What is the physical significance or difference between Convective derivative : $\vec{v} \cdot \nabla $ and the Divergence of velocity $\nabla \cdot \vec{v}$?
I have understood the convective derivative as the measure of change of a function along the direction of velocity. i.e. simply like the dot product. For example, the first component $u \frac{\partial{\rho}}{\partial{x}}$ of the convective derivative applied to density. I think of i as the projection of the change of density along x on the x-component of velocity. Is it right to think so? If there is a better reasoning please let me know.
Now coming to the divergence of velocity, how is this different from the convective derivative? I know that the dot product is commutative. Then how does this have a different meaning? i.e. Why does it take the velocity components into the derivative?
Does the Del operator have a different meaning if the order of the dot product is changed?
I come from a Mechanical engineering background with an interest in fluid dynamics, hence my interest in understanding the physical meanings. I also appreciate mathematical explanations. All perspectives are welcome.
The convective derivative, $\vec{v} \cdot \nabla$, is a differential operator; it acts on whatever you write to its right, to produce $$ (\vec{v} \cdot \nabla) f = \vec{v} \cdot (\nabla f) = v_1 \frac{\partial f}{\partial x_1} + v_2 \frac{\partial f}{\partial x_2} + v_3 \frac{\partial f}{\partial x_3} . $$ But the divergence of the velocity, $\nabla \cdot \vec{v}$, is just a scalar field (i.e., a real-valued function): $$ \nabla \cdot \vec{v} = \frac{\partial v_1}{\partial x_1} + \frac{\partial v_2}{\partial x_2} + \frac{\partial v_3}{\partial x_3} . $$ So they are completely different things. The point is that the operator $\nabla$ acts on what stands immediately to the right of it, so it makes a big difference whether you write the vector field $\vec{v}$ before or after the $\nabla$.