2.14 Diffusion

The momentum equation for a homogeneous, incompressible, Newtonian fluid is presented in Eq. (2.47 ). In the case of zero body force, eqn, and eqn constant, Eq. (2.47 ) becomes

@u @t-+ r (uu) r ( ru) = rp: \relax \special {t4ht=
This is a typical transport equation characterised by:
  • the local rate of change eqn, described in Sec. 2.5 ;
  • advection of eqn by eqn, described in Sec. 2.8 ;
  • diffusion of eqn by eqn;
  • a “source” due to eqn, described in Sec. 2.12 .

Note that, since eqn constant, eqn, where eqn denotes the Laplace operator.

PICT\relax \special {t4ht=

The eqn term is a special form of divergence in which the flux includes the surface normal gradient denoted by the operator eqn where eqn.

The term models diffusion across the surface of the fluid element. Diffusion generally represents the transport of a fluid property — here, momentum — due to fluctuating motions that are not captured by the bulk motion that is represented by the continuum velocity eqn.

PICT\relax \special {t4ht=

Fluctuations include any random motion of particle constituents of matter, e.g. molecules, and turbulent structures. Through these motions, particles can pass across a surface boundary, transporting property eqn through a gradient of eqn (above, left).

Particles carrying higher eqn move into regions of particles with lower eqn and vice versa. Through particle collisions, high values of eqn tend to reduce and low values increase (right).


“Laplacian”11 describes a term of the form eqn where eqn is a diffusivity coefficient.12 A Laplacian term is conservative since all variables are to the right of a divergence, as described in Sec. 2.9 . It is also bounded since it tends to decrease high values and increase low values as shown above.

The Laplacian represents a flux due to eqn across the surface, per unit volume, as eqn

 1 Z r ( r ) = lim ---- (dS rn ): V !0 V S \relax \special {t4ht=

11After Pierre-Simon Laplace, Théorie des attractions des sphéroïdes et de la figure des planètes, 1785 but described in 1761 both by Leonhard Euler and Jean-Baptiste le Rond d’Alembert.
12More precisely, “Laplacian” denotes a term without eqn, i.e. eqn, represented as eqn but we include eqn here because it is generally needed.

Notes on CFD: General Principles - 2.14 Diffusion