The structure of viscoplastic fluid flow during filling of a circular pipe / plane channel

Abstract

The investigation of a viscoplastic fluid flow initiated in a circular pipe/plane channel during its filling in the gravity field at the flow rate specified at the inlet section is carried out. A mathematical formulation of the problem is developed using complete equations of motion, continuity equation, natural boundary conditions on the free surface, and no-slip boundary conditions on the solid wall. The rheological behavior of the medium is described by the Shvedov-Bingham law, which implies the existence of zones with quasi-solid motion (unyielded zones) in the regions of low strain rate. The numerical solution of the problem is based on the finite-difference approach with an application of the finite volume method and SIMPLE algorithm for calculating velocity and pressure fields at the internal nodes of a staggered grid. The method of invariants is used to satisfy the boundary conditions on the free surface. To provide a through computation of the flow with unyielded regions, a regularization of the rheological equation is implemented. The investigation of the free boundary behavior, flow structure, and flow characteristics as a function of the main parameters is carried out. It was found that in the course of time the initially flat free boundary assumes a stationary convex shape, which remains invariant moving through the pipe/channel at an average velocity. In the flow near and far away from the free boundary one can distinguish the fountain flow zones and the one-dimensional flow regions, respectively. The typical flow structures with different number and various locations of unyielded regions in the flow are shown. The topograms of the above mentioned flow structures are plotted as a function of the ratio of viscous and gravity forces and plasticity in the fluid flow. The stable and unstable behavior of the free boundary shape is shown to be related to the values of the constitutive parameters.