---
title: "Viscosity from torque on rotating cylinder"
subject: "Fluid Mechanics"
chapter: 1
key: "semester_3/fluid_mechanics:concentric-cylinders-torque-viscosity"
source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/concentric-cylinders-torque-viscosity
site: BCT Engineering
---

# Viscosity from torque on rotating cylinder

Fluid Mechanics · Chapter 1: Fluid and its physical properties

- Asked in: 2074 Asoj, 4 marks
- Video: [Mechanics of Fluids - Topic2 - Example 1 - Viscosity - Rotating Cylinder in Annulus 1](https://www.youtube.com/watch?v=OKTY7SIBEIM) — Dato Ts Dr Mohd Zamri Yusoff
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/concentric-cylinders-torque-viscosity
- Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-1

## Question

A 15 cm diameter vertical cylinder rotates concentrically inside another cylinder of diameter 15.10 cm; both cylinders are 25 cm high. The space between the cylinders is filled with a liquid whose viscosity is unknown. If a torque of 12 Nm is required to rotate the inner cylinder at 100 rpm, determine the viscosity of the fluid.

## Short answer

For thin-gap linear profile, torque T = μ (ωR/t) 2π R² h. Solving for μ yields μ = T t / (2π ω R³ h) = (12×0.0005)/(2π×10.472×0.075³×0.25) ≈ 0.865 Pa·s (8.65 poise).

## Answer

**Data:** inner radius $R=0.075$ m, outer radius $0.0755$ m, so the gap $t=0.5$ mm $=0.0005$ m. Height $h=0.25$ m, $T=12$ N·m, $N=100$ rpm, so $\omega=\dfrac{2\pi\times100}{60}=10.472$ rad/s.

Surface velocity of the inner cylinder: $V=\omega R=10.472\times0.075=0.7854$ m/s.

For a thin gap the profile is linear: $\tau=\mu\dfrac{V}{t}$. The torque on the inner cylinder (bottom neglected):

$$
T=\tau\,(2\pi Rh)\,R=\mu\frac{\omega R}{t}\,2\pi R^{2}h
$$

$$
\mu=\frac{T\,t}{2\pi\omega R^{3}h}=\frac{12\times0.0005}{2\pi(10.472)(0.075)^{3}(0.25)}
$$

$$
\mu=0.8646\ \text{Pa·s}
$$

**Answer: $\mu\approx0.865\ \text{N·s/m}^2$ (8.65 poise).**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Cylinder diameters are 15 cm and 15.10 cm. What is the gap?
   A. 1 mm
   B. 0.5 mm
   C. 0.05 mm
   D. 5 mm
   Answer: B. The radii are 7.5 and 7.55 cm, so t = 0.05 cm = 0.5 mm. (Key point: Gap thickness)

2. Which relation connects torque and viscosity?
   A. T = mu omega R^3 h t / 2 pi
   B. T = mu omega R t / (2 pi R^2 h)
   C. T = 2 pi mu omega R^3 h / t
   D. T = mu omega R^2 t h
   Answer: C. T = tau (2 pi R h) R with tau = mu omega R/t gives T = 2 pi mu omega R^3 h / t. (Key point: Torque relation)

3. What viscosity results (T = 12 N·m, 100 rpm)?
   A. 0.0865 Pa·s
   B. 8.65 Pa·s
   C. 0.865 Pa·s
   D. 86.5 Pa·s
   Answer: C. mu = T t/(2 pi omega R^3 h) = 0.865 Pa·s (8.65 poise). (Key point: Final viscosity)

## Related in Fluid Mechanics

- [Liquid rise in rotating U-tube](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Au-tube-rotating-liquid-rise) — Fluid Mechanics, chapter 2
- [Model liquid viscosity, Froude and Reynolds](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Amodel-liquid-viscosity-froude-reynolds) — Fluid Mechanics, chapter 10
- [Force to lift plate in liquid gap](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Aplate-lifted-in-gap) — Fluid Mechanics, chapter 1
- [Vertical force on cylinder with hemispherical ends](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Ahemispherical-cylinder-vertical-force) — Fluid Mechanics, chapter 3
- [Torque to overcome bearing friction](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Ajournal-bearing-torque) — Fluid Mechanics, chapter 1
- [Pressure diagram in rotating closed tank](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Aclosed-rotating-tank-pressure-diagram) — Fluid Mechanics, chapter 2

## Related in other subjects

- [Hollow cylinder: heat transfer; temperature at 10 cm](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Ahollow-cylinder-8-12cm-radial-10cm) — Fundamental of Thermodynamics & Heat Transfer, chapter 7, asked 2×
- [Flow between reservoirs (15 cm and 25 cm pipe)](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Areservoirs-pipeline-15-25cm) — Hydro Power, chapter 1
- [Hollow shaft 10/7 cm: power transmitted](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Ahollow-shaft-power-300rpm-13-mean) — Strength of Materials, chapter 6, asked 2×
- [Thin copper shell: pressure from extra 50 cc liquid](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Athin-copper-shell-extra-liquid-50cc) — Strength of Materials, chapter 5, asked 2×