The Bernoulli Equation for an incompressible, steady Fluid Flow:
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 v1 and at a height h1 above the reference level (earth’s surface). It leaves the pipe with a normal velocity v2 at the narrow end B of cross–sectional area a2 and at a height h2 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 cross-sectional 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 cross-sectional 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 work-energy 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.