What Kind of Tensor Product is v=S:∇I?

[jure]

Hi,
I'm reading a book about fluid dynamics and I found some strange product between tensors. It's written like this: v=S:∇I , where S and I are matrices and v is a vector. Symbol : usually denotes Frobenius inner product. In this case we have a product of a matrix with a tensor of rank 3 and the result is vector - so it's not classical Frobenius product.
I would like to know what kind of product this is (especially useful would be index notation).
Thanks for help.

 

[pasmith]

Hi,
I'm reading a book about fluid dynamics and I found some strange product between tensors. It's written like this: v=S:∇I , where S and I are matrices and v is a vector. Symbol : usually denotes Frobenius inner product. In this case we have a product of a matrix with a tensor of rank 3 and the result is vector - so it's not classical Frobenius product.
I would like to know what kind of product this is (especially useful would be index notation).
Thanks for help.

I would guess a double contraction, which would turn a rank 5 tensor into a vector:$$v_i = S_{ijk} \frac{\partial I_k}{\partial x_j}$$

 

[George Jones]

I would guess a double contraction, which would turn a rank 5 tensor into a vector:$$v_i = S_{ijk} \frac{\partial I_k}{\partial x_j}$$

I initially guessed something like this too, but then thought that

S and I are matrices and v is a vector

might indicate that S only has two indices.

 

[George Jones]

Maybe something like $\left( S \cdot \nabla \right) \cdot I$ with component form

$$v_k = S_{ij} \frac{\partial I_{ik}}{\partial x_j}?$$

 

[jure]

Maybe something like $\left( S \cdot \nabla \right) \cdot I$ with component form

$$v_k = S_{ij} \frac{\partial I_{ik}}{\partial x_j}?$$

Yes, something like this. I and S only have 2 indices. How do you know over which indices to sum? Why wouldn't you sum over i and k instead of i and j?