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Question #42105

Difficulty: 8/10 | 🟢 Free

The Question

A dilute gas of constant macroscopic density $\rho$ flows steadily through a horizontal Venturi tube. The inlet has a cross-sectional area $A$ and the throat has a cross-sectional area $A/2$ . The measured pressure drop between the inlet and the throat is $\Delta P$ . Inside the throat, a microscopic flat disc of area $a$ ( $a \ll A$ ) is held stationary, perpendicular to the flow. Because the gas is highly dilute (Knudsen number $\gg 1$ ), it interacts with the disc locally as a stream of discrete, non-interacting particles. Assuming these gas particles collide elastically with the disc, what is the average force exerted by the gas on the disc?

Options

A.
$\frac{16}{3} a \Delta P$
B.
$\frac{8}{3} a \Delta P$
C.
$\frac{4}{3} a \Delta P$
D.
a \Delta P

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Venturi Meter Logic|Collision Impulses|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of macroscopic fluid dynamics (Bernoulli's principle) with microscopic kinetic theory (impulse-momentum theorem).

A common trap is assuming the force on the disc is simply the macroscopic stagnation pressure force ( $F = \frac{1}{2}\rho a v_2^2$ ) or the inelastic momentum flux ( $F = \rho a v_2^2$ ).

However, in the dilute gas regime (Knudsen number $\gg 1$ ), the discrete particles undergo elastic collisions, transferring twice their initial momentum.

Students must correctly couple the macroscopic pressure drop to the microscopic collision rate and apply Newton's Second Law to find the true average reaction force.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Fluids - Class 11 Part 2
Concept	Venturi Meter Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free