The Bernoulli Equation for an incompressible, steady Fluid Flow:
The Bernoulli Equation is a statement derived from conservation of energy and workenergy ideas that come from Newton's Laws of Motion.
Statement:It states that the total energy (pressure energy, potential energy and kinetic energy) of an incompressible and non–viscous fluid in steady flow through a pipe remains constant throughout the flow, provided there is no source or sink of the fluid along the length of the pipe.
Proof of Bernoulli’s Theorem:
This statement is based on the assumption that there is no loss of energy due to friction.
To prove Bernoulli’s theorem, we make the following assumptions:
1. The liquid is incompressible.
2. The liquid is non–viscous.
3. The flow is steady and the velocity of the liquid is less than the critical velocity for the liquid.
Imagine an incompressible and non–viscous liquid to be flowing through a pipe of varying cross–sectional area as shown in Fig. The liquid enters the pipe with a normal velocity v_{1} and at a height h_{1} above the reference level (earth’s surface). It leaves the pipe with a normal velocity v_{2} at the narrow end B of cross–sectional area a_{2} and at a height h_{2} above the earth’s surface.
We examine a fluid section of mass m traveling to the right as shown in the schematic above. The net work done in moving the fluid is
Pressure is the force exerted over the crosssectional area, or P = F/A. Rewriting this as F = PA and substituting into Eq.(1) we find that
 
The displaced fluid volume V is the crosssectional area A times the thickness x. This volume remains constant for an incompressible fluid, so
The energy change between the initial and final positions is given by
Here, the the kinetic energy K = mv²/2 where m is the fluid mass and v is the speed of the fluid. The potential energy U = mgh where g is the acceleration of gravity, andh is average fluid height.
The workenergy theorem says that the net work done is equal to the change in the system energy. This can be written as
Substitution of Eq.(4) and Eq.(5) into Eq.(6) yields
Dividing Eq.(7) by the fluid volume, V gives us
is the fluid mass density. To complete our derivation, we reorganize Eq.(8).
Finally, note that Eq.(10) is true for any two positions. Therefore,
Equation (11) is commonly referred to as Bernoulli's equation. Keep in mind that this expression was restricted to incompressible fluids and smooth fluid flows.
