LA FORMA CORRECTA DE UTILIZAR LA ECUACIÓN DE BERNOULLI RESUMEN

When Bernoulli's equation is intended to apply on a certain case of a fluid flow problem, some restrictions must be met in order to correctly apply this particular equation. The fluid flow must be considered inviscid, incompressible, steady and irrotational. However, if the fluid flow is rotational, Bernoulli`s equation can still be applicable as long as the points of interest are on the same streamline of the fluid flow. Here, we will focus on demonstrate that, in the case of a rotational fluid flow, the points of interest must be on the same streamline and because of that, it can be proceeding with the usage of Bernoulli`s equation. The principle is only applicable for isentropic fluid flows: when the effects of irreversible processes (e.g. turbulence, friction) and non-adiabatic processes (e.g. heat radiation, mass diffusion) are small and can be neglected.


INTRODUCTION
Various forms of Bernoulli's equation can be modeled because of the existence of various types of fluid flow, and therefore Bernoulli's principle can be applied too.
Bernoulli's principle states that, an increase in the speed of a fluid flow occurs simultaneously with a decrease in internal pressure. The principle is named after Daniel Bernoulli published it in his book Hydrodynamic in 1738. Although, Bernoulli deduced that pressure decreases when the flow speed increases, it was Leonhard Euler who derived Bernoulli's equation in its usual form in 1752. All in all, there is a correct way of using Bernoulli`s equation with confidence and which is briefly described to continuation.
The law that explained the phenomenon from the energy conservation point of view was found in his Hydrodynamic work. Later, Euler deduced an equation for an inviscid flow (assuming that viscosity was insignificant) from which Bernoulli's equation arises naturally when considering a stationary case subjected to a conservative gravitational field.

DISCUSSION
To arrive at Bernoulli`s equation, certain assumptions had to be made which limit us the level of applicability. According to Euler equation Eq. 1, defined by Anderson, Jr. (1989), it gives the variation of pressure with respect to speed variation, ignoring shear forces (inviscid fluid flow) and body forces (weight of the air fluid particle is ignored). Only pressure forces were considered.

= −
(1) Integrating Eq. (1) by using a limit integration and considering an incompressible fluid flow (change in density is very small because of low speed), will give us Bernoulli`s equation applicable to points 1 and 2 which are on the same streamline.
However, if the flow is uniform throughout the field, then the constant in Eq.
(2) is the same for all streamlines as defined by Anderson, Jr. (1989).
The assumptions that were made during the derivation of this equation led us to some restrictions that must be implemented in order to use Bernoulli`s equation. But, first of all, we must verify if the flow field in question is possible to exist. This is done by verifying if Continuity Equation is fulfilled.

Continuity Equation in its Vector Form
The continuity equation states that, "the net outflow of mass through the surface surrounding the volume must be equal to the decrease of mass within the volume" (Bertin and Smith 1998, p. 24). This is, when a fluid is in motion, it must move in such a way that mass is conserved as it is stated in Eq. 3 defined by Bertin and Smith (1998).
Where ρ is the fluid density, t is the time, → is the flow velocity vector field

Steady Flow
To see further how mass conservation places restrictions on the velocity field, consider a steady fluid flow. That is, for a relatively low speed flow, the pressure variations are sufficiently small, and because of this, the density change is also small that can be assumed to be constant and so, the density of the fluid flow does not vary with time.

Incompressible
Since density change is very small for low velocity airflows, it can be assumed to be constant ( = ). One way to proof this, is by verifying if we are dealing with low velocity airflows. As a rule of thumb, if its Mach Number is lower than 0,3 or has a velocity less than 300 ft/s or 100m/s (or approximately 200 mph), then the velocity airflow can be assumed to be small and treated as incompressible Anderson, (1989) and Anderson (2003). Let`s consider a 2D velocity flow field at sea level ( = 1.225 3 ) and defined by: Where "u" and "v" are defined in m/s.
Second, we need to find out if the given flow velocity field is rotational or irrotational.
Substituting equations (14) and (15) into Eq. (7) yields: It is clear that, the given velocity flow field is rotational (ω ≠ 0). So, that means that we can still use Bernoulli`s eq. only if the two given points are on the same streamline. So, we need to identify the streamline by using the 2D streamline Eq.  (8) and (9) Now, if we intend to use Bernoulli`s eq, for example to find the static pressure difference between two points in the flow, we must be sure to have these two points on the same streamline.
Consider these two points to be: (-1, 2) and (2, 2). The coordinates of these two points are defined in meters.

CONCLUSION
Throughout this paper, the correct way of using Bernoulli`s equation has been shown. Initially, it has been presented some assumptions for which this equation is valid to apply. These assumptions led to a set of restrictions that must be met in order to apply correctly this equation. However, during the process of the application of Bernoulli`s equation, the analyst has to be sure which restrictions apply for the particular case. According to this, the results must be presented in a similar way as it was done in this paper like the pressure difference between the two points on the same streamline.