I'm considering an incompressible Newtonian fluid with uniform density and try to figure out what's meant by the term advection.
Let $\Omega_0\subseteq\mathbb R^d$ be an (infinitesimal small) bounded domain occupied by the fluid at time $0$. If I'm not terribly wrong, advection is all about measuring the change of a thing injected into the fluid. Let's denote the (unitless) amount of that thing injected at $x\in\Omega_0$ at time $t\ge 0$ by $c(x,t)$.
If I'm not wrong, we want to find an expression for $$\frac\partial{\partial t}c(x,t)$$ in terms of the velocity field $u:\Omega_0\times[0,\infty)\to\mathbb R^d$ (and the pressure $p:\Omega_0\times[0,\infty)\to\mathbb R$?) of the fluid.
Is that the case or am I wrong? And whatever the case is: How can we derive the advection equation?
I don't understand the example of "ink dumped into a river". What's the thing in that case? Ink itself is obviously not a scalar quantity. And, even worse, should the process of pouring ink into water not effect the motion of the water? Is can't see that this obvious fact is taken into account.
Or is everything less complicated an the thing is something which doesn't effect the motion of the fluid?
I could imagine that advection just describes the change of density of massless infinitesimal tiny thingies injected into the fluid, but I'm unsure whether I'm wrong or not.