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Question #43910

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The Question

A vertical capillary tube is constructed from a bimetallic sheet such that its internal radius $r$ expands with temperature $\theta$ according to the relation $\frac{1}{r}\frac{dr}{d\theta} = \alpha$ . The tube is dipped in a large reservoir of liquid with density $\rho$ , surface tension $S$ , and dynamic viscosity $\eta$ . The contact angle is zero. The temperature of the tube is increased at a constant rate $\frac{d\theta}{dt} = C$ . Assume the liquid properties remain constant and the axial velocity profile is locally parabolic. As the radius increases, the liquid column dynamically adjusts its height $h$ . Let $h_q = \frac{2S}{\rho g r}$ be the instantaneous quasi-static equilibrium height. Assuming $\Delta h = h - h_q \ll h_q$ , find the height difference $\Delta h$ at an instant when the radius is $r$ .

Options

A.
$\frac{4 S^2 \alpha^2 C^2}{\rho^2 g^3 r^2}$
B.
$\frac{32 \eta S^2 \alpha C}{\rho^3 g^3 r^4}$
C.
$\frac{16 \eta S^2 \alpha C}{\rho^3 g^3 r^4}$
D.
$\frac{2 S^2 \alpha^2 C^2}{\rho^2 g^3 r^2}$

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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

Capillary Rise Mechanics|Bimetallic Strip Curvature|Binomial Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the interplay between fluid continuity, viscous drag, and inertial forces in a dynamically expanding boundary.

The primary trap is assuming the liquid column moves downwards as a rigid body with uniform velocity $v = h_q \alpha C$ , which leads students to calculate a massive viscous pressure drop (Option B).

However, applying the continuity equation reveals that the radial expansion of the tube requires fluid to flow inwards from the bottom, creating an axial velocity profile $\bar{v}_z(z) = \alpha C(h - 2z)$ .

This profile is perfectly antisymmetric about the midpoint, causing the net viscous drag to integrate to exactly zero.

Consequently, the lag is purely inertial, driven by the local acceleration of the fluid.

Quick Facts About This Question

Question Information

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