Double inner product?

1. Feb 2, 2010

Smed

Hi, I'm having trouble understanding how to perform the following calculation:

$$u=(u,v,w)$$

$$(\nabla u + (\nabla u)^T) : \nabla u$$

I get the following by doing the dot product of the first term and then adding the dot product of the second term, but I'm pretty sure it isn't correct.

$$2\left(\frac{\partial u}{\partial x}\right)^2 + 2\left(\frac{\partial v}{\partial y}\right)^2 + 2\left(\frac{\partial w}{\partial z}\right)^2$$

Could someone please shed some light on how the double inner product should work?
Thanks

2. Feb 2, 2010

HallsofIvy

Your notation here doesn't make sense to me. If you are using "u" to represent a vector, don't use the same "u" to represent one of its components. If you are writing $\nabla u$ as, say, a row vector, then [itex[(\nabl u)^T[/itex] would be a column vector and you cannot add them. In general, a vector and its transpose are in different vector spaces and cannot be added. Finally, I don't know what ":" means. Was that supposed to be $\cdot$?

3. Feb 5, 2010

torquil

I think it is related to the definition in section 1.3.2 found here:
http://www.foamcfd.org/Nabla/guides/ProgrammersGuidese3.html

It is for a pair of rank 2 tensors, and is denoted by a :

The \nabla u's used in the original post are interpreted as second rank tensors, and the double inner product is applied between the terms on each side of the :

Haven't got time to check the calculation myself.

Torquil