Macrostructure Quantification Via a Riemannian Metric

ThisRiemannian metric chapter presents a theory to quantify the formation of spatiotemporal macrostructures (or the non-homogeneous regions of high viscosity at moderate to high fluid inertia) for viscoelastic sub-diffusive flows, by introducing a mathematically consistent decomposition of the polymer conformation tensor, into the so-called Structure tensorStructure Tensor. Our approach bypasses an inherent problem in the standard arithmetic decomposition, namely, the fluctuating conformation tensor fields may not be positive definite and hence, do not retain their physical meaning. Using well-established results in matrix analysis, the space of positive definite matrices is transformed into a Riemannian manifoldRiemannian manifold by defining and constructing a geodesic via the inner product on its Tangent spacetangent space. This geodesic is utilized to define three scalar invariants of the Structure tensorStructure Tensor, which do not suffer from the caveats of the regular invariants (such as trace and determinant) of the polymer conformation tensor. We conclude by considering the problem of formulating perturbative expansions of the structure tensor using this geodesic, such that a constraint on the maximum time, during which the evolution of the perturbative solution can be well approximated by linear theory along the Euclidean manifoldEuclidean manifold is found.