Bernoulli’s Equation For Static Fluids

Bernoulli’s Equation for Static Fluids

Let us first consider the very simple situation where the fluid is static—that is, \({v}_{1}={v}_{2}=0\). Bernoulli’s equation in that case is

\({P}_{1}+\rho {\mathrm{gh}}_{1}={P}_{2}+\rho {\mathrm{gh}}_{2}\text{.}\)

We can further simplify the equation by taking \({h}_{2}=0\) (we can always choose some height to be zero, just as we often have done for other situations involving the gravitational force, and take all other heights to be relative to this). In that case, we get

\({P}_{2}={P}_{1}+\rho {\mathrm{gh}}_{1}\text{.}\)

This equation tells us that, in static fluids, pressure increases with depth. As we go from point 1 to point 2 in the fluid, the depth increases by \({h}_{1}\), and consequently, \({P}_{2}\) is greater than \({P}_{1}\) by an amount \(\rho {\mathrm{gh}}_{1}\). In the very simplest case, \({P}_{1}\) is zero at the top of the fluid, and we get the familiar relationship \(P=\rho \mathrm{gh}\). (Recall that \(P=\mathrm{\rho gh}\) and \(\text{Δ}{\text{PE}}_{\text{g}}=\text{mgh}.\)) Bernoulli’s equation includes the fact that the pressure due to the weight of a fluid is \(\rho \text{gh}\). Although we introduce Bernoulli’s equation for fluid flow, it includes much of what we studied for static fluids in the preceding tutorial.