Complementary principles for Poiseuille flow in a porous pipe

The complementary variational principles for the steady Poiseuille flow of a viscous incompressible liquid in a long straight pipe of arbitrary cross section filled with a porous material are calculated and used to obtain the upper and lower bounds of the flux in selected pipe geometries. It is shown that for a given cross section, axial pressure gradient and permeability, the flux through the porous pipe is less than that through a nonporous pipe, and that for a given pressure gradient Poiseuille flow in the porous medium corresponds to maximum flux through the pipe. The construction of the complementary principles is performed through the selection of trial functions, and the calculation of upper and lower bound is illustrated. Applications of the bounds to the cases of elliptical and rectangular pipe cross sections are then presented.