2.8 Flow in a volume
We presented conservation of momentum, Eq. (2.19 ), in terms of the material derivative . To solve the equation in this form, would require a method that tracks particles of ﬂuid as they move around.
Generally it is easier to solve the equation by ﬁxing the volume of space and solving for the ﬂuid motion through it. To enable this, we replace the material derivative for using the following expression:
Conservation of momentum can then be written
Advection contains a divergence derivative so represents a ﬂux across a surface per unit volume. The example in Eq. (2.27 ) represents the ﬂux of momentum, where the advected property is itself (or , depending whether is associated with bulk ﬂow or the advected property).
Outer product of two vectors
The advection term in Eq. (2.27) includes a product of two vectors . This is the outer product of two vectors7 which produces a tensor by
Inner product of two tensors
The inner product of two tensors, e.g. produces a tensor where the components are (replacing with )
The identity tensor is the tensor equivalent of unity (one) such that for any tensor ,