---
title: "Newton's law of viscosity"
subject: "Hydro Power"
chapter: 1
key: "semester_6/hydro_power:newtons-law-viscosity"
source: https://www.bctengineer.com/q/semester_6/hydro_power/newtons-law-viscosity
site: BCT Engineering
---

# Newton's law of viscosity

Hydro Power · Chapter 1: Introduction to Basic Fluid Mechanics and Hydraulics

- Asked in: 2075 Baisakh, 1+4 marks
- Video: [Newton's Law of Viscosity || Newton's law of viscosity in hindi || Fluid mechanics](https://www.youtube.com/watch?v=5jo4DJZ1cnE) — Gear Institute Mechanical Engineering Videos and Physics by Gear Institute
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/hydro_power/newtons-law-viscosity
- Whole chapter: https://www.bctengineer.com/md/semester_6/hydro_power/chapter-1

## Question

State the Newton's law of viscosity and explain the importance of viscosity in fluid motion.

## Short answer

Newton's law of viscosity: τ = μ du/dy. Viscosity causes friction and head loss, determines flow regime via Reynolds number, enforces the no-slip condition, influences velocity profiles and drag, and dissipates energy.

## Answer

**Newton's law of viscosity:** the shear stress on a fluid layer is directly proportional to the rate of shear strain (velocity gradient) normal to the layer:
$$
\tau = \mu \frac{du}{dy}
$$
where $\mu$ is the coefficient of dynamic viscosity (N·s/m²). Fluids that obey this law are Newtonian fluids.

#### Importance of viscosity in fluid motion
1. **Resistance to flow and head loss:** viscosity causes shear between layers and friction at the walls. This produces head loss in pipes, penstocks and canals ($h_f = \frac{fLV^2}{2gD}$). In hydropower, friction loss in the penstock reduces the net head and power.
2. **Type of flow:** viscosity appears in the Reynolds number $Re = \rho VD/\mu$, which decides whether flow is laminar or turbulent.
3. **No-slip condition and velocity profile:** because of viscosity, fluid touching a wall has zero velocity, giving the parabolic (laminar) or flatter (turbulent) velocity profile and the boundary layer.
4. **Drag and power needs:** viscous drag on bodies and the power needed to pump fluids depend on viscosity.
5. **Lubrication:** a thin oil film separates moving surfaces (bearings of turbines and generators), reducing wear; correct viscosity is essential.
6. **Energy dissipation:** viscosity converts kinetic energy into heat, for example in hydraulic jumps below spillways.
7. **Temperature effect:** viscosity of liquids falls with rise in temperature (weaker cohesion) while that of gases rises (more molecular momentum exchange); this must be considered in design.

## Check yourself

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

1. Newton's law of viscosity states that shear stress is:
   A. Proportional to the velocity
   B. Proportional to the velocity gradient du/dy normal to the layer
   C. Inversely proportional to velocity gradient
   D. Proportional to pressure
   Answer: B. tau = mu du/dy, with mu the dynamic viscosity. (Key point: Newton's law)

2. In a hydropower penstock, viscosity matters because it:
   A. Increases the net head
   B. Causes friction head loss that reduces net head and power
   C. Eliminates turbulence
   D. Makes the flow compressible
   Answer: B. Viscous friction gives head loss h_f, lowering net head and hence power. (Key point: Viscosity and head loss)

3. With rise in temperature, viscosity of liquids and gases respectively:
   A. Rises, rises
   B. Falls, rises
   C. Falls, falls
   D. Rises, falls
   Answer: B. Liquid viscosity falls as cohesion weakens; gas viscosity rises due to more molecular momentum exchange. (Key point: Temperature effect)

4. Because of viscosity, fluid touching a solid wall has:
   A. Maximum velocity
   B. Zero velocity (no-slip), producing a velocity profile and boundary layer
   C. Velocity equal to the mean velocity
   D. Infinite shear stress
   Answer: B. No-slip means zero velocity at the wall, shaping the velocity profile and boundary layer. (Key point: No-slip condition)

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