I'm learning CFD and I can't really understand the Advection Equation and material derivative.
Why material derivative equals zero?
Given \begin{equation} \begin{aligned} \nonumber\frac{\partial{q(t,\vec{x})}}{\partial{t}} &= \frac{\partial{q}}{\partial{t}} + \nabla q \cdot\frac{d\vec{x}}{dt}\\ &= \frac{\partial{q}}{\partial{t}} + \nabla q \cdot\vec{u}\\ &\equiv \frac{Dq}{Dt} \end{aligned} \end{equation}
and the Advection Equation
$$ \frac{Dq}{Dt} = 0 $$
At first I thought $\frac{\partial{q(t,\vec{x})}}{\partial{t}}$ is the change of $q$ at a fixed position $\vec{x}$ at time $t$, but if we set the material derivative to zero, the field would be static.
Then I realized that the value we want to know is $\frac{\partial{q}}{\partial{t}}$, which represents the change of $q$ at a fixed position $\vec{x}$ at time $t$. And $\nabla q$ is a known value from a function of $q$ with respect to position $\vec{x}$.
It is said that material derivative means the quantity is moving around but isn't changing in the Lagrangian viewpoint.
How to understand it?
Definition of material derivative: \begin{equation} \frac{Dq}{Dt} \equiv \frac{\partial{q}}{\partial{t}} + \nabla q \cdot \vec{u} \end{equation} Where $\vec{u}$ is the flow velocity ( for a fluid $\vec{u}=\frac{\vec{dx}}{dt}$ )
Start with continuity equation (with flux $ \ \ \vec{j}=(q\cdot\vec{u})\ \ $ for fluid ): \begin{equation} \begin{aligned} 0 &= \frac{\partial{q}}{\partial{t}} + \nabla \cdot (\vec{j})\\ &= \frac{\partial{q}}{\partial{t}} + \nabla \cdot (q\vec{u})\\ &= \frac{\partial{q}}{\partial{t}} + \nabla q \cdot\vec{u} + q \nabla\cdot\vec{u}\\ \end{aligned} \end{equation} for incompressible fluid $\nabla\cdot\vec{u}=0$ (so the last term disappears ) : \begin{equation} \frac{Dq}{Dt}= \frac{\partial{q}}{\partial{t}} + \nabla q \cdot \vec{u}=0 \end{equation}