  Channel flow is usually described by Darcy law with the Poiseuille flow
profile. However, for incompressible channel flow there is a critical state,
characterized by a critical Reynolds number $Re_c$ and a critical wavevector
mc, beyond which the channel flow becomes unstable in the linear regime. By
obtaining the analytical eigenfunctions of the linearized, incompressible,
three dimensional (3D) Navier-Stokes (NS) equation in the channel geometry,
i.e., the hydrodynamic modes (HMs), we reduce the full NS equation to a system
of coupled autonomous ordinary differential equations (ODEs) by expanding the
velocity in terms of the HMs; time becomes the only independent variable. The
nonlinear term of the NS equation is converted to a third-rank tensor that
couples pairs of the expansion coefficients to effect the time variation on the
third. In the linear regime, the value of $Re_c$ is obtained to five
significant digit accuracy when compared to the Orszag result. We numerically
time evolve the autonomous ODEs at $Re>Re_c$ with a finite set of thermally
excited initial HMs to find a fluctuating equilibrium state with a reduced net
flow rate, accompanied by vortices. Through the perspective of force balance,
interesting features are uncovered in the counter-flow profiles at $Re>Re_c$.
