I am doing a simulation on OpenFOAM where I am looking at a porous media flow though a region. I have no problem with that but what I am wondering is, if there is not obstacle in the region does the governing equation have an actual solution or would I still need to use some numerical method to solve it ? If so what simplifications can I make sense there is no obstacles in the region (if there is any) ? The governing equation is an extension of the Navier-Stokes equations that adds a sink term:
$$ \frac{\partial}{\partial t} (\gamma\rho u_i) + u_j \frac{\partial}{\partial x_j} (\rho u_i) = - \frac{\partial P}{\partial x_i} + \mu \frac{\partial \tau_{ij}}{\partial x_j} + S_i $$
Where the $S_i$ term is defined as: $$ S_i = - (\mu D + \frac{1}{2} \rho |u_{jj}| F)u_i $$
These equations are found in the following paper and I am using the same solver, and sort of configuration for what I am looking at as well.
The variables in the above equation are given as: $$ \begin{eqnarray} &\gamma \in [0,1] \text{ is the porosity of the object} \\ &\rho \text{ is the density} \\ & u \text{ is the flow velocity} \\ & \tau \text{ is the Cauchy-Stress Tensor} \\ & P \text{ is the Pressure} \\ & S_i \text{ is the sink term} \\ & D,F \text{ are Darcy-Forchheimer constants} \\ & \mu \text{ is inversely related to the Reynolds Number} \end{eqnarray} $$