TITLE: Interpretation for a condition in fluid dynamics
QUESTION [5 upvotes]: I have been working with some mathematical models in biology and fluid mechanics. My problem is about
the interpretation of a condition that I found for a vector
representing the velocity of a fluid. The exact question is the next:
Let $\mathbf{u=(}u_{1},u_{2},u_{3})$ be a vector field representing the
velocity of a fluid. After making some accounts with models using partial
differential equations, I found that the matrix
\begin{equation}
-A:=\frac{1}{2}\left(  \frac{\partial u^{j}}{\partial x_{i}}+\frac{\partial
u^{i}}{\partial x_{j}}\right)  _{i,j=1,2,3}%
\
\end{equation}
should be positive definite. Does this condition has a physical interpretation
in fluid dynamics or tensors?
Any comment or reference will be highly appreciate!

REPLY [2 votes]: I don't know if this is the answer that you have been looking for, but let me offer a rather trivial observation.
Your tensor is the Lie derivative of the metric tensor with respect to the vector field $\textbf u$. Loosely speaking, the Lie derivative $\mathcal{L}_\textbf{u}$ has an interpretation as a derivative with respect to "dragging" a tensor along the flow defined by $\textbf{u}$. A simple manipulation shows that for any vector field $\textbf{v}$,
\begin{equation}
\mathcal{L}_\textbf{u} (\textbf{v} \cdot \textbf{v}) = 
2 \textbf{v} \cdot \mathcal{L}_\textbf{u} \textbf{v} - 2 \textbf{v} A \textbf{v} \,,
\end{equation}
where the matrix $A$ is the one you have defined. A flow $\textbf{u}$ with positive definite $(-A)$ has the property that
\begin{equation}
\mathcal{L}_\textbf{u} (| \textbf{v} |^2 ) \geq 
2 \textbf{v} \cdot \mathcal{L}_\textbf{u} \textbf{v}
\end{equation}
for any vector field $\textbf{v}$.
I hope this observation has some use to you.