Darcy-Brinkman Flow Through a Bumpy Channel

The forced flow through a channel with bumpy walls which sandwich a porous medium is studied. The problem models micro-fluidics where, due to the small size of the channel width, the surface roughness of the walls is amplified. The Darcy-Brinkman equation is solved analytically through small perturbations on the ratio of bump amplitude to the half width of the channel. The first- order perturbation solutions give the three-dimensional velocity effects of the bumpiness and the second-order perturbation solutions give the increased resistance due to roughness. The problem depends heavily on the non-dimensional porous medium parameter $$k$$ which represents the importance of length scale to the square root of permeability. Our solutions reduce to the clear fluid limit when $$k$$ is zero and to the Darcy limit when $$k$$ approaches infinity.