I have a two-dimensional flow so $\textbf{u} = (u(x, y, t), v(x, y, t), 0)$ and thus the vorticity of the flow $\pmb{\omega} = (0, 0, w(x, y, t))$
I am interested in the following two expressions:
$(\textbf{u} \cdot \nabla) \pmb{\omega}$
$(\pmb{\omega} \cdot \nabla) \textbf{u}$
Now apparently, the first is not in general $0$, but the second is $0$.
I am struggling to see why. I am also just in general struggling to find meaning in these expressions. I don't understand what operation to do first, why it is written in brackets like that - I mean is the first, for example, equivalent to $\textbf{u} \cdot \nabla \pmb{\omega}$?
And most importantly, how do you compute these things and show that the second is $0$?
$\mathbf{u}\cdot\nabla\omega$ is confusing -- which component of the rank $2$-tensor $\nabla\omega$ are you contracting with $u$? You could mean one of $u_i\partial_i\omega$ or $u_i\nabla\omega_i$ and they are not equal.
(1) is in general nonzero --- its $z$-component is $$u(x,y,t)\frac{\partial}{\partial x}w(x,y,t)-v(x,y,t)\frac{\partial}{\partial y}w(x,y,t).$$
(2) is 0 because $\omega\cdot\nabla=w(x,y,t)\frac{\partial}{\partial z}$ and we know $u$ has no $z$-dependence.