I would like some guidance / clarification on my workings for this.
Here is my attempt at expanding the following,
$$\underline u \cdot [A\nabla^4 \underline u] = \underline u \cdot [A (\nabla \cdot \nabla)(\nabla \cdot \nabla \underline u) ]$$
Let, $\hat{\underline u} = \nabla \underline u $ and working in component form where $\underline u = (u,v)$ and ignoring the $v$ component for now. $A$ is a constant, so can be passed through the $\nabla$'s
Then, $$u \cdot [A (\nabla \cdot \nabla)(\nabla \cdot \nabla u) = u \nabla \cdot [A \nabla(\nabla \cdot \hat{u}) ] $$ $$ = \nabla \cdot [uA\nabla^2 \hat{u}] - A \nabla^2 \hat{u} \cdot \nabla u $$ by the dot product rule. Using the identity, $\nabla^2(\phi\psi) = \phi\nabla^2\psi+2(\nabla \phi)\cdot(\nabla \psi)+(\nabla^2\phi)\psi$ where $\phi = uA$ and $\psi = \hat{u}$.
This gives,
$$ u \cdot [A \nabla^4 u] = \nabla \cdot [\nabla^2(uA\nabla u) - (\nabla^2 uA)\nabla u] - 2A(\nabla^2 u)^2 - A\nabla^2 (\nabla u )^2 $$
Does anyone have any thoughts?