Question

Water flows through a frictionless duct with a cross-section varying as shown in the figure. The pressure $P$ at points along the axis (starting from left end to right end) is represented by

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

When cross-sectional area of duct decreases, the velocity of water increases.
Applying continuity equation,
$Av=\text{constant}$
$\Rightarrow v\propto \frac{1}{A}...\left(i\right)$
Again, according to Bernoulli's theorem, sum of pressure, kinetic energy and potential energy of a fluid per unit volume is constant.
$P+\frac{1}{2}\rho v^2+\rho gh=\text{constant}...\left(ii\right)$
From Eq.$\left(i\right)\&\left(ii\right)$,
As the cross-sectional area decreases, $KE\uparrow$, i.e term $\left(\frac{1}{2}\rho v^2\right)\uparrow$.
Since elevation of fluid is constant throughout, pressure$\left(P\right)$ of fluid decreases as area of cross section decreases.
As we move along the axis, initially pressure remains constant, then decreases as the cross-section narrows and stays constant again.
Hence, $P-x$ graph is described by option $\left(a\right)$