Question

A fluid is flowing through a horizontal pipe of varying cross - section, with speed $v\text{m}{s}^{-1}$ at a point where the pressure is $P\text{Pascal}$. At another point where pressure is $\frac{P}{2}\text{Pascal}$ its speed is $V{\text{ms}}^{-1}.$ If the density of fluid is $\rho {\text{kg/m}}^3$ and the flow is streamline, then $V$ is equal to :

• A. $\sqrt{\frac{P}{\rho }+v}$
• B. $\sqrt{\frac{2P}{\rho }+v^2}$
• C. $\sqrt{\frac{P}{2\rho }+v^2}$
• D. $\sqrt{\frac{P}{\rho }+v^2}$

Answer 1

The correct option is D $\sqrt{\frac{P}{\rho }+v^2}$

Using Bernoulli's equation,

$P_1+\frac{1}{2}\rho v_1^2+\rho g{h}_1=P_2+\frac{1}{2}\rho v_2^2+\rho g{h}_2.....\left(1\right)$

For horizontal pipe,
$h_1=0\text{and}{h}_2=0$

Given,
$P_1=P,P_2=\frac{P}{2}$

Putting in eq. $\left(1\right)$, we get

$\Rightarrow P+\frac{1}{2}\rho v^2=\frac{P}{2}+\frac{1}{2}\rho V^2$

$\Rightarrow \frac{P}{2}+\frac{1}{2}\rho v^2=\frac{1}{2}\rho V^2$

$\therefore V=\sqrt{v^2+\frac{P}{\rho }}$

Hence, option $\left(\text{D}\right)$ is correct.