Deformation of a Bingham viscoplastic fluid in a plane confuser

Abstract

A review is given to and comprehensive numerical-analytic study is carried out of the problem of steady Bingham viscoplastic flow in a plane confuser. The solution is constructed in the first approximation with the yield stress as a small parameter and the solution of the Jeffery-Hamel problem (steady radial motion of an incompressible viscous material in a plane confuser) as the zero-order approximation. The numerical analysis is based on the modified accelerated-convergence method proposed earlier by the authors. The bifurcations of the deformation pattern occurring when the parameters reach some critical values are discussed and commented on. The asymptotic boundaries of the rigid zones that appear at infinity upon perturbation of the yield stress are determined Introduction. Recently, the journal Prikladnaya Mekhanika (International Applied Mechanics) has published review papers on solid mechanics and related subject areas [60, 63, 66, etc.], which is of practical interest for scientists who solve important problems based on recent achievements in mathematics and mechanics. This paper outlines and analyzes the theoretical results from solution of one specific problem.This problem describes the stationary deformation of a Bingham viscoplastic medium in a plane confuser. To solve it, we will use numerical and analytical methods. The solution will be found in the first approximation with the yield stress as a small parameter and the solution of the Jeffery-Hamel problem (steady radial motion of a viscous incompressible medium in a flat-walled converging channel) as the zero-order approximation. We will determine and discuss the main characteristics of the process-asymptotic (first-order) quasi-rigid and deformed zones and analyze their behavior over a wide range of parameters. New mechanical effects will be revealed (bifurcations in the deformation pattern, shape and position of the asymptotic quasi-rigid zones, and the behavior of the velocity of the medium at their boundaries).Moreover, we will also perform a complete numerical and analytic study of the Jeffery-Hamel problem over the whole range of dimensionless parameters-the opening angle of the confuser and the Reynolds number. The numerical analysis is based on the modified accelerated-convergence method proposed by the authors.