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

•Finite-element solution to full and parabolized 3D Navier–Stokes equations.•Implemented in velocity–vorticity formulation.•Parabolized approach requires only 1/30th of the CPU time.•Parabolized approach requires only 1/70th of the RAM usage.•Parabolized approach gives no discernible loss in accuracy.

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.