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We present a theory to quantify the formation of spatiotemporalmacrostructures (or the non-homogeneous regions of high viscosity at moderateto high fluid inertia) for viscoelastic sub-diffusive flows, by introducing amathematically consistent decomposition of the polymer conformation tensor,into the so-called structure tensor. Our approach bypasses an inherent problemin the standard arithmetic decomposition, namely, the fluctuating conformationtensor fields may not be positive definite and hence, do not retain theirphysical meaning. Using well-established results in matrix analysis, the spaceof positive definite matrices is transformed into a Riemannian manifold bydefining and constructing a geodesic via the inner product on its tangentspace. This geodesic is utilized to define three scalar invariants of thestructure tensor, which do not suffer from the caveats of the regularinvariants (such as trace and determinant) of the polymer conformation tensor.First, we consider the problem of formulating perturbative expansions of thestructure tensor using the geodesic, which is consistent with the Riemannianmanifold geometry. A constraint on the maximum time, during which the evolutionof the perturbative solution can be well approximated by linear theory alongthe Euclidean manifold, is found. Finally, direct numerical simulations of theviscoelastic sub-diffusive channel flows (where the stress-constitutive law isobtained via coarse-graining the polymer relaxation spectrum at finer scale,Chauhan et. al., Phys. Fluids, DOI: 10.1063/5.0174598 (2023)), underscore theadvantage of using these invariants in effectively quantifying themacrostructures.

Quantifying macrostructures in viscoelastic sub-diffusive flows