A finite-element method for the weakly compressible parabolized steady 3D Navier-Stokes equations in a channel with a permeable wall

There are numerous scientific and technical applications that require the solution of the steady 3D Navier-Stokes equations in slender channels or ducts; often, this is carried out using commercially available software which is unable to make use of the fact that the equations can be parabolized to give a formulation that, in terms of CPU time and random access memory (RAM) usage, is orders of magnitude cheaper to compute. Here, we implement a velocity-vorticity formulation in a commercial finite-element solver to tackle the weakly compressible parabolized steady 3D Navier-Stokes equations in a channel with a permeable wall - a situation that occurs in polymer electrolyte fuel cells. Benchmarks results, for which the compressibility is present via a fluid density that is a function of channel length, indicate at least a 30-fold saving in CPU time and a 70-fold saving in RAM usage, as compared to full 3D computations, without any discernible loss in accuracy.