Abstract:
The unsteady plane-parallel motion of a incompressible viscoelastic Maxwell medium with constant relaxation time is considered. The equations of motion of the medium and the rheological relation admit an extended Galilean group. The class of solutions of this system which are partially invariant with respect to the subgroup of the indicated group generated by translation and Galilean translation along one of the coordinate axes is studied. The system does not have invariant solutions, and the set of partially invariant solutions is very narrow. A method for extending the set of exact solutions is proposed which allows finding solutions with a nontrivial dependence of the stress tensor elements on spatial coordinates. Among the solutions obtained by this method, the solutions describing the deformation of a viscoelastic strip with free boundaries is of special interest from a point of view of physics.
Citation:
V. V. Pukhnachev, “Exact solutions of the equations of motion for an incompressible viscoelastic Maxwell medium”, Prikl. Mekh. Tekh. Fiz., 50:2 (2009), 16–23; J. Appl. Mech. Tech. Phys., 50:2 (2009), 181–187
\Bibitem{Puk09}
\by V.~V.~Pukhnachev
\paper Exact solutions of the equations of motion for an incompressible viscoelastic Maxwell medium
\jour Prikl. Mekh. Tekh. Fiz.
\yr 2009
\vol 50
\issue 2
\pages 16--23
\mathnet{http://mi.mathnet.ru/pmtf1712}
\elib{https://elibrary.ru/item.asp?id=11839324}
\transl
\jour J. Appl. Mech. Tech. Phys.
\yr 2009
\vol 50
\issue 2
\pages 181--187
\crossref{https://doi.org/10.1007/s10808-009-0025-y}
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https://www.mathnet.ru/eng/pmtf1712
https://www.mathnet.ru/eng/pmtf/v50/i2/p16
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A. A. Kosov, E. I. Semenov, “Multidimensional exact solutions to the reaction-diffusion system with power-law nonlinear terms”, Siberian Math. J., 58:4 (2017), 619–632
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A. A. Kosov, E. I. Semyonov, “Exact radially-symmetric solutions of a class of nonlinear elliptic systems of equations”, Russian Math. (Iz. VUZ), 59:12 (2015), 36–45
Andrei D. Polyanin, Alexei I. Zhurov, “Integration of linear and some model non-linear equations of motion of incompressible fluids”, International Journal of Non-Linear Mechanics, 49 (2013), 77
A. D. Polyanin, A. V. Vyazmin, “Decomposition of three-dimensional linearized equations for Maxwell and Oldroyd viscoelastic fluids and their generalizations”, Theor Found Chem Eng, 47:4 (2013), 321
V. V. Pukhnachev, “Mathematical model of an incompressible viscoelastic Maxwell medium”, J. Appl. Mech. Tech. Phys., 51:4 (2010), 546–554
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