2.6 Forces at a surface

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

PICT\relax \special {t4ht=

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

df = tdS \relax \special {t4ht=
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.

PICT\relax \special {t4ht=

Instead, we require 3 traction vectors eqn, eqn and eqn, defined in planes perpendicular to one another, i.e. eqn, eqn and eqn. This results in the stress tensor eqn 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 eqn and the stress tensor eqn such that

 df = t dS = (n ☐☐☐)dS = dS ☐☐☐: \relax \special {t4ht=


We have defined the stress tensor3, an entity with 9 component values, corresponding to our eqn, eqn and eqn axes (or more specifically base vectors of unit length aligned with our eqn, eqn and eqn 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 eqn, such that the number of component values for 3D space = eqn.

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 eqn and tensor eqn produces a vector whose 3 components are evaluated as follows:

 0 1 axTxx + ayTyx + azTzx a T = B@ axTxy + ayTyy + azTzy CA axTxz + ayTyz + azTzz \relax \special {t4ht=
This inner product tensor is commutative only if eqn is symmetric since eqn — see Eq. (2.24 ) for the transpose eqn.
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