Bernoulli’s Equation

5.1 INTRODUCTION

The dynamic behaviours of fluid are studied considering the cause of the motion i.e., force. We will use Newton’s 2nd law of motion to derive differential equation for dynamic behaviour of ideal fluid (p = constant & µ = 0).

5.2 BERNOULLI’S EQUATION

There are many forces which act on a fluid element. They are

F g = gravity force,

Fµ= pressure force,

Fµ =viscous force,

Ft = turbulence force,

And                             Fe= compressibility force.

Here we will consider gravity and pressure force to derive Bernoulli’s equation.

Fig. 5.1 shows dynamic equilibrium of fluid element under the action of forces as shown in the figure. Here p denotes for intensity of pressure. By using Newton’s 2nd law we get

Σ forces in s direction= mass x acceleration ins direction

= pd A ds x as

where as                           = acceleration ins direction

v = v(s, t)

From.(5.1)

Equation (5.2) is the Euler’s equation of motion. For steady, incompressible flow we can write down

or incompressible flow                              p = const.

or                                    p / p + g z + v2 /2 = const

or                               p / p g + v2 / 2 g + z = const.

Equation (5.3) is known as Bernoulli’s equation.