Question

An incompressible, non viscous fluid is initially at rest in the vertical portion of the pipe shown in Figure P14.79a, where $\mathit{L}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{}\mathit{m}$. When the valve is opened, the fluid flows into the horizontal section of the pipe. What is the fluid’s speed when all the fluid is in the horizontal section as shown in Figure P14.79b? Assume the cross-sectional area of the entire pipe is constant.

Short answer

The fluid’s speed when all the fluid is in the horizontal section is $v=4.43m/s$.

Step-by-step solution

Energy conservation:

The law of conservation of energy states that energy cannot be created or destroyed - only converted from one form of energy to another. This means that the system always has the same amount of energy unless it is added from outside.

By Energy conservation:

${(K+U)}_1={(K+U)}_f$

Where, $K$ is the kinetic energy, $U$is the potential energy, $i$is the initial state, and $f$is the final state.

Determine the speed of fluid:

Consider the known data as below.

The acceleration due to gravity, $g=9.8m/s^2$

The length, $L=2.00m$

Energy for the fluid-Earth system is conserved. Therefore,

$$\begin{gathered} {(K+U)}_i={(K+U)}_{f}0+\frac{mgL}{2}=\frac{1}{2}m{v}^2+0 \\ v=\sqrt{gL} \\ =\sqrt{(9.8m/s^2)(2.00m)} \\ 4.43m/s \end{gathered}$$

Hence, the fluid’s speed when all the fluid is in the horizontal section is

$v=4.43m/s$