Normalization constraint for variational bounds on fluid permeability

A careful reexamination of the formulation of Prager’s original variational principle for viscous flow through porous media has uncovered a subtle error in the normalization constraint on the trial functions. Although a certain surface integral of the true pressure field over the internal surface area always vanishes for isotropic materials, the corresponding surface integral for a given trial pressure field does not necessarily vanish but has nevertheless been previously neglected in the normalization. When this error is corrected, the form of the variational estimate is actually simpler than before and furthermore the resulting bounds have been shown to improve when the constant trial functions are used in either the two-point or three-point bounds.