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 )
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
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: