52.3: Bernoulli’s Equation

Bernoulli's equation was developed by 18th-century Swiss physicist Daniel Bernoulli. Given fluid flow in a pipe that varies in elevation, the equation relates the velocity, pressure, and elevation as the fluid flows through the pipe. It states

$$\frac{P}{\rho g}+\frac{v^{2}}{2 g}+y=\text { constant, }$$

where $P$ is the pressure, $v$ is the fluid velocity, $y$ is elevation, $\rho$ is the fluid density, and $g$ is the acceleration due to gravity. Each term in Bernoulli's equation has units of length and is called a head: the $P /(\rho g)$ term is called the pressure head, the $v^{2} /(2 g)$ term is called the velocity head, and the $y$ term is called the elevation head.

Example $\PageIndex{1}$

Suppose we have a vertical pipe containing a stationary incompressible fluid of density $\rho$. How does the pressure $P$ vary with depth $h$ ?

Solution

Let the pressure at depth $h=0$ be $P_{0}$. Since the fluid is stationary, the fluid velocity $v$ is zero everywhere. Then Bernoulli's equation becomes (with $y=-h$ )

$$\frac{P_{0}}{\rho g}+\frac{0}{2 g}+0 =\frac{P}{\rho g}+\frac{0}{2 g}-h$$
$$\frac{P_{0}}{\rho g} =\frac{P}{\rho g}-h$$
$$P_{0} =P-\rho g h$$
P & =P_{0}+\rho g h,\]

in agreement with Eq. 51.4.5.