Brinkman Channel 2D Example with BGK operator

Brinkman flow in a channel is a classical benchmark problem for validating the Navier--Stokes--Brinkman (NSB) solver, as it combines viscous diffusion and Darcy drag effects characteristic of porous media. The problem considers a steady laminar flow through a channel filled with a porous matrix, where the additional Darcy drag term significantly alters the velocity distribution. The governing one-dimensional Brinkman equation reads

$$\frac{\partial^2 u_x}{\partial y^2} = F_0 u_x, \label{nsb_eq_1}$$

where $u_x$ is the velocity in the streamwise ($x$) direction, and $F_0 = \tfrac{\mu_0}{k\mu}$ is the normalized Darcy coefficient, with $\mu$ the fluid viscosity, $\mu_0$ the effective viscosity, and $k$ the permeability.

The boundary conditions are imposed as a no-slip condition at the lower wall and a symmetric velocity condition at the upper wall,

$$u(y=0) = 0, \qquad u(y=h) = u_m. \label{nsb_bc}$$

The analytical solution of this boundary-value problem is

$$u(y) = k_1 e^{\sqrt{F_0} y} + k_2 e^{-\sqrt{F_0} y}, \label{nsb_solution}$$

with $k_1 = \tfrac{u_m}{e^{\sqrt{F_0} h} - e^{-\sqrt{F_0} h}}$ and $k_2 = -k_1$.

The analytical solution is written in func.lua. The values of $F_0$ is and other parameters are set in params.lua.