## 2.6Forces at a surface

The next law to deﬁne is that of conservation of momentum, i.e. Newton’s second law of motion () for ﬂuids. It involves forces within the ﬂuid so requires a description of forces at a surface bounding a volume .

The force is in the direction of the traction vector with magnitude of (traction surface area) — compare with Eq. (2.3 )

 (2.16)
The tractions at a surface bounding some volume of ﬂuid (or any continuum, e.g. solid) depend on the orientation of the surface. Consequently, we cannot deﬁne forces within a ﬂuid as traction vectors at points within the ﬂuid.

Instead, we require 3 traction vectors , and , deﬁned in planes perpendicular to one another, i.e. , and . This results in the stress tensor with 9 components, consisting of 3 traction vectors, each containing 3 components.

The traction can be calculated at a surface with any orientation by taking the inner product of the unit normal vector and the stress tensor such that

 (2.17)

### Tensors

We have deﬁned the stress tensor3, an entity with 9 component values, corresponding to our , and axes (or more speciﬁcally base vectors of unit length aligned with our , and axes).

In fact, the term “tensor” describes any entity with multiple component values corresponding to the dimensions of space — here 3. A tensor has rank , such that the number of component values for 3D space = .

In this book we use the term “tensor” to mean “tensor of rank 2” unless otherwise noted. A vector is a tensor of rank 1 and a scalar is rank 0.

The inner product of a vector and tensor produces a vector whose 3 components are evaluated as follows:

 (2.18)
This inner product tensor is commutative only if is symmetric since — see Eq. (2.24 ) for the transpose .
3More precisely the “Cauchy stress tensor”, introduced by Augustin-Louis Cauchy in De la pression ou tension dans un corps solide, 1827.

Notes on CFD: General Principles - 2.6 Forces at a surface