Bernoulli's Equation

Consider the same liquid flowing through a pipe. We want to see what is the relationship between pressure, p, height of the pipe above ground, y, and flow velocity, v. The result of this investigation is known as Bernoulli's Equation.

Bernoulli's equation comes directly from the law of conservation of energy. The work done by the forces must equal the gain in kinetic plus potential energy.

W = -F $\cdot$ $\Delta$x = -(p $\cdot$ A) $\cdot$ (v $\cdot$ $\Delta$t)
= $\Delta$K + $\Delta$U =

$\Delta$m v² + $\Delta$m g y + const.

Now we divide both sides by $\Delta$V and arrive at Bernoulli's equation:

p +

r v² + $\rho$g y = constant

or, equivalently:

p₁ +

$\rho$v₁² + $\rho$g y₁ = p₂ +

$\rho$v₂²+$\rho$g y₂

where we have used the definition of the mass density of the liquid, \[ \rm \mathbf{ \rho = \frac{\Delta m}{\Delta V}} \]