Tube Bend Flow Pressure Analysis

For a steady, incompressible fluid flow through a horizontal 90° tube bend, the fluid follows a curved path. This curvature induces a radial acceleration (centripetal acceleration) towards the center of the bend.

• The fluid particles closer to the outer wall travel a longer distance and experience a greater centripetal acceleration compared to those near the inner wall.
• To provide the necessary centripetal force for this acceleration, a pressure gradient must exist across the bend, acting from the outer wall towards the inner wall.
• This means the pressure at the outer wall ($P_o$) must be higher than the pressure at the inner wall ($P_i$).

Pressure Comparison ($P_o$ vs $P_i$)

Based on the fluid dynamics principles for flow in a bend:

• Outer Wall Pressure ($P_o$): Experiences higher pressure due to the greater radial acceleration requirement.
• Inner Wall Pressure ($P_i$): Experiences lower pressure relative to the outer wall.

Therefore, the correct relationship is $P_o > P_i$.

Evaluating Options

• Option 1: $P_o > P_i$ - This is consistent with the physical principles.
• Option 2: $P_o < P_i$ - Incorrect.
• Option 3: $(P_o - P_i)$ is directly proportional to the bend radius. - Incorrect. The pressure difference is typically inversely proportional to the radius of curvature.
• Option 4: $(P_i - P_o)$ is inversely proportional to the average flow velocity. - Incorrect. The pressure difference is related to the velocity squared.

The correct option is the one stating $P_o > P_i$.