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Bulgara  Ceha slovaca  Croata  Engleza  Estona  Finlandeza  Franceza 
Germana  Italiana  Letona  Lituaniana  Maghiara  Olandeza  Poloneza 
Sarba  Slovena  Spaniola  Suedeza  Turca  Ucraineana 
We shall further study, for the most general case, the movement state of a fluid through a volume _{} that is situated in the fluid stream; we shall not take into consideration the interior frictions(i.e.viscosity), so we shall analyse the case of perfect (ideal) fluids that are on varied movement.
The volume _{}is situated in an accelerated system of axes, joint with this system. The equations, which describe the movement of the fluid, will be obtained by applying d’Alembert’s principle for the fluid that is moving through the volume _{}.
The three categories of forces that act upon the fluid that is moving through the volume _{},bordered by the surface _{} (fig.3.1), are:
Fig.3.1
the mass forces _{};
the inertia forces _{};
the pressure forces _{} (with an equivalent effect; these forces replace the action of the negligible fluid outside volume _{}).
According to d’Alembert’s principle, we shall get:
_{} (3.1)
Equation (3.1) represents in fact the general vectorial form of Euler’s equations.
Let’s establish the mathematical expressions of those three categories of forces.
If _{} is the mass unitary force (acceleration) that acts upon the fluid in the volume _{}, the mass elementary force that acts upon the mass_{}, will be:
_{} (3.2)
hence:
_{} (3.3)
As the fluid velocity through the volume _{}is a vectorial function with respect to point and time: _{}, upon the mass _{} that is moving with velocity _{} the elementary inertia will act:
_{} (3.4)
So, the inertia will be:
_{} (3.5)
If _{} is a surface element upon which the pressure p acts, and _{} the versor of the exterior normal (Fig.3.1), the elementary force of pressure is:
_{} (3.6)
Having in mind GaussOstrogradski’s theorem, the resultant of pressure forces will be:
_{} (3.7)
By replacing equations (3.3), (3.5) and (3.7) in the equation (3.1), we shall get:
_{} (3.8)
Hence:
_{} (3.9)
Or
_{} (3.10)
The equation (3.10) – Euler’s equation in a vectorial form for the ideal fluid in a nonpermanent movement.
Projecting this equation on the three axes, we shall obtain:
_{}
_{} (3.11)
_{}
This equation can be obtained by writing in two ways the variation in the unity of time for the mass of fluid that is in the control volume _{}, bordered by the surface _{} (fig.3.1). By splitting from the volume _{}one element _{}, and taking into consideration that the density is a scalar function of point and time, _{}, we can write the total mass of the volume _{}:
_{}
The variation of the total mass in the unity of time will be:
_{}
The second form of writing the variation of mass is obtained by examining the flow of the mass through surface _{} that borders volume_{}.
Denoting by _{}the versor of the exterior normal to the area element _{}, and by _{} the vector of the fluid velocity, the elementary mass of fluid that passes in the unity of time through the area element _{} is:
_{}
In the unity of time through the whole surface _{}will pass, the mass:
_{}
that is the sum of the inlet and outlet mass in volume _{}, by crossing surface _{}.
By equalling equations (3.13) and (3.15), it will result:
_{} (3.16)
According to GaussOstrogradski’s theorem:
_{}
Taking into consideration (3.17), the equation (3.16) will take the form:
_{} (3.18)
hence, successively:
_{} (3.19)
_{} (3.20)
_{} (3.21)
The equation (3.21) represents the equation of continuity for compressible fluids.
In the case of noncompressible fluids (_{}, _{}), the equation of continuity takes the form:
_{} (3.22)
or
_{} (3.23)
It follows that the inlet volume of noncompressible liquid is equal to the outlet one in and from the volume _{}
From a thermodynamically point of view, the state of a system can be determined by the direct measurement of some characteristic physical values, that make up the group of state parameters (e.g. pressure, volume, temperature, density etc.).
Among the state parameters of a thermodynamically system generally there are link relationships, explained by the laws of physics.
