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 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]: Year: Volume: Issue: Page: Find

 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2016, Volume 20, Number 3, Pages 496–507 (Mi vsgtu1508)

Mechanics of Solids

Dual plane problems for creeping flow of power-law incompressible medium

D. S. Petukhov, I. E. Keller*

Institute of Continuous Media Mechanics, Ural Branch of RAS, Perm, 614013, Russian Federation

Abstract: In this paper, we consider the class of solutions for a creeping plane flow of incompressible medium with power-law rheology, which are written in the form of the product of arbitrary power of the radial coordinate by arbitrary function of the angular coordinate of the polar coordinate system covering the plane. This class of solutions represents the asymptotics of fields in the vicinity of singular points in the domain occupied by the examined medium. We have ascertained the duality of two problems for a plane with wedge-shaped notch, at which boundaries in one of the problems the vector components of the surface force vanish, while in the other—the vanishing components are the vector components of velocity, We have investigated the asymptotics and eigensolutions of the dual nonlinear eigenvalue problems in relation to the rheological exponent and opening angle of the notch for the branch associated with the eigenvalue of the Hutchinson–Rice–Rosengren problem learned from the problem of stress distribution over a notched plane for a power law medium. In the context of the dual problem we have determined the velocity distribution in the flow of power-law medium at the vertex of a rigid wedge, We have also found another two eigenvalues, one of which was determined by V. V. Sokolovsky for the problem of power-law fluid flow in a convergent channel.

Keywords: steady-state creep, power-law rheology, duality, variable separation, crack mechanics, flow in convergent channel

 Funding Agency Grant Number Russian Academy of Sciences - Federal Agency for Scientific Organizations 15-10-1-18 This work was supported by the program of the Ural Branch of RAS (project no. 15–10–1–18).

* Author to whom correspondence should be addressed

DOI: https://doi.org/10.14498/vsgtu1508

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Bibliographic databases:

UDC: 539.376
MSC: 74D10, 74G55
Original article submitted 13/VII/2016
revision submitted – 26/VIII/2016

Citation: D. S. Petukhov, I. E. Keller, “Dual plane problems for creeping flow of power-law incompressible medium”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 20:3 (2016), 496–507

Citation in format AMSBIB
\Bibitem{PetKel16} \by D.~S.~Petukhov, I.~E.~Keller \paper Dual plane problems for creeping flow of power-law incompressible medium \jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.] \yr 2016 \vol 20 \issue 3 \pages 496--507 \mathnet{http://mi.mathnet.ru/vsgtu1508} \crossref{https://doi.org/10.14498/vsgtu1508} \zmath{https://zbmath.org/?q=an:06964522} \elib{https://elibrary.ru/item.asp?id=28282245} 

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This publication is cited in the following articles:
1. A. V. Khokhlov, “Indikatory primenimosti i metodiki identifikatsii nelineinoi modeli tipa maksvella dlya reonomnykh materialov po krivym polzuchesti pri stupenchatykh nagruzheniyakh”, Vestnik Moskovskogo gosudarstvennogo tekhnicheskogo universiteta im. N. E. Baumana. Seriya: Estestvennye nauki, 2018, no. 6 (81), 92–112
2. A. V. Khokhlov, “Applicability Indicators and Identification Techniques for a Nonlinear Maxwell–Type Elastoviscoplastic Model Using Loading–Unloading Curves”, Mechanics of Composite Materials, 55:2 (2019), 195–210
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