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

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...