  We present 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 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 manifold by
defining and constructing a geodesic via the inner product on its tangent
space. This geodesic is utilized to define three scalar invariants of the
structure tensor, which do not suffer from the caveats of the regular
invariants (such as trace and determinant) of the polymer conformation tensor.
First, we consider the problem of formulating perturbative expansions of the
structure tensor using the geodesic, which is consistent with the Riemannian
manifold geometry. A constraint on the maximum time, during which the evolution
of the perturbative solution can be well approximated by linear theory along
the Euclidean manifold, is found. Finally, direct numerical simulations of the
viscoelastic sub-diffusive channel flows (where the stress-constitutive law is
obtained via coarse-graining the polymer relaxation spectrum at finer scale,
Chauhan et. al., Phys. Fluids, DOI: 10.1063/5.0174598 (2023)), underscore the
advantage of using these invariants in effectively quantifying the
macrostructures.
