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Vestnik Yuzhno-Ural'skogo Universiteta. Seriya Matematicheskoe Modelirovanie i Programmirovanie, 2014, Volume 7, Issue 3, Pages 69–76
DOI: https://doi.org/10.14529/mmp140307
(Mi vyuru146)
 

Mathematical Modelling

On the Strong Solutions in an Oldroyd-Type Model of Thermoviscoelasticity

V. P. Orlov, M. I. Parshin

Voronezh State University, Voronezh, Russian Federation
References:
Abstract: For the initial-boundary value problem in a dynamic Oldroyd-type model of thermoviscoelasticity, we establish the local existence theorem for strong solutions in the planar case. The continuum under consideration is a plane bounded domain with sufficiently smooth boundary. The corresponding system of equations generalizes the Navier–Stokes–Fourier system by having an additional integral term in the stress tensor responsible for the memory of the continuum. In our proof, we study firstly the initial-boundary value problem for an Oldroyd-type viscoelasticity system with variable viscosity. Then we consider the initial-boundary value problem for the equation of energy conservation with a variable heat conductivity coefficient and an integral term. We establish the solvability of these problems by reducing them to operator equations and applying the fixed-point theorem. For the original thermoviscoelasticity system, we construct an iterative process consisting in a consecutive solution of auxiliary problems. Suitable a priori estimates ensure that the iterative process converges on a sufficiently small interval of time. The proof relies substantially on Consiglieri's results on the solvability of the corresponding Navier–Stokes–Fourier system.
Keywords: Navier–Stokes equation; Oldroyd-type model; thermoviscoelastic; strong solutions; fixed point.
Received: 03.01.2014
Document Type: Article
UDC: 517.958
MSC: 90C30
Language: Russian
Citation: V. P. Orlov, M. I. Parshin, “On the Strong Solutions in an Oldroyd-Type Model of Thermoviscoelasticity”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 7:3 (2014), 69–76
Citation in format AMSBIB
\Bibitem{OrlPar14}
\by V.~P.~Orlov, M.~I.~Parshin
\paper On the Strong Solutions in an Oldroyd-Type Model of Thermoviscoelasticity
\jour Vestnik YuUrGU. Ser. Mat. Model. Progr.
\yr 2014
\vol 7
\issue 3
\pages 69--76
\mathnet{http://mi.mathnet.ru/vyuru146}
\crossref{https://doi.org/10.14529/mmp140307}
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