Bernoulli's Principle

The formula for Bernoulli’s Principle (also known as Bernoulli’s Equation) states that for two points along the streamline of an incompressible, inviscid, steady-flow fluid, the sum of the pressure, kinetic energy per unit volume, and potential energy per unit volume remains constant. It has the following forms:

Pressure Form #

$$P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g h_2$$

Where

• $\small \small P_1$ and $\small \small P_2$ are the pressures at points 1 and 2,
• $\small \small \rho$ is the density of the fluid,
• $\small \small v_1$ and $\small \small v_2$ are the velocities at points 1 and 2,
• $\small \small g$ is the acceleration due to gravity, and
• $\small \small h_1$ and $\small \small h_2$ are the heights at points 1 and 2.

It can be also be written by stating that the sum of the quantities remain constant:

$$P + \frac{1}{2} \rho v^2 + \rho g h = \text{const}$$

Energy Form #

$$\frac{P}{\rho} + \frac{1}{2} v^2 + g h = \text{const}$$

$$\frac{P_1}{\rho} + \frac{1}{2} v_1^2 + g h_1 = \frac{P_2}{\rho} + \frac{1}{2} v_2^2 + g h_2$$

Head Form #

$$\frac{P}{\gamma} + \frac{v^2}{2g} + h = \text{const}$$

$$\frac{P_1}{\gamma} + \frac{v_1^2}{2g} + h_1 = \frac{P_2}{\gamma} + \frac{v_2^2}{2g} + h_2$$

Where

• $\small \small \gamma$ is the specific weight of the fluid.

Sources #

• Wikipedia
• Khan Academy