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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., Comment: 19 pages, 2 figures

We propose a novel family of asymptotically stable, implicit-explicit, adaptive, time integration method (denoted with the $\theta$-method) for the solution of the fractional advection-diffusion-reaction (FADR) equations. This family of time integration method generalized the computationally explicit $L_1$-method adopted by Brunner (J. Comput. Phys. {\bf 229} 6613-6622 (2010)) as well as the fully implicit method proposed by Jannelli (Comm. Nonlin. Sci. Num. Sim., {\bf 105}, 106073 (2022)). The spectral analysis of the method (involving the group velocity and the phase speed) indicates a region of favorable dispersion for a limited range of Peclet number. The numerical inversion of the coefficient matrix is avoided by exploiting the sparse structure of the matrix in the iterative solver for the Poisson equation. The accuracy and the efficacy of the method is benchmarked using (a) the two-dimensional (2D) fractional diffusion equation, originally proposed by Brunner, and (b) the incompressible, subdiffusive dynamics of a planar viscoelastic channel flow of the Rouse chain melts (FADR equation with fractional time-derivative of order $\alpha=\nicefrac{1}{2}$) and the Zimm chain solution ($\alpha=\nicefrac{2}{3}$). Numerical simulations of the viscoelastic channel flow effectively capture the non-homogeneous regions of high viscosity at low fluid inertia (or the so-called `spatiotemporal macrostructures'), experimentally observed in the flow-instability transition of subdiffusive flows., Comment: 29 pages, 9 figures

The temporal and spatiotemporal linear stability analyses of viscoelastic, subdiffusive, plane Poiseuille and Couette flows obeying the Fractional Upper Convected Maxwell (FUCM) equation in the limit of low to moderate Reynolds number ($Re$) and Weissenberg number ($We$), is reported to identify the regions of topological transition of the advancing flow interface. In particular, we demonstrate how the exponent in the subdiffusive power{\color{black}-}law scaling ($t^\alpha$, with $0 {\color{black} < } \alpha \le 1$) of the mean square displacement of the tracer particle, in the microscale [Mason and Weitz, Phys. Rev. Lett. {\bf 74}, 1250-1253 (1995)] is related to the fractional order of the derivative, $\alpha$, of the corresponding non-linear stress constitutive equation in the continuum. The stability studies are limited to two exponents: monomer diffusion in Rouse chain melts, $\alpha=\nicefrac{1}{2}$, and in Zimm chain solutions, $\alpha=\nicefrac{2}{3}$. The temporal stability analysis indicates that with decreasing order of the fractional derivative: (a) the most unstable mode decreases, (b) the peak of the most unstable mode shifts to lower values of $Re$, and (c) the peak of the most unstable mode, for the Rouse model precipitates towards the limit $Re \rightarrow 0$. The Briggs idea of analytic continuation is deployed to classify regions of temporal stability, absolute and convective instabilities and evanescent modes. The spatiotemporal phase diagram indicates an abnormal region of temporal stability at high fluid inertia, revealing the presence of a non-homogeneous environment with hindered flow, thus highlighting the potential of the model to effectively capture certain experimentally observed, flow-instability transition in subdiffusive flows., Comment: 28 pages, 8 figures. arXiv admin note: text overlap with arXiv:2109.08922