# Interpretation for a condition in fluid dynamics

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 $$-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}% \$$ should be positive definite. Does this condition has a physical interpretation in fluid dynamics or tensors?

Any comment or reference will be highly appreciate!

• In the fluids literature, the symmetric part of the velocity gradient is usually called the "rate of deformation tensor" or the "velocity strain" tensor. Your $A$ is the negative of that. Nov 7, 2014 at 8:59
• I would be highly suspicious of such a model. Biological fluids are usually close to incompressible. In that case the matrix A is traceless and therefore definitely NOT positive definite! Nov 7, 2014 at 11:18

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 $\textbf{v}$, $$\mathcal{L}_\textbf{u} (\textbf{v} \cdot \textbf{v}) = 2 \textbf{v} \cdot \mathcal{L}_\textbf{u} \textbf{v} - 2 \textbf{v} A \textbf{v} \,,$$ where the matrix $A$ is the one you have defined. A flow $\textbf{u}$ with positive definite $(-A)$ has the property that $$\mathcal{L}_\textbf{u} (| \textbf{v} |^2 ) \geq 2 \textbf{v} \cdot \mathcal{L}_\textbf{u} \textbf{v}$$ for any vector field $\textbf{v}$.