<s>[INST] Divergence of vector-tensor outer product multiplication I have a material derivative of a tensor quantity $\mathbf{S}$.
$$
\frac{\partial \mathbf{S}}{\partial t} + \vec{v} \cdot \nabla \mathbf{S}
$$
I would like to know if the term $\vec{v} \cdot \nabla \mathbf{S}$ can be rewritten as:
$$
\nabla \cdot \left(\vec{v} \otimes \mathbf{S}\right) - \left(\nabla \cdot \vec{v}\right) \mathbf{S} = \vec{v} \cdot \nabla \mathbf{S}
$$
So that the final expression would be:
$$
\frac{\partial \mathbf{S}}{\partial t} + \nabla \cdot \left(\vec{v}\otimes \mathbf{S}\right) - \left(\nabla  \cdot \vec{v} \right) \mathbf{S}
$$
I know this works for scalar quantities, but I do not know if it holds for tensor quantities.
Can anyone help me with this?
Best Regards
 [/INST] Note that following relations hold:
Scalar $\rightarrow$ Vector:
$$
\operatorname{grad} \phi(\mathbf{x}):=\frac{\mathrm{d} \phi(\mathbf{x})}{\mathrm{d} \mathbf{x}}=: \mathbf{w}(\mathbf{x})
$$
Vector $\rightarrow$ Matrix:
$$
\operatorname{grad} \mathbf{v}(\mathbf{x}):=\frac{\mathrm{d} \mathbf{v}(\mathbf{x})}{\mathrm{d} \mathbf{x}}=: \mathbf{S}(\mathbf{x})
$$
Matrix $\rightarrow$ Tensor (3):
$$
\operatorname{grad} \mathbf{T}(\mathbf{x}):=\frac{\mathrm{d} \mathbf{T}(\mathbf{x})}{\mathrm{d} \mathbf{x}}=: \mathrm{U}^3(\mathbf{x})
$$
Tranpose:
$$
\begin{aligned}
\mathbf{I} \otimes \mathbf{I} & =\left(\mathbf{e}_i \otimes \mathbf{e}_i\right) \otimes\left(\mathbf{e}_j \otimes \mathbf{e}_j\right) \\
(\mathbf{I} \otimes \mathbf{I})^{\stackrel{23}{\text{T}}} & =\mathbf{e}_i \otimes \mathbf{e}_j \otimes \mathbf{e}_i \otimes \mathbf{e}_j \\
(\mathbf{I} \otimes \mathbf{I})^{\stackrel{24}{\text{T}}} & =\mathbf{e}_i \otimes \mathbf{e}_j \otimes \mathbf{e}_j \otimes \mathbf{e}_i
\end{aligned}
$$
Gradient:
$$
\begin{aligned}
\operatorname{grad}(\phi \psi) & =\phi \operatorname{grad} \psi+\psi \operatorname{grad} \phi \\
\operatorname{grad}(\phi \mathbf{v}) & =\mathbf{v} \otimes \operatorname{grad} \phi+\phi \operatorname{grad} \mathbf{v} \\
\operatorname{grad}(\phi \mathbf{T}) & =\mathbf{T} \otimes \operatorname{grad} \phi+\phi \operatorname{grad} \mathbf{T} \\
\operatorname{grad}(\mathbf{u} \cdot \mathbf{v}) & =(\operatorname{grad} \mathbf{u})^{\text{T}} \mathbf{v}+(\operatorname{grad} \mathbf{v})^{\text{T}} \mathbf{u} \\
\operatorname{grad}(\mathbf{u} \times \mathbf{v}) & =\mathbf{u} \times \operatorname{grad} \mathbf{v}+\operatorname{grad} \mathbf{u} \times \mathbf{v} \\
\operatorname{grad}(\mathbf{a} \otimes \mathbf{b}) & =\left[\operatorname{grad} \mathbf{a} \otimes \mathbf{b}+\mathbf{a} \otimes(\operatorname{grad} \mathbf{b})^{\text{T}}\right]^{\stackrel{23}{\text{T}}} \\
