4.1.2. Continuum Model What is the Continuum Model?

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

  • Pressure = Average collision force per unit area. 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. What does the Continuum Model enable?

Enables the conservation laws:

  • Mass
  • Momentum
  • Energy Where can the Continuum Model be applied?

Can be applied to:

  • Finite volume element
  • Infinitesimal differential element What is the Knudsen number?

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


  • \(\lambda\) = mean free path
  • \(L\) = characteristic length (e.g. pipe diameter, boundary layer thickness, shockwave thickness) What is the Mean Free Path?

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


  • \(n\) = 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 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) What does the Continuum Model not permit?

Mass-energy conversion e.g.

  • Nuclear reactions
  • Relativity