# 4.1.2. Continuum Model¶

## 4.1.2.1. What is the Continuum Model?¶

Discontinuous molecular collisions are replaced with continuous statistically averaged values, e.g.

- Pressure = Average collision force per unit area.

## 4.1.2.2. Why is the Continuum Model used?¶

Average fluid properties in the Continuum model are used because:

- Molecular density of fluid is large with respect to characteristic length.
- Interaction forces (potentials) are now known.

## 4.1.2.4. Where can the Continuum Model be applied?¶

Can be applied to:

- Finite volume element
- Infinitesimal differential element

## 4.1.2.5. What is the Knudsen number?¶

\[{Kn = {\lambda \over L}}\]

where:

- \(\lambda\) = mean free path
- \(L\) = characteristic length (e.g. pipe diameter, boundary layer thickness, shockwave thickness)

## 4.1.2.6. What is the Mean Free Path?¶

\[{\lambda = {1 \over {\sqrt{2} n S}}}\]

where:

- \(n\) = number of particles per \(m^3\) from the Boltzmann equation (\(p=nk_bT\))
- \(S\) = surface area of a spherical molecule = \(\pi d^2\)
- \(d\) = molecular diameter or collision diameter

## 4.1.2.7. What are the limitations of the Continuum Model?¶

The Continuum Model is valid for Knudsen Numbers \(\ll 1\) i.e.

Knudsen Number | Meaning |
---|---|

\(Kn > 1\) | Molecular dynamics (e.g. gas in nano flow) |

\(0.1 < Kn < 1\) | Transition/continuum mechanics |

\(Kn \sim 0.01\) | Continuum (NS) + slip boundary |

\(Kn \ll 1\) | Continuum (NS) |

\(Kn \to 0\) | Continuum (Euler) |

## 4.1.2.8. What does the Continuum Model not permit?¶

Mass-energy conversion e.g.

- Nuclear reactions
- Relativity