How can we describe the evolution of a density "injected" into an incompressible Newtonian fluid? Let $d\in\left\{2,3\right\}$ and $\Omega\subseteq\mathbb R^d$ be a bounded domain. The evolution up to time $T>0$ of an incompressible Newtonian fluid with uniform density $\rho_0$ and viscosity $\nu$ is given by the instationary Navier-Stokes equations $$\left\{\begin{matrix}\displaystyle\left(\frac\partial{\partial t}+\boldsymbol u\cdot\nabla\right)\boldsymbol u&=&\displaystyle\nu\Delta\boldsymbol u-\frac 1\rho_0\nabla p+\boldsymbol f&&\text{in }\Omega\times (0,T)\\\nabla\cdot \boldsymbol u&=&0&&\text{in }\Omega\times (0,T)\end{matrix}\right.\;,\tag 1$$ where $\boldsymbol u:\Omega\times [0,T]\to\mathbb R^d$ and $p:\Omega\times [0,T]\to\mathbb R$ are the velocity field and pressure, respectively, and $\boldsymbol f:\Omega\times (0,T)\to\mathbb R^d$ is the sum of all external forces.

Now, I've read that the evolution of a density $\rho:\Omega\times[0,T]\to\mathbb R$ injected into the fluid is described by a PDE of the form $$\left(\frac\partial{\partial t}+\boldsymbol u\cdot\nabla\right)\rho=\kappa\Delta\rho+s\;\;\;\text{in }\Omega\times (0,T)\tag 2$$ where $\kappa\in\mathbb R$ is somehow a diffusion rate and $s:\Omega\times[0,T]$ is an (or the sum of many?) external source(s).



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*How do we obtain $(2)$? I'm searching for a simple, but mathematically coherent derivation

*Obviously, $(2)$ looks very similar to $(1)$. Why did the term $-1/\rho_0\nabla p$ disappear? (Is $(2)$ a special case of a more general variant?)

*What do people mean when they say, that density is injected into a fluid?

 A: I am not a mathematician but an engineer with a specialty in fluid mechanics so forgive me if i skip some mathematical details :)...

What do people mean when they say, that density is injected into a
  fluid?

I think the problem here is the use of the word 'density' which in this context means 'species density' (aka concentration) as opposed to 'mass density'. The variable $\rho$ in equation (2) describes 'species density' as you say the incompressible fluid has uniform 'mass density' $\rho_0$. Then the injection of a mass of a certain species into the fluid at some point makes more sense (e.g. imagine injecting a tracer compound in the fluid).

How do we obtain (2)? I'm searching for a simple, but mathematically coherent derivation

In the same way the Navier-Stokes equations are a representation of the local conservation of momentum of an infinitessimal fluid volume, (2) is a representation of the local conservation of (species) mass. A species concentration can be transported by the velocity field (aka advection, i.e. the $\boldsymbol{u}\cdot\boldsymbol{\nabla}\rho$ term) and by a diffusion process (aka Fick's law, i.e. the $k\Delta\rho$ term). Furthermore, it may be produced or consumed ($s$ term) by e.g. a chemical reaction.
The derivation is fairly straightforward given that the accumulation of the 'species mass' in a infinitessimal fluid volume $dV$ is the result of the in- and outflux $\boldsymbol{f}$ of the 'species mass' at the closed surface $S$ of the volume $V$:
$$d_{t}\int_{V}\rho dV=-\int_{S}\boldsymbol{f}\cdot\boldsymbol{n}dS+\int_{V}sdV$$
Here, $\boldsymbol{n}$ indicates the unit normal to the surface and the convective flux $f$ has a advective and diffusive contribution, respectively:
$$\boldsymbol{f}=\rho\boldsymbol{u}+\boldsymbol{j} = \rho \boldsymbol{u} - k\boldsymbol{\nabla}\rho$$
Using Gauss' divergence theorem the integral equation is transformed:
$$\int_{V}\partial_{t}\rho dV=\int_{V}\left[-\boldsymbol{\nabla}\cdot\boldsymbol{f}+s\right]dV$$
and we obtain the differential form:
$$\partial_{t}\rho =-\boldsymbol{\nabla}\cdot\boldsymbol{f}+s$$
Substituting in the expression for the flux we retrieve the requested 'species density' equation:
$$\partial_{t}\rho+\boldsymbol{\nabla}\cdot\rho \boldsymbol{u}=k\Delta\rho+s$$
where we can subsequently simplify $\boldsymbol{\nabla}\cdot\rho \boldsymbol{u}=\rho\boldsymbol{\nabla}\cdot\boldsymbol{u}+\boldsymbol{u}\cdot\boldsymbol{\nabla}\rho=\boldsymbol{u}\cdot\boldsymbol{\nabla}\rho$ because of the continuity equation $\boldsymbol{\nabla}\cdot\boldsymbol{u}=0$.

Obviously, (2) looks very similar to (1). Why did the term $−1/ρ_0∇p$ disappear? (Is (2) a special case of a more general variant?)

All conservation equations look similar because the starting point, i.e. conservation of mass, momentum, energy, entropy, etc. are similar. For the Navier-Stokes equations, we are interested in the accumulation of momentum in an infinitessimal fluid volume:
$$d_{t}\int_{V}\rho\boldsymbol{u} dV=-\int_{S}\boldsymbol{\sigma}\cdot\boldsymbol{n}dS+\int_{V}fdV$$
I have changed notation slightly where $\sigma$ is now the in- and outflux through the surface $S$ and $f$ is a source or sink of momentum acting on the volume $V$ (aka known as a body force, see Newton's second law), but the structure of the equations are the same. The change of notation was to make the point that the convective flux $\sigma$ is different from the convective flux $f$ in that $\sigma$ is now a rank 2 tensor, however still contains a advective and diffusive contribution:
$$\boldsymbol{\sigma}=\rho\boldsymbol{u}\otimes\boldsymbol{u}+\boldsymbol{\tau}=\rho\boldsymbol{u}\otimes\boldsymbol{u}+p\boldsymbol{I}-\mu\left[\boldsymbol{\nabla}\boldsymbol{u}+\boldsymbol{\nabla}\boldsymbol{u}^{T}\right]$$
Here $\rho\boldsymbol{u}\otimes\boldsymbol{u}$ denotes the advective flux of momentum, whereas $\tau=p\boldsymbol{I}-\mu\left[\boldsymbol{\nabla}\boldsymbol{u}+\boldsymbol{\nabla}\boldsymbol{u}^{T}\right]$ represents the diffusive flux of momentum, most often refered to as the stress tensor. This stress tensor contains two types of stress; normal and shear stress. Normal stresses are caused by mainly the pressure whereas shear stresses are caused by viscosity due to velocity gradients across laminae of fluid (see Newtonian fluids).
The reason why the pressure term is in the Navier-Stokes equations is because a pressure gradient directly leads to changes in momentum, whereas it has no direct effect on the transport of 'species mass'. It instead would be reflected in the convective term of the 'species mass' equation.
