## 2.3Flow through a surface

The concept of ﬂow through a surface appears in many areas of CFD, including ﬂuid dynamics equations, numerical methods, boundary conditions and general ﬂow calculations. When we talk about something that travels through a surface, the term ﬂux is generally used.2

To quantify the ﬂux of some property, we multiply the area of surface by the property at the surface. If the property is a vector, we take the component normal to the surface. For example, the ﬂux associated with velocity through a surface segment of area would be .

As shown in the ﬁgure, can be calculated from by the inner product with normal vector of unit length, expressed as

 (2.5)
The ﬂux associated with is
 (2.6)
It is a good habit to write ﬁrst since the order matters with a tensor, e.g. stress , introduced in Sec. 2.6 , i.e.

### Inner product of two vectors

The normal component of velocity is described in Eq. (2.5 ) by the inner, or “dot”, product of and . It is calculated for vectors and as shown in the ﬁgure below, where denotes the magnitude of the vector and is the internal angle between the two vectors.

The inner product of two vectors is a scalar invariant, since the magnitudes and angle are the same irrespective of the co-ordinate system used. It is calculated from vector components as follows:

The inner product of two vectors is commutative, i.e. . It is distributive, i.e. with an additional vector ,

Scalar multiplication and inner products are associative, i.e.

An inner product of a vector with itself is simply the square of the vector magnitude, i.e.

 (2.7)

2Joseph Fourier used the term ﬂuxion in relation to ﬂow of heat in Théorie analytique de la chaleur, 1822; James Clerk Maxwell used the term ﬂux in A Treatise on Electricity and Magnetism, 1873.

Notes on CFD: General Principles - 2.3 Flow through a surface