Question

Water is flowing through a channel (lying in a vertical plane) as shown in the figure. Three sections $A,B$ and $C$ are shown. Section $B$ and $C$ have equal areas of cross- section. If $P_A,P_B$ and $P_C$ are the pressures at $A,B$ and $C$ respectively, then

• A. $P_A<P_B=P_C$
• B. $P_A<P_B<P_C$
• C. $P_A>P_B=P_C$
• D. $P_A>P_B>P_C$

Answer 1

The correct option is B $P_A<P_B<P_C$

Applying Bernoulli's principle and equation of continuity.
Comparing points $A$ and $B$ (the vertical elevation $h$ is same for both),
$A_A{V}_A=A_B{V}_B$ (equation of continuity)
$\because A_A<A_B$
$\Rightarrow V_A>V_B$
$P_A+\frac{1}{2}\rho V_A^2+\rho gh=P_B+\frac{1}{2}\rho V_B^2+\rho gh$ (Bernoulli's equation)
$\because V_A>V_B$
In accordance with Bernoulli's equation,
$\frac{1}{2}\rho V_A^2>\frac{1}{2}\rho V_B^2$
$\therefore P_A<P_B...\left(1\right)$
Now comparing $C$ and $B$
Area of cross-section $A_B=A_C$
$\Rightarrow V_B=V_C=V$, from equation of continuity
$P_B+\frac{1}{2}\rho V^2+\rho g{h}_B=P_C+\frac{1}{2}\rho V^2+\rho g{h}_C$
(Applying Bernoulli's equation between points $B$ and $C$)
$\Rightarrow P_B+\rho g{h}_B=P_C+\rho g{h}_C$
$\because h_B>h_C$
$\Rightarrow P_B<P_C...\left(2\right)$
Using Eq.$\left(1\right)$ and $\left(2\right)$ we get,
$P_A<P_B<P_C$