## 2.14Diﬀusion

The momentum equation for a homogeneous, incompressible, Newtonian ﬂuid is presented in Eq. (2.47 ). In the case of zero body force, , and constant, Eq. (2.47 ) becomes (2.49)
This is a typical transport equation characterised by:
• the local rate of change , described in Sec. 2.5 ;
• advection of by , described in Sec. 2.8 ;
• diﬀusion of by ;
• a “source” due to , described in Sec. 2.12 .

Note that, since constant, , where denotes the Laplace operator. The term is a special form of divergence in which the ﬂux includes the surface normal gradient denoted by the operator where .

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 . 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 through a gradient of (above, left).

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

### Laplacian

“Laplacian”11 describes a term of the form where 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 across the surface, per unit volume, as  (2.50)

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 , i.e. , represented as but we include here because it is generally needed.

Notes on CFD: General Principles - 2.14 Diﬀusion 