tensor calculus - What does a colon mean in hydrodynamics equations?

In some hydrodynamics book I saw a notation like $e:e$ where $e$ is a matrix (shear stress tensor). This double dot product is in a scalar equation, so the result of this operation must be scalar. I found this article on Wiki, however it is about dyadics, which are 1-rank matrices (and the shear stress tensor, as far as I understand, is not generally a first rank matrix).

My guess is that $e:e=\sum_{ij}e_{ij}e_{ij}$, but I want to be sure.

Answer

The double inner product expands to be (for second rank tensors that you encounter in hydrodynamics):

$$\mathbf{a}\mathbf{:}\mathbf{b} = a_{ij}b_{ij} = a_{11}b_{11} + a_{12}b_{12} + ...$$

So it behaves just like you would expect a vector dot product to behave. You add up the product of all of the values with the same indexing.

You can do the same operation with a second rank and third rank tensor (which may come up in fluids or structures, but mostly is just for educational value here):

$$\mathbf{c}\mathbf{:}\mathbf{d} = c_{ij}d_{ijk} \neq \mathbf{d}\mathbf{:}\mathbf{c} = d_{ijk}c_{jk}$$

which would give you a vector as a result.