About dynamic features of viscous polymer flow in equal-channel multiple angular extrusion through a die with a movable wall

Technological processes of Equal Channel Multiple Angular Extrusion have contemporary applications in many fields of solid state physics and materials science. Equal-channel multiple angular extrusion dynamics essentially depends on rheology of processed materials but there has been insufficient study of viscous polymer flows in dies with movable walls, which the present research addresses. The present work is focused on the phenomenological description of equal-channel multiple angular extrusion technological processes with the introduction of a numerical mathematical simulation for viscous flows of physical polymeric materials models through a two-turn Segal die with a movable wall. The numerical finite-difference model for plane viscous Newtonian flow of an incompressible continuous medium in a multiple angular region with a movable entrance die wall, based on the formulation and the numerical solution of the boundary value problem for the Navier-Stokes equations in the curl transfer form has been derived. The numerical integration of the curl transfer equation is implemented using a finite-difference approximation with an alternating-direction implicit method. The presence of a movable entrance wall of the multiple angular die, moving parallel to the extrusion direction, was taken into account through the introduction of an appropriate boundary condition for the curl function, defined for the nodes of the movable wall. The proposed difference form of the boundary condition includes the dimensionless velocity of the movable entrance wall. Derived numerical solutions show that the movable entrance wall of the die, which determines the transportation motion in the system, has a major influence on the viscous material flow features near the movable die wall. It was found that in the case when the wall moves toward the viscous flow, the streamline nearest the movable wall "accelerates" in comparison with the case of the fixed die wall. From a hydrodynamic point of view, the observable effect is that the polymer flow rate remains constant, because the average punching velocity is the same. The layer of viscous material adjacent to the movable wall moves with near die wall velocity in the punching direction. Therefore, the area of effective cross-section decreases and the flow rate increases. The rotational interpretation is that within the layer of viscous material adjacent to the movable wall, a negative macroscopic rotation is formed, which narrows the effective cross-section of the viscous flow. Another case was analyzed in which the movable die wall moves in the direction of punching. It was found that for such a pressing mode near the movable die wall a positive macroscopic rotation is formed. So, near the central axis of the entrance die channel the viscous material moves backward. In this case the effective cross-section is compressed to the opposite fixed die wall, near which the viscous flow velocity essentially increases. A numerical estimation of the influence of the direction of movement of a movable entrance die wall on computational flow lines, stream and curl functions, and viscous flow velocity fields has been carried out within the scope of the developed model for a movable entrance wall of the die. The proposed hydrodynamic approach extends the ideas concerning dynamics of the macroscopic rotation formation within the volume of the viscous physical model of a polymeric material during equal-channel multiple angular extrusion through a two-turn Segal die with a movable wall.