Question

A horizontal pipe carries water in a streamline flow. At point along the pipe, where the cross-sectional area is \(A_1\), the velocity of water is \(V_1\) and the pressure is \(P_1\). What is the pressure of water at another point where the cross-sectional area is \(A_2\)?

• A. \(P_1-\frac{ρV_2^2}{2A_1^2}(A_2^2-A_1^2)\)
• B. \(P_1+\frac{ρV_1^2}{2A_2^2}(A_2^2-A_1^2)\)
• C. \(P_1ρV_1^2(A_1^2-A_2^2)\)
• D. \(P/ρV_1^2A_2^2\)

Hint:

Use the equation of continuity to get V2, then Bernoulli for a horizontal pipe.

Answer 1

The Correct Option is B

Step 1: Energy per volume
Along a streamline the sum $P+\frac12\rho v^2$ is constant.

Step 2: Eliminate V2
Replace $V_2^2$ with $V_1^2A_1^2/A_2^2$ and take the difference: $P_2 - P_1 = \frac12\rho V_1^2\frac{A_2^2-A_1^2}{A_2^2}$. Option (B).

Final Answer:
Option B. \[ \boxed{\text{(B)}\ P_2=P_1+\frac{\rho V_1^2}{2A_2^2}(A_2^2-A_1^2)} \]