Does this equation have an analytical solution?

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: $$\begin{eqnarray} &\gamma \in [0,1] \text{ is the porosity of the object} \\ &\rho \text{ is the density} \\ & u \text{ is the flow velocity} \\ & \tau \text{ is the Cauchy-Stress Tensor} \\ & P \text{ is the Pressure} \\ & S_i \text{ is the sink term} \\ & D,F \text{ are Darcy-Forchheimer constants} \\ & \mu \text{ is inversely related to the Reynolds Number} \end{eqnarray}$$

• You would have to provide some more definitions here, like what is $\gamma,\rho,u$ ... And just my intuition, if this is basically Navier-Stokes but "more complicated", an analytical solution is highly improbable. – Sisyphus Mar 31 '18 at 20:24
• My apologies I will add that in now, essentially it is the characteristics of the liquid. – Robert Mar 31 '18 at 20:27
• For the geometries you have, no. Maybe if you have an squared section duct and a linear equation (assume small perturbations of variables w.r.t. the base flow) you could (improbable more than possible) reach an analytical solution. – HBR Mar 31 '18 at 20:35
• Well the reason why I am wondering is because I am looking at having the equation without an obstacle in the domain.... essentially I want to compare how the fluid flows with and without the obstacles in the domain, where I will use Open-FOAM to solve the system with an obstacle in the domain. Because of this I figured the geometry would be simplified. Also to the domain i'm interested in is strictly a rectangle domain. – Robert Mar 31 '18 at 20:42