In the case of homogenous systems, there is only one implicit relationship, which carries out the link among the three state parameters, in the form of:
_{}. (3.24)
Adding to vectorial equations (3.10) and (3.21) the equation of state, we get three equations with three unknowns _{}, that enable us solve the problems of motion and repose for the ideal fluids.
Bernoulli’s equation is obtained by integrating Euler’s equation written under a different form (Euler – Lamb), that stresses the rotational or nonrotational nature of the ideal fluid (see the relation (1.25)).
Euler – Lamb’s equation:
_{}. (3.25)
Considering the case when the mass force derives from a potential U, thus being a conservative force (the mechanical energykinetic and potentialwill be constant):
_{}. (3.26)
In the case of compressible fluids, when _{}, we insert the function:
_{}. (3.27)
Thus:
_{}. (3.28)
The equation (3.25) takes the form:
_{}. (3.29)
The equation (3.29) can be easily integrated in certain particular cases.
In the case of permanent motion _{}, and:
along a stream line:
_{} , (3.30)
along a whirl line:
_{}, (3.31)
in the case of potential motion _{}:
_{}, (3.32)
in the case of helicoid motion (the velocity vector _{} is parallel to the whirl vector):
_{}. (3.33)
Multiplying by _{} the equation (3.29), we shall get under the conditions of permanent motion (_{}):
_{}. (3.34)
Since _{}, we shall get:
_{}. (3.35)
The determined is zero for one of the four cases above. By integrating in these cases we shall get Bernoulli’s equation:
_{}. (3.36)
If the fluid is a noncompressible one then _{}.
If the axis Oz of the system is vertical, upwards directed, the potential U is:
_{}. (3.37)
It results the well known Bernoulli’s equation as the load equation:
_{}. (3.38)
The kinetic load _{} represents the height at which it would rise in vacuum a material point, vertically and upwards thrown, with an initial velocity v, equal to the velocity of the particle of liquid considered.
The piezometric load _{}is the height of the column of liquid corresponding to the pressure p of the particle of liquid.
The position load z represents the height at which the particle is with respect to an arbitrary chosen reference plane.
Bernoulli’s equation, as an equation of loads, may be expressed as follows: in the permanent regime of an ideal fluid, noncompressible, subjected to the action of some conservative forces, the sum of the kinetic, piezometric and position loads remains constant along a streamline.
Multiplying (3.38) by _{} we get the equation of pressures:
_{}, (3.39)
where:
_{} dynamic pressure;
_{} piezometric (static) pressure;
_{} position pressure.
Multiplying (3.38) by the weight of the fluid G, we get the equation of energies:
_{} (3.40)
where
_{}  kinetic energy;
_{}  pressure energy;
_{}  position energy.
Going back to the relation (3.38) and considering C = H (fig.3.2):
_{} (3.41)
Fig.3.2
The sum of all the terms of Bernoulli’s equation represents the total energy (potential and kinetic) with respect to the unit of weight of the moving liquid.
This energy measured to a horizontal reference plane NN, arbitrarily chosen is called specific energy and it remains constant during the permanent movement of the ideal noncompressible fluid that is under the action of gravitational and pressure forces.
Let’s consider the flow of an ideal noncompressible fluid through the channel between two concentric pipes that revolve around an axis Oz with angular velocity _{} (fig.3.3.).
Fig.3.3
In the equation (3.38) v is replaced by w, which represents the relative velocity of the liquid to the channel that is revolving with the velocity _{}.
Upon the liquid besides the gravitational acceleration g, the acceleration _{} acts as well.
The unitary mass forces decomposed on the three axes will be:
_{} (3.42)
In this case, the potential U will be:
_{} (3.43)
By adding (3.43) to Bernoulli’s equation, we get:
_{} (3.44)
or
_{} (3.45)
In the theory of hydraulic machines we use the following denotations:
v – absolute velocity;
w – relative velocity;
u – peripheral velocity.
The equation (3.45) written for two particles on the same streamline is:
_{} (3.46)
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