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Spatiotemporal linear stability of viscoelastic subdiffusive channel flows: a fractional calculus framework

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Abstract

The temporal and spatiotemporal linear stability analyses of viscoelastic, subdiffusive, plane Poiseuille flow 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 of the power-law scaling (\(t^\alpha \), with \(0 < \alpha \le 1\)) in viscoelastic microscale models (Mason and Weitz in Phys Rev Lett 74:1250–1253, 1995) is related to the fractional order of the time 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, , and in Zimm chain solutions, . The temporal stability analysis indicates that with decreasing order of the fractional derivative (a) the most unstable mode decreases with decreasing values of \(\alpha \), (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.

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Acknowledgements

T. Chauhan, D. Bansal, and S. Sircar acknowledge the financial support of the Grant CSIR 09 /1117 (0012) /2020-EMR-I and DST ECR/2017/000632, respectively.

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  1. Tanisha Chauhan and Diksha Bansal have contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, IIIT Delhi, Okhla Phase III, New Delhi, Delhi, 110020, India

    Tanisha Chauhan, Diksha Bansal & Sarthok Sircar

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  1. Tanisha Chauhan
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TC and DB performed the simulation and analysis. SS formulated the problem. All authors reviewed the manuscript.

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Correspondence to Sarthok Sircar.

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Appendix A: Viscoelastic dispersion relation

Appendix A: Viscoelastic dispersion relation

The expressions, \(M_i\), utilized in the viscoelastic dispersion relation outlined in Sect. 2.2, is given as

$$\begin{aligned} M_1&= (y\!-\!y^2) \left( \!Re (-\textrm{i} \omega )^\alpha \! +\! \textrm{i} k Re (y\!-\!y^2\!) \! -\! { 2} \nu (\textrm{i} k)^{2}\right) \! +\! { 2} \nu , \nonumber \\ M_2&= Re ( 1-2y) (y-y^2) - { \nu } (1-2y) (\textrm{i} k), \nonumber \\ M_3&= (y\!-\!y^2)\left( \! Re (-\textrm{i} \omega )^\alpha \!+\! \textrm{i} k Re (y\!-\!y^2)\! -\! { \nu } (\textrm{i}k)^{2}\!\right) \!+\! { 4} \nu , \nonumber \\ M_4&= (-\textrm{i} \omega )^\alpha + \textrm{i} k (y-y^2) + \frac{1}{We}, \nonumber \\ M_5&= { -\textrm{i} k} \left( 1-2y\right) (y-y^2) - { \frac{1}{ We}} \left( 1-2y \right) , \nonumber \\ M_6&=- { 2} (y-y^2) - { 2} \textrm{i} k We (1-2y)^2 (y-y^2) - \left( 1- 2y \right) ^2 - {\frac{1}{ We}} (y-y^2)(\textrm{i} k ) , \nonumber \\ M_7&= - {8 } We \left( 1-2y \right) (y-y^2) - { 4} We (1-2y)^3 -\frac{{ 2}(1-2y)}{We}. \end{aligned}$$
(A1)

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Chauhan, T., Bansal, D. & Sircar, S. Spatiotemporal linear stability of viscoelastic subdiffusive channel flows: a fractional calculus framework. J Eng Math 141, 8 (2023). https://doi.org/10.1007/s10665-023-10282-7

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