Question

A steady incompressible laminar flow passes through a 90 degree tube bend placed on a horizontal surface. In horizontal diametric plane, two pressure taps \(P_i\) and \(P_o\) are provided across the cross-section at inner and outer walls of the bend, respectively. Which one of the following options is correct?

• A. \(P_o > P_i\)
• B. \(P_o < P_i\)
• C. \((P_o - P_i)\) is directly proportional to the bend radius.
• D. \((P_i - P_o)\) is inversely proportional to the average flow velocity.

Hint:

Think about what force keeps fluid moving along a curved path and where that force must come from.

Answer 1

The Correct Option is A

Flow around a bend needs a centripetal force to turn each fluid particle, and in a pipe that force comes only from a pressure gradient set up across the bend's cross section, from the inner wall toward the outer wall.

1. $P_o > P_i$: correct. Supplying the inward force needed for curved motion means pressure must rise from the inner wall to the outer wall, since fluid at the outer wall sits at a larger radius.
2. $P_o < P_i$: wrong, this would push fluid outward instead of supplying the inward force the curved path needs.
3. $(P_o - P_i)$ directly proportional to bend radius: wrong, the radial pressure gradient formula ties the difference to velocity squared and radius together, not a plain direct proportion with radius alone.
4. $(P_i - P_o)$ inversely proportional to average velocity: wrong, this gets both the sign and the trend backwards, since the gradient formula shows the difference grows with velocity squared.

The physically consistent result is that the outer wall pressure is higher than the inner wall pressure, option A.