5.11 Bernoulli's Equation

5.11 Bernoulli's Equation

The sum of pressure, K.E. per unit volume and P.E. per unit volume of an ideal fluid throughout its steady flow remains constant.

As the fluid moves through a pipe of varying cross-section and height, the pressure will change along the pipe. Bernoulli's equation is the fundamental equation in fluid dynamics that relates pressure to fluid speed and height.

In deriving Bernoulli's equation, we assume that the fluid is incompressible, non-viscous and flows in a steady state manner. Let us consider the flow of the fluid through the pipe in time $t$, as shown in Fig. 5.17.

The force on the upper end of the fluid is $P_{1}A_{1}$, where $P_{1}$ is the pressure and $A_{1}$ is the area of cross-section at the upper end. The work done on the fluid, by the fluid behind it, in moving it through a distance $\Delta x_{1}$, will be:

Similarly, the work done on the fluid at the lower end is:

where $P_{2}$ is the pressure, $A_{2}$ is the area of cross-section of lower end and $\Delta x_{2}$ is the distance moved by the fluid in same time interval $t$. The work $W_{2}$ is taken to be $-$ve as this work is done against the fluid force. The net work done will be:

or

If $v_{1}$ and $v_{2}$ are the velocities at the upper and lower ends respectively, then:

From equation of continuity, $A_{1}v_{1} = A_{2}v_{2}$

Hence $A_{1}v_{1}t = A_{2}v_{2}t = V$ (volume)

So

If $m$ is the mass and $\rho$ is the density, then $V = m/\rho$. So, Eq. (5.18) becomes:

A part of this work is utilized by the fluid in changing its K.E. and a part is used in changing its gravitational P.E.

Change in K.E.:

Change in P.E.:

where $h_{1}$ and $h_{2}$ are the heights of the upper and lower ends respectively.

Applying the law of conservation of energy to this volume of the fluid, we have:

Rearranging Eq. (5.22), we have:

This is Bernoulli's equation and is often expressed as: