An incompressible fluid is flowing through a vertical pipe with a constriction. The wide section is $2.00 \mathrm{~cm}$ in diameter and is at the top of the pipe. The pressure of the fluid in the wide section at the top is $200 \mathrm{kPa}$. The velocity of the fluid in the wide section is $4.00 \mathrm{~m} / \mathrm{s}$. The narrow section is $1.00 \mathrm{~cm}$ in diameter and is located 2.00 m below the wide section. What is the pressure of the fluid in the narrow section? (density of the fluid is $1,000 \mathrm{~kg} / \mathrm{m}^{3}$ )
A) $116 \mathrm{kPa}$
B) $80.0 \mathrm{kPa}$
C) $105 \mathrm{kPa}$
D) $99.6 \mathrm{kPa}$
E) $72.0 \mathrm{kPa}$

Question

Quizplus · Accepted Answer

To find the pressure in the narrow section, we can use Bernoulli's equation, which for an incompressible fluid is given by $P_1 + \frac{1}{2}\rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho gh_2$. Given that the fluid is incompressible, the volume flow rate must be constant, so $A_1v_1 = A_2v_2$, where $A$ is the cross-sectional area and $v$ is the velocity. From this, we can find $v_2$, and then use Bernoulli's equation to solve for $P_2$, the pressure in the narrow section. Given the diameters, we can calculate the areas ($A_1 = \pi(0.01)^2$ m$^2$ and $A_2 = \pi(0.005)^2$ m$^2$), find $v_2$, and then solve for $P_2$, which results in approximately $99.6 \mathrm{kPa}$.

An Incompressible Fluid Is Flowing Through a Vertical Pipe with a Constriction