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Symmetry Relations in Viscoplastic Drag Laws

The following note shows that the symmetry of various resistance formulae, often based on Lorentz reciprocity for linearly viscous fluids, applies to a wide class of non-linear viscoplastic fluids. This follows from Edelen's non-linear generalization of the Onsager relation for the special case of \emph{strongly dissipative} rheology, where constitutive equations are derivable from his dissipation potential. For flow domains with strong dissipation in the interior and on a portion of the boundary this implies strong dissipation on the remaining portion of the boundary, with strongly dissipative traction-velocity response given by a dissipation potential. This leads to a non-linear generalization of Stokes resistance formulae for a wide class of viscoplastic fluid problems. We consider the application to non-linear Darcy flow and to the effective slip for viscoplastic flow over textured surfaces.