Delta Vgh_2)$ $- (\frac {\rho \Delta Vv_1^2} {2} + \rho \Delta Vgh_1)$   (3)

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

$\frac {\rho v_1^2} {2} + \rho gh_1 + p_1 = \frac {\rho v_2^2} {2} + \rho gh_2 + p_2$ (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:

$\frac{\rho v_1^2}{2}+p_1 = \frac{\rho v_2^2}{2}+p_2$ (5)

Visualization of (5) and the continuity equation.

(The information has been taken from 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 ).