What important thing happens when the speed of a moving fluid increases?

Bernoulli's Principle states that as the speed of a moving fluid increases, the pressure within the fluid decreases and vice versa, provided the fluid is incompressible and there is no loss of energy due to factors like viscosity and friction. This relation can be applied to a streamline (a path along which a fluid particle flows) and is mathematically expressed as: \(\textit{P} + \frac{1}{2}\rho\textit{v}^2 + \rho\textit{gh} = \textit{constant}\) Here, - \(P\) is the pressure, - \(\rho\) is the fluid density, - \(v\) is the fluid velocity, - \(g\) is the acceleration due to gravity, and - \(h\) is the height above a reference level. In this exercise, to understand the important thing that happens when the fluid's speed increases, we will focus on the first two terms of the equation: \(\textit{P} + \frac{1}{2}\rho\textit{v}^2\).

We can rewrite the first two terms of Bernoulli's equation as a sum of the pressure and kinetic energy per unit volume: \(\textit{P} + \frac{1}{2}\rho\textit{v}^2 = \textit{constant}\) Think of the sum of these two terms as the "total mechanical energy per unit volume" of the fluid. As the fluid's velocity increases, it gains kinetic energy, which should compensate for the loss in pressure, to maintain the same total mechanical energy per unit volume: \(\textit{Increased kinetic energy} = \frac{1}{2}\rho\textit{(v + \Delta v)}^2 - \frac{1}{2}\rho\textit{v}^2\) Clearly, when the fluid's velocity increases (i.e., \(\Delta v > 0\)), this increased kinetic energy must be accounted for by a decrease in the fluid's pressure: \(\textit{Decreased pressure} = \textit{P} - \textit{P'}\)

The important thing that happens when the speed of a moving fluid increases is a decrease in pressure within the fluid, as described by Bernoulli's principle. This principle has practical implications in diverse areas such as flow measurement devices, lift production in airfoils, and the functioning of everyday devices like shower curtains being pushed away due to the flow of water.