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

Step 1: Set up the force balance across the bend.
When a fluid flows around a curved bend, each fluid element needs a net inward (centripetal) force to keep moving along the curved path. In a horizontal bend this force can only come from a pressure difference across the cross section.

Step 2: Write the radial pressure gradient.
For flow along a curved streamline of local radius \(r\), the radial equilibrium condition gives \(\dfrac{dp}{dr} = \dfrac{\rho V^2}{r}\), where \(V\) is the local flow speed. Since \(\rho\), \(V^2\), and \(r\) are all positive, pressure must rise as \(r\) increases.

Step 3: Identify which tap sits at the larger radius.
The outer wall of the bend sits farther from the center of curvature, so it lies at a larger radius than the inner wall. By Step 2, pressure at the larger radius must be higher.

Final Answer:
The outer wall pressure exceeds the inner wall pressure in a curved bend. \[ \boxed{P_o > P_i} \]