TY - JOUR

T1 - Numerical Analysis of the Blow-Up of One-Dimensional Polymer Fluid Flow with a Front

AU - Bryndin, L. S.

AU - Semisalov, B. V.

AU - Beliaev, V. A.

AU - Shapeev, V. P.

N1 - This work was supported by the Russian Science Foundation, agreement no. 23-21-00499.

PY - 2024/1

Y1 - 2024/1

N2 - One-dimensional flows of an incompressible viscoelastic polymer fluid that are qualitatively similar to the solutions of Burgers’ equation are described on the basis of mesoscopic approach for the first time. The corresponding initial boundary-value problem is posed for the system of quasilinear differential equations. The numerical algorithm for solving it is designed and verified. The algorithm uses the explicit fifth-order scheme to approximate unknown functions with respect to time variable and the rational barycentric interpolations with respect to space variable. A method for localization of singular points of the solution in the complex plain and for adaptation of the spatial grid to them is implemented using the Chebyshev-Padé approximations. Two regimes of evolution of the solution to the problem are discovered and characterized while using the algorithm: regime 1—a smooth solution exists in a sufficiently large time interval (the singular point moves parallel to the real axis in the complex plane); regime 2—the smooth solution blows up at the beginning of evolution (the singular point reaches the segment of the real axis where the problem is posed). We study the influence of the rheological parameters of fluid on the realizability of these regimes and on the length of time interval where the smooth solution exists. The obtained results are important for the analysis of laminar-turbulent transitions in viscoelastic polymer continua.

AB - One-dimensional flows of an incompressible viscoelastic polymer fluid that are qualitatively similar to the solutions of Burgers’ equation are described on the basis of mesoscopic approach for the first time. The corresponding initial boundary-value problem is posed for the system of quasilinear differential equations. The numerical algorithm for solving it is designed and verified. The algorithm uses the explicit fifth-order scheme to approximate unknown functions with respect to time variable and the rational barycentric interpolations with respect to space variable. A method for localization of singular points of the solution in the complex plain and for adaptation of the spatial grid to them is implemented using the Chebyshev-Padé approximations. Two regimes of evolution of the solution to the problem are discovered and characterized while using the algorithm: regime 1—a smooth solution exists in a sufficiently large time interval (the singular point moves parallel to the real axis in the complex plane); regime 2—the smooth solution blows up at the beginning of evolution (the singular point reaches the segment of the real axis where the problem is posed). We study the influence of the rheological parameters of fluid on the realizability of these regimes and on the length of time interval where the smooth solution exists. The obtained results are important for the analysis of laminar-turbulent transitions in viscoelastic polymer continua.

KW - Burgers equation

KW - Chebyshev–Padé approximation

KW - mesoscopic model

KW - one-dimensional flow

KW - polymer fluid

KW - rational approximation

KW - rheology

KW - trajectory of a singular point in the complex plane

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85188231065&origin=inward&txGid=20d3af52713b70e445f245a95e826583

UR - https://www.mendeley.com/catalogue/e96d3f46-e599-3817-b47c-852cde24f139/

U2 - 10.1134/S0965542524010068

DO - 10.1134/S0965542524010068

M3 - Article

VL - 64

SP - 151

EP - 165

JO - Computational Mathematics and Mathematical Physics

JF - Computational Mathematics and Mathematical Physics

SN - 0965-5425

IS - 1

ER -