# Computing material derivated of tensor quantity

I would like to compute the material derivated of a tensor quantity, in the context of the finite volume method (FVM):

The equation is:

$$\frac{\mathrm{d} \textbf{T}}{\mathrm{d} t} = \frac{\partial \textbf{T}}{\partial t} + \vec{v} \cdot \nabla \textbf{T} = \underbrace{\frac{\partial \textbf{T}}{\partial t} + \nabla \cdot \left(\textbf{T} \vec{v}\right) - \textbf{T} \left(\nabla \cdot \vec{v}\right)}_{is \: this \: true \: for \: tensors?}$$

I am thinking of solving this by solving each component entry in the tensor. Is this the correct procedure? In python, I used to solve this with solve_ivp but that considers a moving reference frame and in the FVM the frame is stationary.

Any tricks/things to be aware of when solving for a tensor quantity?

No, the right-hand side of your equation is not valid. I will assume that $$\mathbf{T}$$ is a second-order tensor. If that's the case we need that the time derivative is second-order. The factor $$\nabla \mathbf{T}$$ gives you a third-order quantity that is later turned into a second-order one via projection over $$v$$. Now, the factor $$\mathbf{T} v$$ is a vector and is turned into a scalar after applying the divergence.
• I think whether it's valid or not depends on what the notation is supposed to mean. If $\mathbf{T}v$ is the outer product $\mathbf{T}\otimes v$, which has rank equal to $\text{rank}(\mathbf{T}) + \text{rank}(v)$, then I think it's correct, but your answer assumes that it means the contraction of $\mathbf{T}$ with $v$ along some index. Either way I agree that writing it in index notation will probably help. May 27, 2022 at 22:50
• $\nabla \cdot (s \vec {v}) = s (\nabla \cdot \vec{v}) + \vec{v} \cdot \nabla s$ $\Leftrightarrow \nabla \cdot (s \vec {v}) - s (\nabla \cdot \vec{v})= \vec{v} \cdot \nabla s$