Fundamentals of Equations of Fluid Flow: Bernoulli’s Theorem and it’s Application!

#### Bernoulli’s Theorem:

This theorem is a form of the well-known principle of conservation of energy.

The theorem is stated as follows:

In a steady continuous flow of a frictionless incompressible fluid, the sum of the potential head, the pressure head and the kinetic head is the same at all points.

Proof of Bernoulli’s Theorem:

Consider a liquid flowing through a pipe as shown in Fig. 7.2. Let the pipe be filled with the liquid and be under pressure. Consider the quantity of the liquid between two sections 1-1 and 2-2.

#### Euler’s Equation of Motion:

The net force that acts on a fluid body is called the inertial force. The inertial force is the resultant of all the forces acting on a fluid body. In general a fluid body may be subjected to the following forces namely.

#### Applicability of Bernoulli’s Equation:

Bernoulli’s equation is applicable subject to the following assumptions:

(i) The flow of the fluid is steady.

(ii) The flow is frictionless. No tangential stresses exist.

(iii) The fluid is incompressible,

(iv) The flow is continuous.

#### Inter-Conversion of Potential, Pressure and Kinetic Heads:

As per Bernoulli’s equation, the sum of the potential, pressure and kinetic heads is the same at all points in a fluid having a steady incompressible flow. As a fluid mass moves to different positions, the total energy of the fluid mass remains the same following the principle of conservation of energy. Whereas the potential, pressure and kinetic energies will individually vary, the total energy will remain the same.

Consider the tank shown in Fig. 7.4. Consider the points A, B and C.

The point A is on the liquid surface. The points B and C are at the datum level. At C a small opening is provided through which the liquid is discharged in the form of a jet.

At the point A, the total energy is entirely due to position i.e., at A there is only potential energy. At the point B there exists only pressure energy. At the outlet point C there exists only kinetic energy. Thus, we find that the potential head at A is converted into pressure head at B and to kinetic head at C.

As the jet rises up it loses the kinetic head and gains the potential head. As the jet reaches the highest point D it has only potential head. In the actual cases a loss of energy head occurs in the system and hence the level at D is lower than the level at A. The difference of level between A and D represents the loss of head in the system.

#### Kinetic Energy Correction Factor α:

As a liquid flows through a pipe line, at any section of the pipe line the velocity is not uniform. If v is the average velocity at the section, then Bernoulli’s equation can be modified to the form-

#### Control Volume:

A control volume means a definite extent of space with well-defined boundaries. When a fluid is in motion, it may at times be convenient to consider a certain control volume and study the quantity of the fluid entering and leaving the control volume in a definite period of time.

Consider a liquid flowing through a pipe. (Fig. 7.6.)

Consider the sections 1-1 and 2-2. Let the space within the pipe between the two sections be selected as the control volume.

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