Eur. Phys. J. Plus (2018) 133: 173
https://doi.org/10.1140/epjp/i2018-11999-6

Regular Article

Exact solutions of the Navier-Stokes equations generalized for flow in porous media

¹ Department of Civil Engineering, Monash University, Clayton, VIC, Melbourne, Australia
² Department of Mechanical and Aerospace Engineering, Monash University, Clayton, VIC, Melbourne, Australia

^* e-mail: edoardo.daly@monash.edu

Received: 4 December 2017
Accepted: 1 April 2018
Published online: 4 May 2018

Abstract

Flow of Newtonian fluids in porous media is often modelled using a generalized version of the full non-linear Navier-Stokes equations that include additional terms describing the resistance to flow due to the porous matrix. Because this formulation is becoming increasingly popular in numerical models, exact solutions are required as a benchmark of numerical codes. The contribution of this study is to provide a number of non-trivial exact solutions of the generalized form of the Navier-Stokes equations for parallel flow in porous media. Steady-state solutions are derived in the case of flows in a medium with constant permeability along the main direction of flow and a constant cross-stream velocity in the case of both linear and non-linear drag. Solutions are also presented for cases in which the permeability changes in the direction normal to the main flow. An unsteady solution for a flow with velocity driven by a time-periodic pressure gradient is also derived. These solutions form a basis for validating computational models across a wide range of Reynolds and Darcy numbers.