I'm working on a problem, which involves the concentration of magnetic particles which interact with a flow field and a magnetic field.
Derived by the classical convection diffusion equation, I want to solve the steady state concentration in the domain.
The PDE reads as follws: $$ 0 = \nabla \bullet (c \nabla \varphi - c a_{mag} \nabla H D - \nabla c D ( 1 + \frac{\nu c}{1- \nu c}))$$
Where c is the concentration of the particles;
$\varphi$ is a scalar field, describing the potential of the flow;
$a_{mag}$ is a constant with magnetic properties of the particles;
H is a scalar field, which is the magnetic field strength;
D is a constant describing the diffusion coefficient;
$\nu $ is also a constant, which is the maximum possible concentration of the particles due to their finite size.
The domain is rectangular with zero flux on two sides and inlet and outlet on the other two.
Is there a analytical solution $c(x,y)$ for this problem? My PDE class was some time ago so any general tips for solving this kind of problem is much appreciated.