Extrusion of a viscoplastic material with small yield limit from a plane convergent channel

Steady-state deformation of a viscoplastic medium in a plane convergent channel is investigated on the basis of the Bingham-Il'yushin model for the case of small shear yield limit of the material. The solution is constructed in the first approximation with respect to the small yield limit, with the solution of the corresponding Jeffery-Hamel problem for viscous fluid flow being the generating solution. We apply the numerical-analytical accelerated convergence method that has been developed by the authors and enables one to construct solutions with high degree of accuracy in a broad range of the system parameters. We determine the major characteristics of the flow process, such as velocities, pressures, quasi-rigid zones, and flow zones, investigate these characteristics depending on the parameters of the problem, and give an interpretation of the results. We established a number of new properties of the system under consideration related, for example, to the number and arrangement of the quasi-rigid zones, flow bifurcations, and the drop in the velocity of particles at the boundaries of the quasi-rigid zones. Technological operations in the production of thin sheets that are utilized, for example, in manufacturing laminated materials are based on the inertialess (quasi-static) extrusion of primary components from draw plates and convergent channels. The minimization of the material expenditure required for these operation is provided by an appropriate choice of the geometry of the domain (the angle between the walls of the channel and the profile of the walls) and the pressure distribution at infinity. Normally, the optimization of the geometry and the pressure distribution is associated with the requirement for the laminar flow to be radial (pseudo-radial for curved walls). In this case, the corresponding velocity component must vanish. Another broad area related to technological operations that involve flow of continuous media out of small holes and slots is associated with mixing. In contrast to extrusion, mixing operations require conditions which break laminar character of the flow. This area is closely related to hydrodynamic stability and various theories of turbulence. A. A. Il'yushin and A. Yu. Ishlinskii were the first to study the stability of deformation of viscoplastic bodies systematically. Il'yushin [I] has formulated the general problem of stability of a plane flow in terms of perturbations of the flow function and the velocity potential and, in addition, investigated the tension/compression of a strip and the spread of a cylinder. (In the latter case, the eccentricity of the boundary surfaces of the cylinder was taken into account). Ishlinskii [2, 3] has investigated the stability of viscoplastic flow for a strip, a round rod, and a plate. The fundamental and applied results obtained in the cited papers were extremely topical, especially during the war time, and played an important role in the optimization of technological processes in the defense industry. For a modern review of studies related to 2D and 3D viscoplastic flows and their stability, see [4]. In what follows, we study the slow steady-state deformation of a viscoplastic material in a plane convergent channel with an arbitrary angle between the walls, including angles close to the straight one. The plastic properties of the medium are substantially less pronounced as compared with the viscous properties. For this reason, we constructed only the first approximation to the solution in terms of the small yield limit. We determine the number and boundaries of quasi-rigid zones, as well as the velocity and pressure profiles for small, medium, and large angles between the walls of the channel. These characteristics depend on geometrical and physical parameters.