Asymptotic solution for the Darcy–Brinkman–Boussinesq flow in a pipe with helicoidal shape

We study the fluid flow and heat transfer in a helical pipe filled with a sparsely packed porous medium. Motivated by the engineering applications, pipe's thickness and the distance between two coils of the helix have the same small order of magnitude, whereas the fluid inside the pipe is assumed to be cooled (or heated) by the exterior medium. After writing the dimensionless Darcy–Brinkman–Boussinesq system in curvilinear coordinates, we employ the multi-scale expansion technique to formally derive the effective model valid for small Brinkman–Darcy number. The obtained asymptotic solution is given in the explicit form which is important with regards to numerical simulations. Comparison with our previous results on the straight-pipe flow is also provided.