Question

An incompressible liquid flows through a horizontal pipe having cross-sectional areas \( A \) at one end and \( 2A \) at the other end. If the pressure and velocity of the liquid at the lower cross-section are \( P \) and \( v \), then these values at the other end are

• A. \( \frac{v}{2},\, P + \frac{3}{8}\rho v^2 \)
• B. \( v,\, P + \frac{1}{8}\rho v^2 \)
• C. \( \frac{v}{4},\, P + \frac{1}{4}\rho v^2 \)
• D. \( v,\, P + \frac{1}{2}\rho v^2 \)
• E. \( 2P + \rho v^2 \)

Hint:

Always apply continuity first, then Bernoulli.

Answer 1

The Correct Option is A

Concept: Continuity + Bernoulli equation.

Step 1: Continuity.
\[ Av = 2A v_2 \Rightarrow v_2 = \frac{v}{2} \]

Step 2: Bernoulli.
\[ P + \frac{1}{2}\rho v^2 = P_2 + \frac{1}{2}\rho \left(\frac{v}{2}\right)^2 \]

Step 3: Solve.
\[ P_2 = P + \frac{1}{2}\rho v^2 - \frac{1}{8}\rho v^2 = P + \frac{3}{8}\rho v^2 \]