Formula Bernoulli equation Static pressure Mass density Height Flow velocity
$$\mathit{\Pi} ~=~ \mathit{\Pi}_{\text{s}} ~+~ \frac{1}{2} \rho \, v^2 ~+~ \rho \, g \, h$$ $$\mathit{\Pi} ~=~ \mathit{\Pi}_{\text{s}} ~+~ \frac{1}{2} \rho \, v^2 ~+~ \rho \, g \, h$$ $$\mathit{\Pi}_{\text{s}} ~=~ \mathit{\Pi} ~-~ \frac{1}{2} \rho \, v^2 ~-~ \rho \, g \, h$$ $$\rho ~=~ \frac{ \mathit{\Pi} - \mathit{\Pi}_{\text{s}} }{ \frac{1}{2} v^2 + g \, h }$$ $$h ~=~ \frac{1}{g} \, \left( \frac{ \mathit{\Pi} - \mathit{\Pi}_{\text{s}} }{\rho} - \frac{1}{2}\,v^2 \right)$$ $$v ~=~ \sqrt{ \frac{2}{\rho} \, \left( \mathit{\Pi} - \mathit{\Pi}_{\text{s}} \right) ~-~ 2g\,h}$$
Total pressure
\( \mathit{\Pi} \) Unit \( \mathrm{Pa} \) Constant total pressure of a stationary, non-viscous, incompressible fluid or gas along a streamline. The total pressure here is the sum of the dynamic pressure \(\frac{1}{2} \rho \, v^2\), hydrostatic pressure \(\rho \, g \, h\) and the static pressure \(\mathit{\Pi}_{\text{st}}\).
The Bernoulli equation results from the law of conservation of energy.
Static pressure
\( \mathit{\Pi}_{\text{s}} \) Unit \( \mathrm{Pa} \) Static pressure is the pressure that a body would feel if it were moving with the flow. The static pressure results from the potential energy of the flowing fluid.
Mass density
\( \rho \) Unit \( \frac{ \mathrm{kg} }{ \mathrm{m}^3} \) Mass per volume of the fluid or gas.
Height
\( h \) Unit \( \mathrm{m} \) Height of the considered streamline of a fluid or gas above the ground (or other specified zero point).
Flow velocity
\( v \) Unit \( \frac{\mathrm m}{\mathrm s} \) Flow velocity is the average velocity of the directed motion of the fluid or gas.
Gravitational acceleration
\( g \) Unit \( \frac{\mathrm{m}}{\mathrm{s}^2} \) On Earth, the gravitational acceleration is approximately \( g = 9.8 \, \frac{\mathrm{m}}{\mathrm{s}^2} \).