Symmetry Relations in Viscoplastic Drag Laws

TL;DR

This paper demonstrates that symmetry relations in viscous resistance formulas extend to non-linear viscoplastic fluids using Edelen's dissipation potential, generalizing classical linear results to complex non-linear flows.

Contribution

It introduces a non-linear generalization of resistance formulas for viscoplastic fluids based on Edelen's dissipation potential, extending symmetry principles beyond linear viscous fluids.

Findings

Symmetry relations apply to a wide class of non-linear viscoplastic flows.

Derived non-linear resistance formulas for viscoplastic fluids.

Applications include non-linear Darcy flow and slip over textured surfaces.

Abstract

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.