Joe Goddard

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 nonlinear viscoplastic fluids. This follows from Edelens nonlinear generalization of the Onsager relation for the special case of 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 nonlinear generalization of Stokes resistance formulae for a wide class of viscoplastic fluid problems. We consider the application to nonlinear Darcy flow and to the effective slip for viscoplastic flow over textured surfaces.

Remarks on isotropic extension of anisotropic constitutive functions via structural tensors

In their original formulation of the method of isotropic extension via structural tensors, which is meant for applications to the derivation of coordinate-free representation formulas for anisotropic constitutive functions, both Boehler and Liu start with the assumption that the invariance group of structural tensors is the symmetry group that defines the anisotropy of the constitutive function in question. As a result, the method (with structural tensors of order not higher than two) is applicable only when the anisotropy is characterized by a cylindrical group or belongs to the triclinic, monoclinic, or rhombic crystal classes. In this note we present a reformulation of the method in which the aforementioned assumption of Boehler and of Liu is relaxed, and we show by examples in finite elasticity and anisotropic linear elasticity that the method of isotropic extension via structural tensors could be applicable beyond the original limitations.

Radiative transfer and flux theory

The fundamental phenomenological equations of radiative transfer, e.g., Lambert’s cosine rule and the radiant transport equation, are derived from an analysis based on the Cauchy flux theory of continuum mechanics. For the classical case, where the radiance is distributed regularly over the unit sphere, it is shown that Lambert’s rule follows from a balance law for the transfer of radiative power in each direction u on the sphere, together with the appropriate Cauchy postulates and the additional assumption that the corresponding flux vector field ju be parallel to u. The standard radiant transport equation follows from the additional assumption that radiant flux is given as the advection of radiant energy density. A theory is also presented for the singular limit of isolated rays, where the distribution of radiance on the sphere reduces to a Borel measure.