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Question #41986

Difficulty: 8/10 | 🟢 Free

The Question

A fluid flows steadily through a horizontal pipe of constant cross-sectional area and length $L$ . The pipe rotates with a constant angular velocity $\omega$ about a vertical axis passing through its entrance ( $r=0$ ). Due to an externally maintained temperature profile, the density of the fluid at a radial distance $r$ from the axis is strictly given by $\rho(r) = \rho_0 \left(1 - \frac{2r}{3L}\right)$ , where $\rho_0$ is the density at the entrance. Both ends of the pipe are open to the atmosphere. Find the exit velocity $v_L$ of the fluid at $r=L$ .

Options

A.
$\frac{\sqrt{5}}{2} \omega L$
B.
$\sqrt{\frac{3}{2}} \omega L$
C.
$\sqrt{\frac{5}{2}} \omega L$
D.
$\frac{3}{2\sqrt{2}} \omega L$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Bernoulli’s Energy Balance|Work-Energy Theorem for Rotation|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of the Work-Energy Theorem (Euler's equation) for fluid flow in a non-inertial frame with variable density.

A common lethal mistake is blindly applying the standard Bernoulli equation $P + \frac{1}{2}\rho v^2 - \frac{1}{2}\rho \omega^2 r^2 = \text{constant}$ , which is only valid for fluids with uniform density.

Students falling for this trap will incorrectly use local density values in the pseudo-potential term, leading to Option B.

Another trap is incorrectly integrating the kinetic energy gradient as $\Delta(\frac{1}{2}\rho v^2)$ instead of $\int \rho v dv$ , leading to Option C.

The correct approach requires recognizing that the constant mass flux $J = \rho v$ factors out of the convective acceleration integral, while the centrifugal force must be explicitly integrated over the density gradient.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Fluids - Class 11 Part 2
Concept	Bernoulli’s Energy Balance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free