Sometimes I encounter PDE's with a term like this
$\nabla \cdot c(\nabla \textbf{v} + (\nabla \textbf{v})^T)$
An example are the Navier-Stokes equations. Oftentimes this can be further simplified to $c\nabla^2\textbf{v}$.
I understand gradients, divergence and the Laplacian. But somehow I cannot grasp the role of this transpose. Is this only an artifact from the derivation of the equation, or is there a deeper meaning that I am missing?
Best Answer
I think it helps to write out the Cartesian components of this expression: \begin{equation} c \sum_{k=1}^3 \partial_k \left(\partial_k v_i + \partial_i v_k\right) \end{equation} where $i$ and $k$ run over $\{1, 2, 3\}$, and where \begin{equation} \partial_i \equiv \frac{\partial }{\partial x^i} \end{equation} The meaning of the transpose is that the indices on the partial derivative and the vector are switched.