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# Bernoulli's Principle

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 Sub Topics Bernoulli's principle is one of the fundamental theorem in fluid dynamics. This theorem is related to the flow of fluids. Fluids are nothing but the liquids or gases. These objects have an attractive force between the molecule. If the density of the fluid is constant while flowing, this type flow is incompressible flow. If the density is varying accordingly, the flow become compressible. The Bernoulli's principle is only applicable for the incompressible fluids. The details about the topic is given in the next sections.

## Definition

Bernoulli's principle states that, in an ideal and incompressible fluid, when the flow is steady and continuous, the sum of the pressure head, velocity head and elevation head of fluid is constant at all points in the flow system.

The limitations of Bernoulli's theorem are;

• It is useful only to the fluid is ideal or frictionless and incompressible, so that the only forces present are the pressure forces and gravity forces
• It is useful if the flow is steady, continuous and one-dimensional

## Equation

Let us consider an ideal fluid moving through a pipe. The fluid is flowing from left to right. Let denote the pressure and the speed of the fluid at the far left as P1 and v1 and suppose the left end of the pipe is at a height h1. Likewise, the pressure, fluid speed and height at the far right are P2, v2 and h2.
Bernoulli's equation is given by,

It tells how the pressure and speed vary from place to place within a moving fluid.

## Experiments

The Bernoulli's theorem is based on the principle of law of conservation of energy. According to this theorem, for an ideal and steady fluid flow along a streamline, the total energy per unit weight of the flowing fluid is constant.

The set up consists of a horizontal converging-diverging duct having constant width but varying depth. The duct is made of transparent perspex sheets, which are joined together to from duct of required shape.A number of piezometers are fitted on the duct at equal intervals for measuring the pressure heads at different gauge points. The duct is connected to two tanks, one at the upstream end (inlet tank) and the other at the downstream end (outlet tank). The inlet tank is fitted with a piezometer for indicating the water level in the tank. The outlet tank is provided with an outlet value for controlling the outflow. The set up is placed on a hydraulic bench. Water is supplied to the inlet tank by a supply pipeline provided with an inlet valve and connected to a constant overhead water tank.

Procedures:
• Open the inlet valve gradually to fill the inlet tank. The water level will start rising in various piezometers. Remove the air bubbles in the piezometers,if any. Open the exit valve and adjust the inflow and the outflow so that the water level in the piezometers is constant, i.e, let the flow becomes steady.
• Measure the levels of water in various piezometers with respect to an arbitrary selected suitable horizontal plane as datum.
• Measure the discharge, in the discharge measurement tank.
• Repeat the above steps for one more run by regulating the supply valve.
• Calculate the areas of the duct at various gauge points using similar triangles or any other method.