Abstract
An initial boundary value problem with a free boundary for a third-order integro-differential equation for the unsteady flow of an aqueous polymer solution in a strip is studied. Questions concerning the solvability of the problem and the qualitative behavior of the solution in dependence on the initial data are investigated: time-local resolvability is established under natural conditions on the input data and conditions of global solvability are found for the problem of strip narrowing. Conditions for the bowing-up of the solution in finite time in the model problem of strip expansion are presented. The problem of small perturbations of the flow of a strip of viscous fluid with respect to a parameter proportional to the relaxation viscosity is also consider.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. A. Toms, “Some Observations on the Flow of Linear Polymer Solutions through Straight Tubes at Large Reynolds Number”, Proceedings of First International Congress on Rheology, Amsterdam, 1948, 135–141.
V. V. Pukhnachev, O. A. Frolovskaya, and A. G. Petrova, “Polymer Solutions and Their Mathematical Models”, Izv. Vyssh. Ucheb. Zaved. Severo-Kavkazskii Region, 2 (2020), 84–93 (Russian).
O. A. Frolovskaya, “Motion of an Aqueous Polymer Solution with a Free Boundary”, Prikl. Mekh. Tekhn. Fiz., (2021).
V. A. Pavlovskii, “On Theoretical Description of Weak Aqueous Solutions of Polymers”, Dokl. Akad. Nauk SSSR, 200:4 (1971), 809–812.
V. K. Andreev, Yu. A. Gaponenko, O. N. Goncharova, and V. V. Pukhnachev, Mathematical Models of Convection, De Gruyter, Berlin and Boston, 2020.
L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, 2014.
O. A. Frolovskaya and V. V. Pukhnachev, “Analysis of the Models of Motion of Aqueous Solutions of Polymers on the Basis of their Exact Solutions”, Polymers., 10:684 (2018).
A. G. Petrova, “On the Unique Solvability of the Problem of the Flow of an Aqueous Solution of Polymers near a Critical Point”, Math. Notes, 106:5 (2019), 784–793.
L. A. Lusternik, V. I. Sobolev, Elements of Functional Analysis, Halsted Press, New York, 1974.
V. V. Pukhnachev and E. N. Zhuravleva, “Viscous Flows with Flat Free Boundaries”, Eur. Phys. J. Plus., 135:554 (2020).
Acknowledgments
The authors thank O.A. Frolovskaya for the opportunity to read her paper [3] in the manuscript.
Funding
The research was financially supported by the Russian Foundation for Basic Research (project 19-01-0096).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Petrova, A.G., Pukhnachev, V.V. Free Boundary Problem in a Polymer Solution Model. Russ. J. Math. Phys. 28, 96–103 (2021). https://doi.org/10.1134/S1061920821010106
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1061920821010106