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Question #42232

Difficulty: 8/10 | 🟢 Free

The Question

A viscous, incompressible fluid of viscosity $\eta$ flows steadily through a horizontal pipe consisting of two cylindrical sections in series. The first section has radius $R$ and length $L$ , while the second section has radius $R/2$ and length $L$ . In the first section, the fully developed laminar velocity profile is given by $v_1(r) = v_0 \left(1 - \frac{r^2}{R^2}\right)$ , where $r$ is the radial distance from the central axis. Assuming fully developed laminar flow is maintained in the second section as well, what is the total viscous drag force exerted by the fluid on the cylindrical walls of the entire pipe?

Options

A.
$20 \pi \eta L v_0$
B.
$40 \pi \eta L v_0$
C.
$10 \pi \eta L v_0$
D.
$12 \pi \eta L v_0$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Pascal’s Law Limitations|Equation of Continuity|Vector Cross-Product

Why This Question Is Tricky

Common Mistakes

This problem tests the synthesis of the Equation of Continuity and viscous shear stress in Poiseuille flow.

The 'Lethal Collision' occurs when students incorrectly calculate the volumetric flow rate by assuming a uniform velocity profile ( $Q = \pi R^2 v_0$ ) instead of integrating the parabolic profile, which leads to an incorrect maximum velocity in the second section and yields the trap answer of $40 \pi \eta L v_0$ .

Additionally, it forces students to abandon hydrostatic assumptions (Pascal's Law) and correctly apply Newton's law of viscosity to find the drag force via the velocity gradient at the pipe walls, rather than erroneously using Bernoulli's equation to find a pressure drop that ignores viscous shear.

Quick Facts About This Question

Question Information

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