---
title: "Fluid properties and flow description methods"
subject: "Fluid Mechanics"
chapter: 1
key: "semester_4/fluid_mechanics_mechanical:fluid-newtonian-lagrangian-eulerian"
source: https://www.bctengineer.com/q/semester_4/fluid_mechanics_mechanical/fluid-newtonian-lagrangian-eulerian
site: BCT Engineering
---

# Fluid properties and flow description methods

Fluid Mechanics · Chapter 1: Definition and Analysis method

- Asked in: year not given, 4+4 marks
- Video: [Understanding Viscosity](https://www.youtube.com/watch?v=VvDJyhYSJv8) — The Efficient Engineer
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/fluid_mechanics_mechanical/fluid-newtonian-lagrangian-eulerian
- Whole chapter: https://www.bctengineer.com/md/semester_4/fluid_mechanics_mechanical/chapter-1

## Question

(a) Define a fluid. Differentiate between a Newtonian and a non-Newtonian fluid with examples, and state how temperature affects the viscosity of liquids and gases.
(b) Differentiate between the Lagrangian and Eulerian methods of describing fluid motion, and between a system and a control volume.

## Short answer

A fluid deforms continuously under shear. Newtonian fluids have constant viscosity (e.g., water); non-Newtonian viscosity varies with shear rate (e.g., blood). Temperature rise decreases liquid viscosity but increases gas viscosity. Lagrangian tracks individual particles; Eulerian observes fixed points. A system is a fixed mass; a control volume is a fixed region allowing mass flow across its surface.

## Answer

#### (a) Fluid, Newtonian and non-Newtonian fluids
A **fluid** is a substance that deforms continuously under the action of a shear stress, however small. Liquids and gases are fluids.

A **Newtonian fluid** obeys Newton's law of viscosity: shear stress is directly proportional to the rate of shear strain (velocity gradient).

$$
\tau = \mu \frac{du}{dy}
$$

| Basis | Newtonian fluid | Non-Newtonian fluid |
|---|---|---|
| $\tau$ vs $du/dy$ | Straight line through origin | Curved line, or has a yield stress |
| Viscosity $\mu$ | Constant at given temperature | Changes with shear rate (apparent viscosity) |
| Examples | Water, air, kerosene, mercury | Blood, paint, toothpaste, polymer solutions |
| Types | Single type | Pseudoplastic, dilatant, Bingham plastic |

**Effect of temperature:**
- Liquids: viscosity **decreases** as temperature rises, because cohesive forces between molecules weaken.
- Gases: viscosity **increases** as temperature rises, because molecular momentum exchange between layers increases.

#### (b) Lagrangian and Eulerian methods
| Basis | Lagrangian | Eulerian |
|---|---|---|
| Idea | Follow an individual fluid particle along its path | Observe a fixed point in space as the fluid passes |
| Variables | Position and velocity as functions of time for each particle | Velocity, pressure as functions of $(x, y, z, t)$ |
| Acceleration | Directly $d\vec V/dt$ | Needs material derivative (local + convective) |
| Use | Particle tracking, sprays, dispersion | Most fluid mechanics, velocity fields |

Fluid mechanics mostly uses the Eulerian method because it is easier to handle mathematically.

| Basis | System | Control volume |
|---|---|---|
| Definition | Fixed identifiable mass of fluid | Fixed region in space chosen for analysis |
| Boundary | Moves and deforms with the fluid | Fixed (usually); called control surface |
| Mass crossing boundary | None | Mass can cross the surface |
| Used with | Lagrangian view | Eulerian view, Reynolds transport theorem |

## Check yourself

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

1. Which statement correctly defines a fluid?
   A. A substance that cannot resist any normal stress
   B. A substance that deforms continuously under any shear stress, however small
   C. A substance that has no fixed volume
   D. A substance that flows only when the shear stress exceeds a yield value
   Answer: B. A fluid keeps deforming under the action of shear stress however small; this is what separates fluids from solids. (Key point: Definition of a fluid)

2. A fluid is Newtonian if:
   A. its density is constant
   B. shear stress is directly proportional to velocity gradient, so viscosity is constant at a given temperature
   C. shear stress is independent of the rate of shear strain
   D. its viscosity changes with shear rate
   Answer: B. Newton's law of viscosity, tau = mu du/dy, gives a straight line through the origin with constant mu. (Key point: Newtonian fluid behaviour)

3. When temperature rises, the viscosity of a liquid and of a gas respectively:
   A. increases, increases
   B. decreases, decreases
   C. increases, decreases
   D. decreases, increases
   Answer: D. In liquids cohesion weakens, so viscosity falls; in gases momentum exchange between layers grows, so viscosity rises. (Key point: Effect of temperature on viscosity)

4. What is the key difference between a system and a control volume?
   A. A system allows mass to cross its boundary, a control volume does not
   B. A system is a fixed mass whose boundary moves with the fluid; a control volume is a fixed region through which mass can cross
   C. A system is used in the Eulerian view, a control volume in the Lagrangian view
   D. There is no difference; both are fixed regions in space
   Answer: B. A system is an identifiable fixed mass (Lagrangian); a control volume is a fixed region with a control surface across which mass flows (Eulerian). (Key point: System vs control volume)

## Related in Fluid Mechanics

- [Navier-Stokes equations for Newtonian fluid](https://www.bctengineer.com/question.md?key=semester_4%2Ffluid_mechanics_mechanical%3Anavier-stokes-newtonian-fluid) — Fluid Mechanics, chapter 4
- [Oil flow between fixed plates problem](https://www.bctengineer.com/question.md?key=semester_4%2Ffluid_mechanics_mechanical%3Aoil-flow-between-plates-numerical) — Fluid Mechanics, chapter 6
- [Acceleration of a fluid particle](https://www.bctengineer.com/question.md?key=semester_4%2Ffluid_mechanics_mechanical%3Aparticle-acceleration-unsteady-field) — Fluid Mechanics, chapter 3
- [Types of flow, streamline, pathline, streakline](https://www.bctengineer.com/question.md?key=semester_4%2Ffluid_mechanics_mechanical%3Aflow-types-streamline-pathline-streakline) — Fluid Mechanics, chapter 3
- [Hagen-Poiseuille flow in a circular tube](https://www.bctengineer.com/question.md?key=semester_4%2Ffluid_mechanics_mechanical%3Ahagen-poiseuille-derivation-circular-tube) — Fluid Mechanics, chapter 6
- [Laminar oil flow in a pipe](https://www.bctengineer.com/question.md?key=semester_4%2Ffluid_mechanics_mechanical%3Alaminar-oil-pipe-pressure-drop) — Fluid Mechanics, chapter 6

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