# Tensor gradient and scalar product

1. Jun 5, 2010

### 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 .

Last edited: Jun 5, 2010