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

• Main Authors: Dmitriy S. Petukhov, Ilya E. Keller
• Format: Article
• Language: English
• Published: Samara State Technical University 2016-09-01
• Series: Vestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki
• Subjects: steady-state creep; power-law rheology; duality; variable separation; crack mechanics; flow in convergent channel
• Online Access: http://mi.mathnet.ru/eng/vsgtu1508

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.