ISSN 0025-6544, Mechanics of Solids, 2009, Vol. 44, No. 5, pp. 792–812. c  Allerton Press, Inc., 2009. Original Russian Text c  E.A. Muravleva, L.V. Muravleva, 2009, published in Izvestiya Akademii Nauk. Mekhanika Tverdogo Tela, 2009, No. 5, pp. 164–188. Unsteady Flows of a Viscoplastic Medium in Channels E. A. Muravleva * and L. V. Muravleva ** Lomonosov Moscow State University, GSP-2, Leninskie Gory, Moscow, 119992 Russia Received July 24, 2008 Abstract—We numerically study the nonstationary Poiseuille problem for a Bingham–Il’yushin viscoplastic medium in ducts of various cross-sections. The medium acceleration and deceleration problems are solved by using the Duvaut–Lions variational setting and the ﬁnite-diﬀerence scheme proposed by the authors. The dependence of the stopping time on internal parameters such as density, viscosity, yield stress, and the cross-section geometry is studied. The obtained results are in good agreement with the well-known theoretical estimates of the stopping time. The numerical solution revealed a peculiar characteristic of the stagnant zone location, which is speciﬁc to unsteady ﬂows. In the annulus, disk, and square, the stagnant zones arising shortly before the ﬂow cessation surround the entire boundary contour; but for other domains, the stagnant zones go outside the critical curves surrounding the stagnant zones in the steady ﬂow. The steady and unsteady ﬂows are studied in some domains of complicated shape. DOI: 10.3103/S0025654409050173 Key words: viscoplastic Bingham-Ilyushin medium, unsteady ﬂow, variational method. 1. INTRODUCTION Viscoplastic ﬂow problems attract the attention not only of mechanical engineers but also of mathe- maticians, numerical analysts, chemists, geophysicists, and rheologists. A comprehensive survey on the topic can be found in the recent papers [1, 2]. In [3], such problems were solved for a more complicated viscoelastoplastic medium. In [4], ﬂows of a viscoplastic ﬂuid in a porous medium were studied. One of the best known problems is the problem on a viscoplastic ﬂow in a duct. In [5–7], the existence and uniqueness theorems were proved for the solution of the problem on the steady ﬂow in pipes of arbitrary cross-section, and a qualitative investigation of the ﬂow character was performed. The further mathematical study of ﬂow in pipes based on variational inequalities is contained in the monographs [8, 9]. In [10], some exact solutions of the problem of viscoplastic unsteady ﬂow in a circular pipe were obtained for a given law of variation in the core boundary. The numerical simulation of the steady ﬂow in circular and square pipes was performed in [2, 11–15]. A detailed survey of papers published before 2005 and concerned with unsteady ﬂows can be found in [1]. In the recent papers [16–18], the one-dimensional problems of the ﬂow cessation were solved numerically for the plane and axisymmetric Poiseuille ﬂows and the plane Couette ﬂow. In [19, 20], numerical simulation of unsteady modes of the Couette ﬂow in an annular gap was implemented. (This is a one-dimensional problem as well.) Since a detailed survey is not the aim of the present paper, we have noted only the most famous papers. We consider an unsteady ﬂow of an incompressible viscoplastic medium in a cylindrical pipe with cross-section Ω and boundary Γ under the action of the pressure diﬀerence C . It is required to ﬁnd the velocity v satisfying the equation ρ ∂v ∂t − μ∇ 2 v − τ s ∇·  ∇v |∇v|  = C in Ω × (0,T ), v =0 on Γ × (0,T ), v   t=0 = v 0 . (1.1) * E-mail: catmurav@gmail.ru ** E-mail: lvmurav@gmail.ru 792