Tensor gradient and scalar product

[zyroph]

Hi all,

I need to evaluate the following equation :

$$\mathbf{n} \cdot [\mathbf{\sigma} + \mathbf{a} \nabla\mathbf{\sigma}]\cdot\mathbf{n}$$

where $$\mathbf{n}$$ is the normal vector, $$\mathbf{a}$$ a vector, and $$\sigma$$ the stress tensor such that :

$$\mathbf{\sigma} \cdot \mathbf{n} = -p\cdot\mathbf{n} + \mu [\nabla \mathbf{u} + (\nabla\mathbf{u})^T]\cdot \mathbf{n}$$

Actually, the first term (in the first equation) is not an issue , since it can be found in any serious book But I'm getting lost with the second one.

I work much more in numerics than in maths, and my knowledge on the topic is very limited so I will be very grateful for any help, i.e.

$$\mathbf{n} \cdot \mathbf{a} \nabla\mathbf{\sigma} \cdot\mathbf{n}$$

Any clue, simplification, explanation would be welcome .

Thanks in advance

 

[fresh_42]

If you mean "in coordinates" by "evaluate", then you should gather the definitions of all your vectors and the operations and calculate. If it means algebraically changing the expression, then you will find useful formulas on Wikipedia: https://en.wikipedia.org/wiki/Vector_calculus_identities