\operatorname{grad}(\mathbf{T} \mathbf{v}) & =(\operatorname{grad} \mathbf{T})^{\stackrel{23}{\text{T}}} \mathbf{v}+\mathbf{T} \operatorname{grad} \mathbf{v} \\
\operatorname{grad}(\mathbf{T} \cdot \mathbf{S}) & =(\operatorname{grad} \mathbf{T})^{\stackrel{13}{\text{T}}} \mathbf{S}^{\text{T}}+(\operatorname{grad} \mathbf{S})^{\stackrel{13}{\text{T}}} \mathbf{T}^{\text{T}}
\end{aligned}
$$
Divergence:
$$
\begin{aligned}
\operatorname{div}(\mathbf{u} \otimes \mathbf{v})&=\mathbf{u} \operatorname{div} \mathbf{v}+(\operatorname{grad} \mathbf{u}) \mathbf{v} \\
\operatorname{div}(\phi \mathbf{v})&=\mathbf{v} \cdot \operatorname{grad} \phi+\phi \operatorname{div} \mathbf{v} \\
\operatorname{div}(\mathbf{T} \mathbf{v})&=\left(\operatorname{div} \mathbf{T}^{\text{T}}\right) \cdot \mathbf{v}+\mathbf{T}^{\text{T}} \cdot \operatorname{grad} \mathbf{v} \\
\operatorname{div}(\operatorname{grad} \mathbf{v})^{\text{T}}&=\operatorname{grad} \operatorname{div} \mathbf{v} \\
\operatorname{div}(\mathbf{u} \times \mathbf{v})&=(\operatorname{grad} \mathbf{u} \times \mathbf{v}) \cdot \mathbf{I}-(\operatorname{grad} \mathbf{v} \times \mathbf{u}) \cdot \mathbf{I} \\
&=\mathbf{v} \cdot \operatorname{rot} \mathbf{u}-\mathbf{u} \cdot \operatorname{rot} \mathbf{v} \\
\operatorname{div}(\phi \mathbf{T})&=\mathbf{T} \operatorname{grad} \phi+\phi \operatorname{div} \mathbf{T} \\
\operatorname{div}(\mathbf{T} \mathbf{S})&=(\operatorname{grad} \mathbf{T}) \mathbf{S}+\mathbf{T} \operatorname{div} \mathbf{S} \\
\operatorname{div}(\mathbf{v} \times \mathbf{T})&=\mathbf{v} \times \operatorname{div} \mathbf{T}+\operatorname{grad} \mathbf{v} \times \mathbf{T} \\
\operatorname{div}(\mathbf{v} \otimes \mathbf{T})&=\mathbf{v} \otimes \operatorname{div} \mathbf{T}+(\operatorname{grad} \mathbf{v}) \mathbf{T}^{\text{T}} \\
\operatorname{div}(\operatorname{grad} \mathbf{v})&=\mathbf{0} \\
\operatorname{div}\left(\operatorname{grad} \mathbf{v} \pm(\operatorname{grad} \mathbf{v})^{\text{T}}\right)&=\operatorname{div} \operatorname{grad} \mathbf{v} \pm \operatorname{grad} \operatorname{div} \mathbf{v} \\
\operatorname{div} \operatorname{rot} \mathbf{v}&=0 \\
\end{aligned}
$$
Rotation:
$$
\begin{aligned}
\operatorname{rot} \operatorname{rot} \mathbf{v}&=\operatorname{grad} \operatorname{div} \mathbf{v}-\operatorname{div} \operatorname{grad} \mathbf{v} \\
\operatorname{rot} \operatorname{grad} \phi&=\mathbf{0} \\
\operatorname{rot} \operatorname{grad} \mathbf{v}&=\mathbf{0} \\
\operatorname{rot}(\operatorname{grad} \mathbf{v})^{\text{T}}&=\operatorname{grad} \operatorname{rot} \mathbf{v} \\
\operatorname{rot}(\phi \mathbf{v})&=\phi \operatorname{rot} \mathbf{v}+\operatorname{grad} \phi \times \mathbf{v}-\mathbf{v} \otimes \mathbf{u})
\end{aligned}
$$
</s>