# 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.

## 4.1.2.3. What does the Continuum Model enable?¶

Enables the conservation laws:

• Mass

• Momentum

• Energy

## 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