Dimensionless numbers play an important role in analysing fluid dynamics and heat and mass transfer problems. They provide a method by which complex phenomena can be characterised, often by way of a simple, single number comparison. This article provides a brief overview of the derivation and use of dimensionless numbers.

## Definitions

## Introduction

In the subsequent sections a simple system is utilised to explore the dimensionless numbers through:

- Derivation using dimensional analysis
- Usefulness in system scaling
- Application in analytical and empirical relationships

## Dimensional Analysis

This gives us the equations of exponents as follows:

## Analytical and Empirical Relationships

## Further Reading

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## 5.1: Dimensionless groups

## 5.1.1 Finding dimensionless groups

\[E_{total} = \frac{1}{2} mv^{2} + \frac{1}{2} k x^{2}.\]

That news is welcome, but how do we find these groups?

Are there other dimensionless groups?

\[\frac{ar}{v^{2}}=\textrm{dimensionless constant,}\]

because there are no other independent dimensionless groups to use on the right side.

Why can’t we use ar / v 2 on the right side?

\[\frac{ar}{v^{2}} =3(\frac{ar}{v^{2}})-1.\]

Its solution is ar/v 2 =1/2, which is another example of our general form

\[\frac{ar}{v^{2}}=\textrm{dimensionless constant.}\]

But what if we use a more complicated function?

\[\frac{ar}{v^{2}}=(\frac{ar}{v^{2}})^{2}-1.\]

\[\frac{ar}{v^{2}}=\left \{ \begin{array}{ll} \phi \\-1/\phi, \\ \end{array} \right. \]

Using that form, which we cannot escape, the acceleration of the train is

\[a \sim \frac{(60 m s^{-1})^{2}}{2 \times 10^{3}m} = 1.8 m s^{-2}.\]

explains why tall people, with a longer leg length, generally walk faster than short people.

\[v_{max} \sim \sqrt{10 m s^{-2} \times 1m} \approx 3 m s^{-1}.\]

## 5.1.2 Counting dimensionless groups

But acceleration is not a fundamental dimension, so how can we use it?

- number of independent dimensions

_____________________________________

number of independent dimensionless groups

Exercise \(\PageIndex{1}\): Bounding the number of independent dimensionelss groups

Why is the number of independent dimensionless groups never more than the number of quantities?

Exercise \(\PageIndex{2}\): Counting dimensionless groups

How many independent dimensionless groups do the following sets of quantities produce?

b. impact speed v of a free-falling object: v , g , and h (initial drop height).

c. impact speed v of a downward-thrown free-falling object: v , g , h , and v 0 (launch velocity).

Exercise \(\PageIndex{3}\): Using angular frequency instead of speed

\[\textrm{dimensionless group proportional to } a= \textrm{dimensionless constant}\]

Exercise \(\PageIndex{4}\): Impact speed of a dropped object

Exercise \(\PageIndex{5}\): Speed of gravity waves on deep water

a. Explain why these quantities produce one independent dimensionless group.

b. What is the group proportional to v ?

Exercise \(\PageIndex{6}\): Using period instead of speed

a. Explain why there is still only one independent dimensionless group.

b. What is the independent dimensionless group proportional to a ?

c. What dimensionless constant is this group equal to?

## Fluid Mechanics and Hydraulic Machines by S. C. Gupta

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## 8.1 Introduction

Get Fluid Mechanics and Hydraulic Machines now with the O’Reilly learning platform.

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