Chapter 10, Problem 37P

To determine

To prove: The Bernoulli’s equation is reduced to the hydrostatic variation of pressure when there is no flow.

Explanation of Solution

Given:

  $v_1=v_2=0$

Calculation:

Bernoulli’s equation represents the relationship between the dynamic pressure, static pressure and manometric pressure. Its general form is,

  $P+\frac{1}{2}\rho v^2+\rho gh=\text{constant}$

According to the given information $v_1=v_2=0$ , the Bernoulli’s equation is evaluated. Let subscript 1 represents the initial point and subscript 2 represents the final point. So, the equation becomes,

  $\begin{array}{l}{P}_1+\frac{1}{2}\rho v_1{}^2+\rho g{h}_1=P_2+\frac{1}{2}\rho v_2{}^2+\rho g{h}_2\\ P_1+\rho g{h}_1=P_2+\rho g{h}_2\left(\because v_1=v_2=0\right)\\ P_1-P_2=\rho g{h}_2-\rho g{h}_1\\ -\Delta P=\rho g\left(h_2-h_1\right)\\ \Delta P=-\rho g\Delta h\end{array}$

The change in depth is just opposite to the change in y -coordinate, so the above equation becomes

  $\Delta P=\rho g\Delta h$